**Non-associative Structures and Other Related Structures**

Special Issue Editor **Florin Felix Nichita**

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*Special Issue Editor* Florin Felix Nichita Simion Stoilow Institute of Mathematics of the Romanian Academy Romania

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This is a reprint of articles from the Special Issue published online in the open access journal *Axioms* (ISSN 2075-1680) (available at: https://www.mdpi.com/journal/axioms/special issues/ non-associative structures).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. *Journal Name* **Year**, *Article Number*, Page Range.

**ISBN 978-3-03936-254-7 (Hbk) ISBN 978-3-03936-255-4 (PDF)**

c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications.

The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND.

### **Contents**

### **About the Special Issue Editor**

**Florin Felix Nichita** (Ph.D.—SUNY Buffalo, 2000/2001) is currently Senior Scientific Researcher at the "Simion Stoilow" Institute of Mathematics of the Romanian Academy, and a collaborator of MDPI journals. His primary research interests are in the areas of topology, noncommutative algebra, geometry, etc. With about 60 published works, and many invited talks/visitor positions, his attention has recently been attracted to poetry and literature.

### **Preface to "Non-associative Structures and Other Related Structures"**

Leonhard Euler (1707–1783) was born on 15 April 1707, in Basel, Switzerland. Euler's formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. When its variable is the number pi, Euler's formula evaluates to Euler's identity.

Bertrand Russell wrote that mathematics can exalt "as surely as poetry". This is especially true of one equation: Euler's identity, the brainchild of Leonhard Euler, the Mozart of mathematics. More than two centuries after Euler's death, it is still regarded as a conceptual diamond of unsurpassed beauty. Called Euler's identity, or God's equation, it includes just five numbers but represents an astonishing revelation of hidden connections. It ties together everything from basic arithmetic to compound interest, the circumference of a circle, trigonometry, calculus, and even infinity. In David Stipp's book, *A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics* (2017), Euler's identity formula becomes a contemplative stroll through the glories of mathematics. The result is an ode to this magical field.

On the other hand, according to Melissa Hogenboom, the most beautiful equation is the Yang-Baxter equation (20 January 2016).

http://www.bbc.com/earth/story/20160120-the-most-beautiful-equation-is-the-yang-baxter-equation. In this book, we consider connections between Euler's formula and the Yang-Baxter equation.

This discussion includes also dual numbers. Dual numbers and their higher-order variants are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, a novel point of view is presented in this book.

Other interesting topics include non-associative algebras with metagroup relations; branching functions for admissible representations of affine Lie Algebras; super-Virasoro algebras; UJLA structures; etc.

Two more special issues will continue our investigations:

*Axioms*: Non-associative Structures, Yang–Baxter Equations and Related Topics

*Sci*: Mathematics and Poetry, with a View Towards Machine Learning

**Florin Felix Nichita** *Special Issue Editor*

### *Editorial* **Non-Associative Structures and Other Related Structures**

### **Florin F. Nichita**

Simion Stoilow Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania<EMAIL_ADDRESS>
Received: 7 April 2020; Accepted: 10 April 2020; Published: 13 April 2020

In January 2019, MDPI published a book titled *Hopf Algebras, Quantum Groups and Yang–Baxter Equations*, based on a successful special issue. We hope that a book titled *Non-Associative Structures and Other Related Structures* will be published soon.

Non-associative algebras are currently a fashionable research direction. There are two important classes of non-associative structures: Lie structures and Jordan structures. Various Jordan structures play an important role in quantum group theory and in fundamental physical theories. In recent years, several attempts to unify (non-)associative structures have led to interesting results. The UJLA structures are not the only structures that realize such a unification. Associative algebras and Lie algebras can be unified at the level of Yang–Baxter structures.

Several papers published in the open access journal Axioms deal with the Yang–Baxter equation. This equation first appeared in theoretical physics, in a paper (1968) by the Nobel laureate C.N. Yang, and in statistical mechanics, in R.J. Baxter's work (1971). Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative descent theory, quantum computing, non-commutative geometry, etc. At the Kyoto International Mathematics Congress (1990), three of the four Fields Medalists were awarded prizes for their work related to the Yang–Baxter equation. Many scientists have used the axioms of various algebraic structures (quasi-triangular Hopf algebras, Yetter-Drinfeld categories, Lie (super)algebras, algebra structures, Boolean algebras, etc.) or computer calculations in order to produce solutions for the Yang–Baxter equation. However, the full classification of its solutions remains an open problem. The Yang–Baxter equation can also be interpreted in terms of logical circuits and, in logic, it represents a kind of compatibility condition when working with many logical sentences in the same time.

It is interesting to note that several special issues published by AXIOMS led to new solutions for the Yang–Baxter equations. In addition, the topics of these special issues were advertised at many conferences (Boston, Bucharest, Brasov, Caen, Galati, Iasi, Malta, Sofia, etc.). Moreover, at the 14th International Workshop on Differential Geometry and Its Applications, hosted by the Petroleum Gas University from Ploiesti, between 9–11 July 2019, the AXIOMS sponsored the "Best Poster Award" for best presented papers to support young scholars (including post-docs up to 35 years old). The winners were announced by the chairs during the workshop.

The authors who contributed to the special issue *Non-associative Structures and Other Related Structures* put in a lot of work in order to write high quality papers: [1–7]. Some other works related to this special issue are the following: [8–12]. Further comments and references for related articles will appear in our next special issues.

**Funding:** This research received no external funding.

**Acknowledgments:** We would like thank the authors who contributed to this special issue, the referees and the editorial staff of Axioms.

**Conflicts of Interest:** The author declares no conflict of interest.

### **References**

c 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Dual Numbers and Operational Umbral Methods**

### **Nicolas Behr 1,\*, Giuseppe Dattoli 2, Ambra Lattanzi 2,3 and Silvia Licciardi 2**

Received: 22 May 2019; Accepted: 26 June 2019; Published: 2 July 2019

**Abstract:** Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus.

**Keywords:** dual numbers; operational methods; umbral image techniques

### **1. Introduction**

Dual numbers (DNs), introduced during the second half of the 19th century [1–5], can be viewed as abstract entities, similar to ordinary complex numbers, and are defined as

$$z = x + \epsilon y, \qquad (x, y) \in \mathbb{R} \tag{1}$$

where the corresponding "imaginary" unit or *dual number unit (DNU)* is a nilpotent number,

$$
\epsilon^2 = 0 \qquad \text{and} \qquad \epsilon \neq 0. \tag{2}
$$

Dual numbers were originally introduced within the context of geometrical studies, and later exploited to deal with problems in pure and applied mechanics [6,7]. For instance, it has been demonstrated in [8–10] how to formulate the equations of rigid body motion in terms of just three "dual" equations instead of their six "real" counterparts (thereby realizing an equivalence between spherical and spatial kinematics). This approach has been extended in [11–13] to a treatment of rigid body motion in terms of a certain variant of "hyper-dual" numbers, implemented in contrast to our approach via sets of "ordinary" dual numbers together with certain algebraic relations that are motivated from the specific requirements within the relevant field of robotics and of mechanics. More recently, as further discussed in the present paper, the importance of a different kind of higher-order dual numbers has been recognized in numerical analysis to reduce round-off errors [14]. We believe that the use of dual numbers in the applied sciences is not as widespread as it could be, and that many new fields of research would benefit from their relevant introduction. An important domain in which they may bring significant novelties is that of the perturbative techniques in classical and quantum mechanics.

The main contribution of this paper consists in fixing the underlying algebraic rules of the dual numbers in the wider context of umbral and operational calculus. The paper is organized as follows: Section 2 delivers a basic mathematical introduction to dual numbers. Section 3 is devoted to the description of the computational procedure based upon dual numbers and umbral calculus. In Section 4, we provide insight into how this powerful method can be applied to deal with problems arising in different contexts. For illustration, we consider the Schrödinger and the heat equation, cornerstones in their respective fields of physics. Section 5 provides a conclusion with further considerations for future works.

### **2. Higher-Order Dual Numbers**

The DN algebraic rules [15,16], summarized below, are a straightforward consequence of the previous identity in Equation (2) (with *z* = *x* + *y* and *w* = *u* + *v*):

#### **Component-wise algebraic addition**

$$\begin{aligned} z + w &= x + u + \epsilon(y + v) \\ \mathbf{Product} \\ z \cdot w &= xu + \epsilon(xv + yu) \\ \text{Inverse} \\ z^{-1} &= \frac{1}{x} \left( 1 - \epsilon \frac{y}{x} \right) \quad (x \neq 0) \\ \mathbf{Power} \\ z^{n} &= x^{n} \left( 1 + n\epsilon \frac{y}{x} \right) \quad (n \in \mathbb{Z}\_{\ge 0}, \ x \neq 0) \end{aligned} \tag{3}$$

While the addition operation is entirely analogous to the component-wise addition operation on two-dimensional vectors, the last three operations (product, inverse and power) characterize the distinguishing special algebraic properties of dual numbers (DNs). The multiplication is commutative, associative and distributive, thus DNs form a two-dimensional associative and commutative algebra over the real numbers.

We now extend this traditional dual number formalism as motivated by the following type of problem. Consider the Taylor expansion up to some order *k* (denoted ≈*k*) of an at least *k*-fold continuously differentiable function *f* around a point *x*,

$$f(\mathbf{x} + \mathbf{y}) \approx\_k \sum\_{m=0}^k \frac{y^m}{m!} f^{(m)}(\mathbf{x})\,. \tag{4}$$

Following the *automatic differentiation* paradigm [17–19], since in practice the function *f* will be implemented in some algorithmic from, it may be advantageous to formulate truncations such as Equation (4) in terms of *generalized (or higher-order) dual numbers*. To this end, let us introduce the families of square matrices *-*ˆ *k* ±, 1ˆ *k* and 0ˆ *k* with entries (for *i*, *j* = 1, . . . , *k*)

$$(\left(\_{k}\mathfrak{E}\_{\pm}\right)\_{i,\mathfrak{j}} := \delta\_{\mathfrak{j},i\pm 1}, \qquad (\_{k}\mathfrak{I})\_{i,\mathfrak{j}} := \delta\_{i,\mathfrak{j}}, \qquad (\_{k}\mathfrak{I})\_{i,\mathfrak{j}} := 0,\tag{5}$$

where *δi*,*j* denotes the Kronecker symbol. It is straightforward to verify that for all *k* ≥ 2 and -≥ 0

$$\left( \,\_k \mathfrak{k}\_{\pm}^{\ell} \right)\_{\mathbf{i}, \mathbf{j}} = \delta\_{\mathbf{j}, \mathbf{i} \pm \ell} \quad \Rightarrow \quad \left( \,\_k \mathfrak{k}\_{\pm} \right)^k = \mathbf{0} \,. \tag{6}$$

Then, under the assumptions in Equation (4), endowing the function *f*(*x*) suitably with a component-wise action on square matrices, we find (for *k* ≥ 2)

$$f\left(\mathbf{x}\_{k}\hat{\mathbf{1}} + \mathbf{y}\_{k}\hat{\mathbf{e}}\_{\pm}\right) = \sum\_{m=0}^{k-1} \frac{1}{m!} y^{m} f^{(m)}(\mathbf{x}) \, \_{k} \hat{\mathbf{e}}\_{\pm}^{m} \,. \tag{7}$$

For example, setting *k* = 2, reproduces the well-known dual number identity [19]

$$f\left(\mathbf{x}\_2\hat{\mathbf{1}} + \mathbf{y}\_2\hat{\mathbf{e}}\_+\right) = f(\mathbf{x})\_2\hat{\mathbf{1}} + yf'(\mathbf{x})\_2\hat{\mathbf{e}}\_+ = \begin{pmatrix} f(\mathbf{x}) & yf'(\mathbf{x}) \\ 0 & f(\mathbf{x}) \end{pmatrix}.\tag{8}$$

For illustration, setting *k* = 3, we obtain

$$f\left(\mathbf{x}\_{3}\hat{\mathbf{1}} + \mathbf{y}\_{3}\hat{\mathbf{e}}\_{+}\right) = f(\mathbf{x})\,\_{3}\hat{\mathbf{1}} + \mathbf{y}f'(\mathbf{x})\,\_{3}\hat{\mathbf{e}}\_{+} + \frac{1}{2}\mathbf{y}^{2}f''(\mathbf{x})\,\_{3}\hat{\mathbf{e}}\_{+}^{2} = \begin{pmatrix} f(\mathbf{x}) & \mathbf{y}f'(\mathbf{x}) & \frac{1}{2}\mathbf{y}^{2}f''(\mathbf{x})\\ 0 & f(\mathbf{x}) & \mathbf{y}f'(\mathbf{x})\\ 0 & 0 & f(\mathbf{x}) \end{pmatrix} . \tag{9}$$

It may be verified that, e.g., for the choice "+" in Equation (5), the first row of the resulting matrices in Equation (7) contain the terms of the Taylor expansion up to order *k* − 1. More explicitly, introducing the auxiliary notations for the row vector *ke*1| and the column vector |*k***1** of length *k* ≥ 2,

$$\left| \langle \:^{k} \mathbf{c}\_{1} \right| := \left( 1, 0, \ldots, 0 \right), \quad \left| \:^{1} \mathbf{1} \right\rangle := \begin{pmatrix} 1 \\ \vdots \\ 1 \end{pmatrix},$$

allows us to define the *order k evaluation operation* acting on some function *F*( *-*ˆ *k* +) (which is itself assumed to act entry-wise) depending on a generalized dual number *-*ˆ *k* + as

$$
\langle F(\!\_{k}\hat{\mathfrak{e}}\_{+})\!\rangle\_{k} := \langle \!\_{k}\mathfrak{e}\_{1} \vert F(\!\_{k}\hat{\mathfrak{e}}\_{+})\!\rangle\_{k}\mathbf{1}\rangle \;. \tag{11}
$$

We thus find that

$$\left< f\left(\mathbf{x}\_{k}\mathbf{\hat{l}} + \mathbf{y}\_{k}\mathbf{\hat{e}}\_{+}\right) \right>\_{k} = \sum\_{m=0}^{k-1} \frac{y^{m}}{m!} f^{(m)}(\mathbf{x}) \,. \tag{12}$$

Recently, expansions such as Equation (7) have received considerable interest in the field of numerical analysis [20]. Referring to Fike [19] for an overview, various alternative types of "numbers" have been studied for the purpose of finding optimized numerical schemes for computing *k*th-order derivatives of functions. For example, it has been demonstrated that the use of so-called *hyper-dual numbers* results in first- and second-derivative calculations that are exact, regardless of the step size [14].

For later convenience, motivated by the identity (for *k* ≥ 2)

$$\exp\left({}\_k\hat{\mathbf{e}}\_+\mathbf{x}\right) = \sum\_{r=0}^{k-1} \frac{x^r}{r!} {}\_k\hat{\mathbf{e}}\_+^r,\tag{13}$$

we may introduce the so-called *truncated exponential polynomials* [21] *en*(*x*) defined through the series

$$c\_n(\mathbf{x}) := \sum\_{r=0}^n \frac{\mathbf{x}^r}{r!} \, , \tag{14}$$

which may be expressed in terms of generalized dual numbers as

$$\begin{pmatrix} e\_n(x) \\ e\_{n-1}(x) \\ \vdots \\ e\_1(x) \\ 1 \end{pmatrix} := \exp\left(\_{n+1}\mathfrak{e}\_+ x\right)|\_{n+1} \mathbf{1}\rangle \,. \tag{15}$$

One may thus easily verify (via Equations (13) and (14)) the property

$$
\varepsilon\_n'(\mathbf{x}) = \varepsilon\_{n-1}(\mathbf{x}) \,. \tag{16}
$$

Having provided a matrix-based extension of ordinary to *k*th-order dual numbers of arbitrary order *k* ≥ 2, we now proceed to develop a computational procedure embedding dual numbers with other techniques inspired by the operational umbral formalism.

### **3. Umbral-Type Methods and Dual Numbers**

Starting from this section, we employ the notational simplification of writing  for the *dual number unit (DNU) -*ˆ *k* ± of generalized dual numbers (cf. Equation (5)), making the order *k* ≥ 2 of the DN explicit only via the analog of the notation in Equation (11), and masking the matrix nature of *-*ˆ *k* ±. Thus, for some function *F* ≡ *F*(*-*), we write

$$F \leadsto\_k G \quad \colon \Leftrightarrow \quad G = \left( F|\_{a^{k+1} \to 0} \right)|\_{a \to 1} \tag{17}$$

for the truncation of *F* via setting *k*+1 = 0 and afterwards *-* = 1. It is straightforward to verify that this formal definition may be implemented in terms of the matrix representations introduced in Section 2 via use of Equation (11) as

$$G = \langle F(\_{k+1}\mathfrak{k}\_+) \rangle\_{k+1} \,. \tag{18}$$

Consider then the *dual complex parameter*

$$
\hat{z} \equiv \hat{z}(a, b) := a + \epsilon b \,. \tag{19}
$$

Following the principles of umbral calculus, we treat the dual complex parameter *z*ˆ as an ordinary algebraic quantity in calculations of integrals, derivatives and other operations, delaying the evaluation of *z*ˆ via performing the operation *k* to the very end of the computations. Note that albeit the term umbral calculus has been introduced in the seminal papers by Roman and Rota [22], in the following we make reference to the formalism developed in [23] which enriches the original formalism with the wealth of techniques derived from the operational calculus [22,23]. We now illustrate the computational benefits of this approach via a number of examples.

### *3.1. Dual Shifted Gaussians*

We first consider a Gaussian-type function explicitly containing in its argument the dual complex parameter in Equation (19), whence the *dual-shifted Gaussian* function

$$f(\mathbf{x}) = e^{-a\mathbf{x}^2 + \frac{a}{2}(a,b)\cdot \mathbf{x}}.\tag{20}$$

Assuming for instance third-order dual numbers (i.e., *-*3 = 0), we may write the above function in more conventional terms as

$$f(\mathbf{x}) \stackrel{\sim\_{\sim\_{2}}}{\sim\_{2}} e^{-\mathbf{a}\mathbf{x}^{2} + \mathbf{a}\mathbf{x}} \left[1 + b\mathbf{x} + \frac{1}{2}(b\mathbf{x})^{2}\right],\tag{21}$$

which is easily recognized as the product of a shifted Gaussian with a second-degree polynomial.

In full analogy to the umbral operational methods of Licciardi [23], it is then straightforward to calculate the following integral of the function *f* of Equation (20) via the standard Gaussian integral formula

$$\int\_{-\infty}^{+\infty} f(\mathbf{x})d\mathbf{x} = \sqrt{\frac{\pi}{a}} \, e^{\frac{\pm (a,b)^2}{4a}} = \sqrt{\frac{\pi}{a}} \, e^{\frac{\sigma^2}{4a} + \frac{ab}{2\pi}a + \frac{b^2}{4a}a^2}.\tag{22}$$

*Axioms* **2019**, *8*, 77

The term on the right has in fact a definite meaning, since the use of the generating function of the *two-variable Hermite polynomials* [24]

$$\sum\_{n=0}^{\infty} \frac{t^n}{n!} H\_n(x, y) = e^{xt + yt^2} \tag{23a}$$

$$H\_n(\mathbf{x}, \mathbf{y}) = e^{\mathbf{y} \cdot \partial\_x^2} \mathbf{x}^n = n! \sum\_{r=0}^{\lfloor \frac{n}{2} \rfloor} \frac{\mathbf{x}^{n-2r} \mathbf{y}^r}{(n-2r)! r!} \tag{23b}$$

permits to cast the right-hand side of Equation (22) into the form

$$\sqrt{\frac{\pi}{a}} \, e^{\frac{\mathbf{p}^2}{4a} + \frac{\mathbf{p}^2}{2a}c + \frac{\mathbf{p}^2}{4a}c^2} = \sqrt{\frac{\pi}{a}} \, e^{\frac{\mathbf{p}^2}{4a}} \sum\_{m \ge 0} \frac{\epsilon^m}{m!} H\_m \left( \frac{ab}{2a}, \frac{b^2}{4a} \right) \quad \leadsto\_k \sqrt{\frac{\pi}{a}} \, e^{\frac{\mathbf{p}^2}{4a}} H \, \mathcal{E}\_k \left( \frac{ab}{2a}, \frac{b^2}{4a} \right) \,. \tag{24}$$

Here, *Hek*(*x*, *y*) denotes the *Hermite-based truncated exponential polynomial* [25–27] defined as

$$h\_H c\_k(x, y) := \sum\_{r=0}^k \frac{1}{r!} \, H\_r \left( x, y \right) \,. \tag{25}$$

#### *3.2. Another Form of Dual Gaussian*

Let us consider as a further example

$$\mathbf{g}(\mathbf{x}) := \mathbf{e}^{-\sharp(a,b)\mathbf{x}^{\mathbbm{2}}} \tag{26}$$

and the following infinite integral (for Re(*a*) > 0)

$$\int\_{-\infty}^{+\infty} \mathbf{g}(\mathbf{x})d\mathbf{x} = \sqrt{\frac{\pi}{2}} = \sqrt{\frac{\pi}{a+\epsilon b}} \quad \leadsto\_{k} \sqrt{\frac{\pi}{a}} \sum\_{r=0}^{k} \binom{-\frac{1}{2}}{r} \left(\frac{b}{a}\right)^{r} . \tag{27}$$

Here, by invoking the operation *k* , we obtain a finite series, thus obviating the need to impose any condition on the relevant convergence range.

#### *3.3. Examples From Symbolic Calculus*

The calculus of higher-order dual numbers may be further refined via combining it with the wealth of techniques available from the theory of special functions and symbolic calculus as put forward in [23,28–32]. Consider for illustration the following identity, known from the theory of two-variable Hermite polynomials [33],

$$
\partial\_{\mathbf{x}}^{\boldsymbol{\eta}} \boldsymbol{\varepsilon}^{\mathbf{ax}^{2}} = H\_{\mathbb{H}}(2\boldsymbol{\alpha}\mathbf{x}, \boldsymbol{a}) \boldsymbol{\varepsilon}^{\mathbf{ax}^{2}} \tag{28}
$$

which allows to simplify the task of calculating successive derivatives of the dual Gaussian introduced in Equation (26), such as in the computation

$$\partial\_x^n e^{-\sharp x^2} = H\_\text{ll}(-2\mathfrak{z}\mathbf{x}, -2)e^{-\sharp x^2} = n! \sum\_{r=0}^{\lfloor \frac{n}{2} \rfloor} \sum\_{s \ge 0} \frac{(-1)^{n-r+s} 2^{n-2r} x^{n-2(r-s)}}{(n-2r)! r! s!} 2^{n-r+s} \cdot 2^{n-r+s} \cdot 2^{n-r+s} \ge 0$$

Here, the first step follows from (28) and the second by invoking (23b).

Another interesting type of calculus concerns infinite integrals involving rational functions such as

$$\Phi(\mathbf{x}; a, b) := \frac{1}{1 + 2\mathbf{x}^2} \leadsto\_k \frac{1}{1 + a\mathbf{x}^2} \sum\_{r=0}^k \left( -\frac{b\mathbf{x}^2}{1 + a\mathbf{x}^2} \right)^r. \tag{29}$$

For example, the infinite integral

$$\int\_{-\infty}^{+\infty} \frac{1}{1 + 2\mathbf{x}^2} d\mathbf{x} = \frac{\pi}{\sqrt{\hat{z}}} \tag{30}$$

may be easily transformed into truncated form in full analogy to the calculation summarized in Equation (27).

#### *3.4. Umbral Image Type Techniques*

Referring to Behr et al. [34] for the precise technical details (compare also [33]), suffice it here to provide the following definition for the action of the *formal integration operator* ˆ I on the formal variable *v* (for *α* ∈ C):

$$\hat{\mathbb{1}}(\upsilon^{\mathfrak{a}}) := \frac{1}{\Gamma(\mathfrak{a})}.\tag{31}$$

Then, an interesting variant of the example presented in Equation (29) may be obtained as

$$\mathbb{1}\left[\int\_{-\infty}^{+\infty} v\Phi(\mathbf{x}; a, vb)d\mathbf{x}\right] = \mathbb{1}\left[\frac{v\pi}{\sqrt{z(a, v\#)}}\right] \stackrel{\sim\_k}{\sim} \sqrt{\frac{\pi}{a}} \sum\_{r=0}^k \frac{1}{\Gamma(\frac{1}{2} - r)(r!)^2} \left(\frac{b}{a}\right)^r. \tag{32}$$

In summary, the combination of the concept of higher order dual numbers with techniques from symbolic and umbral-image type calculus appears to offer a large potential in view of novel tools of computation. To corroborate this claim, we now present some first high-level results in this direction.

### **4. Dual Numbers and Solution of Heat- and Schrödinger-Type Equations**

Before entering the main topic of this section, let us recall a few useful "operational rules", starting with the *Glaisher identity* [27,35]

$$e^{\pi \frac{d^2}{dx^2}} e^{-ax^2} = \frac{1}{\sqrt{1+4\pi a}} e^{-\frac{ax^2}{1+4\pi a}}\,\_{\prime} \tag{33}$$

which can also be understood as the solution of the heat equation with a Gaussian as initial function. It proves particularly useful in the following to note that, according to the definition of the Hermite polynomials *Hn*(*x*, *y*) as given in Equation (23b), an alternative interpretation of Equation (33) is provided in terms of the *double-lacunary exponential generating function* H2,0(*λ*; *x*, *y*) of the polynomials *Hn*(*x*, *y*), where we employ notations as in [36]

$$e^{\pi \frac{d^2}{dx^2}} e^{-ax^2} = \sum\_{n \ge 0} \frac{(-a)^n}{n!} H\_{2n}(\mathbf{x}, \boldsymbol{\tau}) = \mathcal{H}\_{2,0}(-a; \mathbf{x}, \boldsymbol{\tau}) \,. \tag{34}$$

By specializing Equation (33) to *α* = *z*ˆ (with *z*ˆ = *a* + *b* the dual complex parameter in Equation (19)), we obtain the operational identity

$$
\varepsilon^{\tau \frac{d^2}{dx^2}} \varepsilon^{-\frac{d}{dx}^2} = \frac{1}{\sqrt{1 + 4\tau \varepsilon}} \varepsilon^{-\frac{\pm x^2}{1 + 4\tau \varepsilon}}.\tag{35}
$$

Via the simple factorizations

$$\begin{split} 1 + 4\sharp\tau &= \gamma(a,\tau)\gamma\left(\frac{bc}{\gamma(a,\tau)},\tau\right), \\ \frac{\hat{z}}{1 + 4\sharp\tau} &= \frac{a}{\gamma(a,\tau)} + \frac{b\varepsilon}{[\gamma(a,\tau)]^2\gamma\left(\frac{bc}{\gamma(a,\tau)},\tau\right)}, \\ \gamma(c,\tau) &= 1 + 4c\tau, \end{split} \tag{36}$$

we may transform the identity in Equation (35) as

$$e^{\tau \frac{d^2}{dx^2}} e^{-\frac{a}{2}x^2} = \mathcal{H}\_{2,0} \left( -\frac{bc}{[\gamma(a,\tau)]^2}; \mathbf{x}, \tau \gamma(a,\tau) \right) \mathcal{H}\_{2,0} (-a; \mathbf{x}, \tau) \,. \tag{37}$$

By re-inserting the definition of the first double-lacunary exponential generating function, using the Glaisher-identity in Equation (33) for the second one and finally truncating to order *k*, we eventually arrive at the compact result

$$\varepsilon e^{\frac{\pi^2}{4a^2}} e^{-\varepsilon x^2} \underset{\leadsto k}{\leadsto k} \frac{e^{-\frac{a\pi^2}{\gamma(a,\tau)}}}{\sqrt{\gamma(a,\tau)}} \sum\_{n=0}^{k} \frac{1}{n!} \left(\frac{b}{\gamma(a,\tau)^2}\right)^n H\_{2n}\left(\mathbf{x}, \tau \boldsymbol{\gamma}(a,\tau)\right) \; , \qquad \gamma(a,\tau) = 1 + 4a\tau . \tag{38}$$

For example, by evaluating the above expression for second order dual numbers, one finds

$$e^{\frac{\pi^2}{4x^2}}e^{-\frac{a}{2}x^2} \stackrel{\sim}{\sim}\_2 \frac{e^{-\frac{a\pi^2}{\gamma(a,\tau)}}}{\sqrt{\gamma(a,\tau)}} \left(1 - \frac{b}{\gamma(a,\tau)^2}H\_2(\mathbf{x},\tau\gamma(a,\tau)) + \frac{b^2}{2\gamma(a,\tau)^4}H\_4(\mathbf{x},\tau\gamma(a,\tau))\right). \tag{39}$$

The above result may be interpreted as the solution of the heat-type equation

$$
\partial\_{\mathbf{r}} F(\mathbf{x}, \mathbf{r}) = \partial\_{\mathbf{x}}^2 F(\mathbf{x}, \mathbf{r}) \,, \quad F(\mathbf{x}, 0) = e^{-\frac{\mathbf{r} \cdot \mathbf{x}^2}{2}} \,. \tag{40}
$$

An analogous problem has been addressed in [33] within the framework of a different method. The techniques we have envisaged may be further exploited to treat the *paraxial propagation* of the so-called *flattened distributions*, introduced in [37] to study the laser field evolution in optical cavities employing super-Gaussian mirrors [38]. These cavities shape beams whose transverse distribution is not reproduced by a simple Gaussian, but by a function exhibiting a *quasi-constant flat-top*, expressible through a function of the type

$$E(\mathbf{x}; p) := e^{-|\mathbf{x}|^{p}}, \qquad p \in \mathbb{Z}\_{\geq 0}. \tag{41}$$

The paraxial propagation of these beams has less obvious properties than, e.g., Laguerre or Hermite Gauss modes [38]. To overcome this drawback, Gori introduced the so-called *flattened beams* [37], which permit a fairly natural expansion in terms of Gauss Laguerre/Hermite modes, thus providing a straightforward solution to the corresponding paraxial wave equation.

Invoking our formalism as developed thus far, we may approximate the aforementioned Gori beams in the form

$$E(\mathbf{x}; p) \approx \mathbf{Y}(\mathbf{x}; a|m) := e^{-a\mathbf{x}^2} e\_m(\mathbf{x}^2) \,. \tag{42}$$

Here, *em*(*x*) denotes the truncated exponential polynomials introduced in Equation (14), and both parameters *α* and *m* depend on *p* (see [33] for further details). Recalling from Equation (19) the definition *z*ˆ(*a*, *b*) := *a* + *b* of the dual complex parameter, the r.h.s. of Equation (42) may be equivalently expressed as

$$\epsilon^{-\frac{\epsilon}{2}(a,-1)\times^2} \stackrel{\sim}{\sim^+} \chi(\mathfrak{x}; a|m) \, , \tag{43}$$

whence as an instance of a *dual Gaussian* as described in Section 3.2. The problem of the relevant propagation can accordingly be reduced to that of an ordinary Gaussian mode, namely to the solution of the Schrödinger type equation

$$i\partial\_{\mathsf{T}}\Psi(\mathsf{x},\mathsf{t})=-\partial\_{\mathsf{x}}^{2}\Psi(\mathsf{x},\mathsf{t})\ ,\quad\Psi(\mathsf{x},0)=\mathsf{Y}(\mathsf{x};a|m)\ .\tag{44}$$

Consequently, by invoking the operational identity in Equation (35), the paraxial evolution of a flattened beam may be expressed in the form

$$\Psi(\mathbf{x},\tau) = e^{i\tau \partial\_x^2} e^{-\frac{\mathbf{i}\cdot(a\_r-1)\mathbf{x}^2}{2}} = \frac{1}{\sqrt{1+4i\tau \mathbf{i}(a\_r-1)}} e^{-\frac{\mathbf{i}(a\_r-1)\mathbf{x}^2}{1+4i\tau \mathbf{i}(a\_r-1)}},\tag{45}$$

which reproduces indeed the known solution of our problem (compare [33]).

In a forthcoming paper, we will discuss this specific application in further detail by applying the method to the problem of designing super-Gaussian optical systems.

#### **5. Weyl Formula and Modified Hermite Polynomials**

The wide flexibility of the method we propose is corroborated by the following further example, relevant to the use of operational ordering tools. Let us consider an evolution equation of the form

$$
\partial\_{\mathsf{T}} F(\mathsf{x}, \mathsf{\tau}) = [\gamma \partial\_{\mathsf{x}} - 2\mathsf{x}] F(\mathsf{x}, \mathsf{\tau}) \,, \quad F(\mathsf{x}, 0) = f(\mathsf{x}) \,. \tag{46}
$$

The relevant procedure for combining differential calculus with the umbral formalism is described in [39]. Following this approach, the solution of Equation (46) can be expressed as

$$F(\mathbf{x}, \mathbf{\tau}) = e^{\mathbf{\tau}(\gamma \partial\_{\mathbf{x}} - \underline{\mathbf{c}}\, \mathbf{x})} f(\mathbf{x}) \,. \tag{47}$$

To evaluate the solution of Equation (47) explicitly, we need to suitably "factorize" the exponential operator. This so-called *disentanglement operation* may be implemented via the Weyl formula [40]

$$e^{\hat{X} + \hat{Y}} = e^{-\frac{1}{2} \langle \hat{X}, \hat{Y} \rangle} e^{\hat{X}} e^{\hat{Y}} \,, \tag{48}$$

which is applicable whenever the identities [*X*ˆ , [*X*ˆ ,*Y*ˆ]] = [[*X*ˆ ,*Y*ˆ],*Y*ˆ]] = 0 hold. Applying the Weyl formula in Equation (48) to our solution in Equation (47), if we let *X*ˆ = −*τzx*ˆ and *Y*ˆ = *τγ∂x* (resulting in [*X*ˆ ,*Y*ˆ] = *τ*2*γz*ˆ, and with higher-order commutators vanishing), we obtain

$$F(\mathbf{x}, \tau) = e^{-\frac{1}{2}\tau^2 \gamma \hat{z}} e^{-\hat{z}x\tau} f(\mathbf{x} + \gamma \tau) \,. \tag{49}$$

Thus, the solution at any desired truncation order *k* may be obtained by invoking the dual number evaluation operation *k* of Equation (17).

As already mentioned above, the Weyl formula applies in the example presented because the algebraic structure of the argument of the exponential in Equation (47) satisfies a special property: the commutators of the associated generators reduce to a constant after the first commutation bracket. A more interesting extension is given by the case in which the generators are embedded into a solvable Lie algebra. In this case, the combined use of the dual number formalism and of the Wei-Norman ordering method [41] leads to new and interesting results. They deserve a separate treatment that will be reported in a forthcoming paper.

As a final example, we define *modified Hermite polynomials Hn*(*x*, *z*ˆ), whence ordinary two-variable Hermite polynomials *Hn*(*x*, *y*) as introduced in Equation (23b) evaluated at *y* = *z*ˆ, with *z*ˆ ≡ *z*ˆ(*a*, *b*) the dual complex parameter of Equation (19),

$$H\_n(\mathbf{x}, \boldsymbol{\varepsilon}) = \boldsymbol{\varepsilon}^{\sharp \partial\_{\mathbf{x}}^2} \mathbf{x}^n. \tag{50}$$

It is straightforward to verify that these modified polynomials inherit all the relevant properties from the polynomials *Hn*(*x*, *y*), such as the recurrences

$$\begin{aligned} \partial\_{\mathbf{x}} H\_{\mathbf{n}}(\mathbf{x}, \mathbf{\hat{z}}) &= n H\_{\mathbf{n}-1}(\mathbf{x}, \mathbf{\hat{z}}) \ , \\ H\_{\mathbf{n}+1}(\mathbf{x}, \mathbf{\hat{z}}) &= \mathbf{x} H\_{\mathbf{n}}(\mathbf{x}, \mathbf{\hat{z}}) + 2 \mathbf{z} \partial\_{\mathbf{x}} H\_{\mathbf{n}}(\mathbf{x}, \mathbf{\hat{z}}) \ , \end{aligned} \tag{51}$$

and we find that they satisfy the second order differential equation

$$2\sharp\partial\_{\mathbf{x}}^{2}H\_{\mathrm{ll}}(\mathbf{x},\mathbf{\hat{z}}) + \mathbf{x}\partial\_{\mathbf{x}}H\_{\mathrm{ll}}(\mathbf{x},\mathbf{\hat{z}}) = nH\_{\mathrm{ll}}(\mathbf{x},\mathbf{\hat{z}}) \,. \tag{52}$$

The explicit form of these truncated polynomials is easily obtained. For example, by using third order dual numbers, which implies

$$
\epsilon^{\sharp \partial\_x^2} \stackrel{\sim}{\sim} \sim\_2 \epsilon^{a\partial\_x^2} \left( 1 + b\partial\_x^2 + \frac{1}{2} b^2 \partial\_x^4 \right) \, , \tag{53}
$$

we find the explicit formula

$$H\_n(\mathbf{x}, \mathbf{\hat{z}}) \leadsto\_{\mathbf{2}} H\_n(\mathbf{x}, a) + b\partial\_a H\_n(\mathbf{x}, a) + \frac{1}{2}b^2 \bar{\partial}\_a^2 H\_n(\mathbf{x}, a) \,. \tag{54}$$

where we have invoked the well-known identity

$$
\partial\_\mathbf{x}^2 H\_\mathbf{n}(\mathbf{x}, \mathbf{y}) = \partial\_\mathbf{y} H\_\mathbf{n}(\mathbf{x}, \mathbf{y}) \,. \tag{55}
$$

### **6. Final Comments**

The method we have outlined in this paper offers many computational advantages to treat problems where truncated expansions (not necessarily of Taylor type) of functions are involved. At its core, the umbral formalism and the notion of higher order dual numbers allow delaying the explicit expansions to later stages in a given calculation, thus opening the possibility to exploit numerous efficient computation strategies from the theory of operational calculus and special functions.

The technique we have introduced in this paper is amenable for new applications in various different fields. We have presented herein the solution of parabolic equations in transport problems, and within such a context a fairly important example has been provided by treating the propagation of flattened beams [33,37] in optics. For brevity, we have just outlined the procedure in terms of a one-dimensional computation. The relevant extension to the three-dimensional case does not require any particular conceptual effort, but only a consistent numerical implementation. In a forthcoming investigation, we will further extend the method and study its potential for treating perturbative problems in classical and quantum mechanics.

**Author Contributions:** Conceptualization, G.D.; methodology, N.B. and G.D.; validation, N.B., G.D. and S.L.; formal analysis, N.B., G.D., A.L., and S.L.; writing—original draft preparation, N.B., G.D., and A.L.; and writing—review and editing, N.B., A.L., and S.L.

**Funding:** The work of N.B. is supported by funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 753750. A.L. was supported by the NCN research project OPUS 12 No. UMO-2016/23/B/ST3/01714 and by the NAWA project: Program im. Iwanowskiej PPN/IWA/2018/1/00098. S.L. was supported by an *Enea-Research Center Individual Fellowship*.

**Acknowledgments:** N.B. would like to thank the LPTMC (Paris 06) and ENEA Frascati for warm hospitality.

**Conflicts of Interest:** The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

#### **References**

c 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Branching Functions for Admissible Representations of Affine Lie Algebras and Super-Virasoro Algebras**

### **Namhee Kwon**

Department of Mathematics, Daegu University, Gyeongsan, Gyeongbuk 38453, Korea<EMAIL_ADDRESS>
Received: 2 May 2019; Accepted: 17 July 2019; Published: 19 July 2019

**Abstract:** We explicitly calculate the branching functions arising from the tensor product decompositions between level 2 and principal admissible representations over sl2. In addition, investigating the characters of the minimal series representations of super-Virasoro algebras, we present the tensor product decompositions in terms of the minimal series representations of super-Virasoro algebras for the case of principal admissible weights.

**Keywords:** branching functions; admissible representations; characters; affine Lie algebras; super-Virasoro algebras

### **1. Introduction**

One of the basic problems in representation theory is to find the decomposition of a tensor product between two irreducible representations. In fact, the study of tensor product decompositions plays an important role in quantum mechanics and in string theory [1,2], and it has attracted much attention from combinatorial representation theory [3]. In addition, recent studies reveal that tensor product decompositions are also closely related to the representation theory of Virasoro algebra and *W*-algebras [4–6].

In [6], the authors extensively study decompositions of tensor products between integrable representations over affine Lie algebras. They also investigate relationships among tensor products, branching functions and Virasoro algebra through integrable representations over affine Lie algebras.

In the present paper we shall follow the methodology appearing in [6]. However, we will focus on admissible representations of affine Lie algebras. Admissible representations are not generally integrable over affine Lie algebras, but integrable with respect to a subroot system of the root system attached to a given affine Lie algebra. Kac and Wakimoto showed that admissible representations satisfy several nice properties such as Weyl-Kac type character formula and modular invariance [5,7]. In their subsequent works, they also established connections between admissible representations of affine Lie algebras and the representation theory of *W*-algebras [4,8]. In addition, Kac and Wakimoto expressed in ([9], Theorem 3.1) the branching functions arising from the tensor product decompositions between principal admissible and integrable representations as the *q*-series involving the associated dominant integral weights and string functions.

One of the main results of this paper is the explicit calculations of the branching functions appearing in ([9], Theorem 3.1). We are particularly interested in the calculations of the branching functions obtained from certain tensor product decompositions of level 2 integrable and principal admissible representations over sl2 (see Theorem 4). We shall see that these branching functions connect the representation theory of affine Lie algebras with the representation theory of super-Virasoro algebras.

We usually apply the theory of modular functions for calculations of string functions [10]. However, in the current work we shall not use the tools of modular functions for the calculations of the string functions appearing in ([9], Theorem 3.1). Instead, we shall use both the invariance properties of

string functions under the action of affine Weyl group and the character formula whose summation is taken over maximal weights (see Theorem 5). It seems like that this approach provides a simpler way for computations of the string functions in our cases.

We would like to point out that in ([5], Corollary 3(c)) the authors expressed the branching functions in terms of theta functions. We shall show that our expressions for the branching functions appearing in Theorem 4 are actually same as those of ([5], Corollary 3(c)) through the investigations of the characters of the minimal series representations of super-Virasoro algebras. Comparing our calculations of the branching functions over sl2 with the characters of the minimal series representations of super-Virasoro algebras, we also present the tensor product decompositions between level 2 integrable and principal admissible representations in terms of the minimal series representations of super-Virasoro algebras (see Theorem 6). This generalizes the decomposition formula appearing in ([6], Section 4.1(a)) to the case of principal admissible weights.

### **2. Preliminaries**

Let *A* = *aij* 1≤*i*,*j*≤*n* be a symmetrizable generalized Cartan matrix and g the Kac-Moody Lie algebra associated with *A*. Let h be a Cartan subalgebra of g. Fix the set of simple roots Π = {*α*1, ··· , *αn*} of h and simple coroots Π∨ = *α*∨ 1 , ··· , *α*∨ *n* of h∗, respectively. Assume that Π and Π∨ satisfy the condition *αj α*∨ *i* = *aij*. We denote by ( | ) the non-degenerate invariant symmetric bilinear form on g. Write Δ, Δ+ and Δ− for the set of all roots, positive roots and negative roots of g, respectively. Put Δ*re* = {*α* ∈ Δ | (*α*|*α*) > 0} and Δ*im* = {*α* ∈ Δ | (*α*|*α*) ≤ 0}. For each *i* = 1, ··· , *n*, we define the fundamental reflection *rαi* of h∗ by

$$r\_{\mathfrak{a}\_i}(\lambda) = \lambda - \lambda \left(\mathfrak{a}\_i^{\vee}\right) \mathfrak{a}\_i \ (\lambda \in \mathfrak{h}^\*)\ .$$

The subgroup *W* of *GL* (h∗) generated by all fundamental reflections is called the *Weyl group* of g.

Among symmetrizable Kac-Moody Lie algebras, the most important Lie algebras are affine Lie algebras whose associated Cartan matrices are called *affine Cartan matrices*. It is known that every affine Cartan matrix is a positive semidefinite of corank 1. Each affine Cartan matrix is in one-to-one correspondence with the affine Dynkin diagram of type *X*(*r*) *n* , where *X* = *A*, *B*, *C*, *D*, *E*, *F* or *G* and *r* = 1, 2 or 3. The number *r* is called the *tier number* (see [11,12] for details). Given an affine Cartan matrix *A* = *aij* 0≤*i*,*j*≤*l* , two (*l* + 1)-tuples *a*∨ *i* 0≤*i*≤*l* and (*ai*)0≤*i*≤*l* of positive integers are uniquely determined by the conditions

$$\begin{array}{ll} 1. & \left(a\_0^\vee, a\_1^\vee, \dots, a\_l^\vee\right)A = \mathbf{O},\\ 2. & A \begin{pmatrix} a\_0\\ a\_1\\ \vdots\\ a\_l\\ a\_l \end{pmatrix} = \mathbf{O},\\ 3. & \gcd\left(a\_0^\vee, a\_1^\vee, \dots, a\_l^\vee\right) = \gcd\left(a\_0, a\_1, \dots, a\_l\right) = 1, \end{array}$$

where **O** is the zero vector. We call (*ai*)0≤*i*≤*l* (resp. *a*∨ *i* 0≤*i*≤*l* ) the *label* (resp. *colabel*) of the affine matrix *A*. The corresponding positive integer *h* = ∑*l i*=0 *ai* (resp. *h*∨ = ∑*l i*=0 *a*∨ *i* ) is called the *Coxeter number* (resp. *dual Coxeter number*). Notice that the element *K* = ∑*l i*=0 *a*∨ *i α*∨ *i* satisfies *αi*(*K*) = 0 for 0 ≤ *i* ≤ *l*, and we call this element the *central element*. Through the non-degenerate bilinear form ( | ) defined on g, the central element *K* corresponds to *δ* = ∑*l i*=0 *aiαi* in h∗.

Suppose that g is the affine Lie algebra associated to an affine Cartan matrix *A* = *aij* 0≤*i*,*j*≤*l* , and let h be a Cartan subalgebra of g. The Cartan subalgebra h is (*l* + 2)-dimensional, and we can decompose h and h∗ as follows:

$$
\mathfrak{h} = \mathfrak{h} \oplus \mathbb{C} \mathbb{K} \oplus \mathbb{C}d,
$$

$$
\mathfrak{h}^\* = \mathfrak{h}^\* \oplus \mathbb{C}\delta \oplus \mathbb{C}\Lambda\_0.
$$

where h = ∑*l i*=1 C*α*∨ *i* and h ∗ = ∑*l i*=1 C*αi*.

The lattice *Q* = ∑*l i*=0 Z*αi* and *Q*∨ = ∑*l i*=0 Z*α*∨ *i* are called the *root lattice* and *coroot lattice*, respectively. Set

$$M = \begin{cases} \overline{Q}^{\vee} & \text{if } r = 1 \text{ or } A = A\_{2l}^{(2)} \\ \overline{Q} & \text{if } r \ge 2 \text{ and } A \ne A\_{2l}^{(2)} \end{cases}$$

For an element *α* ∈ *Q*, we define *tα* ∈ *GL* (h∗) by

$$t\_a\left(\lambda\right) = \lambda + \left(\lambda\left|\delta\right|\right)a - \left\{\frac{\left|a\right|^2}{2}\left(\lambda\left|\delta\right| + \left(\lambda\left|a\right|\right)\right)\delta\left(\lambda \in \mathfrak{h}^\*\right)\right.$$

We call *tα* (*α* ∈ *Q*) the *translation operator*. It is known that the Weyl group *W* of the affine Lie algebra g is also given by *W tM*, where *W* = *rαi* |1 ≤ *i* ≤ *l* and *tM* = {*tα*|*α* ∈ *M*} .

For a non-twisted affine Lie algebra (i.e., *r* = 1), recall that

$$\Delta^{im} = \{ n\delta | n \in \mathbb{Z} - \{ 0 \} \}$$

and

$$\Delta^{rc} = \left\{ n\delta + \mathfrak{a} | n \in \mathbb{Z}, \ \mathfrak{a} \in \overline{\Delta} \right\} \ ,$$

where Δ is the set of all roots of the finite-dimensional simple Lie algebra associated with the finite Cartan matrix *A* = *aij* 1≤*i*,*j*≤*l* .

Set

$$\begin{array}{rcl} P & = & \left\{ \lambda \in \mathfrak{h}^\* | \lambda \left( \mathfrak{a}\_i^{\vee} \right) \in \mathbb{Z} \text{ for } 0 \le i \le l \right\}, \\ P^{\text{ur}} & = & \left\{ \lambda \in P | \lambda (K) = m \right\}, \\ P\_{+} & = & \left\{ \lambda \in \mathfrak{h}^\* | \lambda \left( \mathfrak{a}\_i^{\vee} \right) \in \mathbb{Z}\_{\geq 0} \text{ for } 0 \le i \le l \right\}, \\ P\_{+}^{\text{ur}} & = & P^{\text{ur}} \cap P\_{+}. \end{array}$$

An element in *P* (reps. *P*+) is called an *integral weight* (resp. a *dominant integral weight*). Let *ρ* be the dominant integral weight defined by *ρ α*∨ *i* = 1 for 0 ≤ *i* ≤ *l*. The element *ρ* is called the *Weyl vector* of g. It is sometimes convenient to choose the Weyl vector satisfying the additional condition *ρ*(*d*) = 0, and we get *ρ* = *ρ* + *h*∨Λ0 in this case.

Define the fundamental weights Λ*i* ∈ h∗ (0 ≤ *i* ≤ *l*) by Λ*i α*∨ *j* = *δij* (0 ≤ *j* ≤ *l*) and Λ*i* (*d*) = 0. Similarly, we define the fundamental coweights Λ∨ *i* ∈ h (0 ≤ *i* ≤ *l*) by Λ∨ *i* |*αj* = *δij* (0 ≤ *j* ≤ *l*) and Λ∨ *i* |*d* = 0. Let Λ*i* and Λ∨ *i* be the restrictions of Λ*i* and Λ∨ *i* to h ∗ and h, respectively. Put *P* = ∑*l i*=1 ZΛ*i* and *P*∨ = ∑*l i*=1 ZΛ∨ *i* , and let us introduce a lattice

$$
\widetilde{M} = \begin{cases}
\overline{P}^\vee & \text{if } r = 1 \text{ or } A = A\_{2l}^{(2)} \\
\overline{P} & \text{if } r \ge 2 \text{ and } A \ne A\_{2l}^{(2)}
\end{cases}
$$

Then, the group *W*" = *W tM*" is called the *extended affine Weyl group* of g.

### **3. Branching functions for admissible weights**

Let g be the Kac-Moody Lie algebra associated to a symmetrizable generalized Cartan matrix *A*, and h a Cartan subalgebra of g. An element *λ* ∈ h∗ satisfying conditions

is called an *admissible weight*. When *λ* is an admissible weight, the corresponding irreducible highest weight g-module *L* (*λ*) is called an *admissible* g*-module* or *admissible representation*. Write

$$\Delta\_{\lambda}^{\vee r\varepsilon} = \left\{ \mathfrak{a}^{\vee} \,|\, \mathfrak{a} \in \Delta^{r\varepsilon} \text{ and } \left\langle \lambda + \rho, \mathfrak{a}^{\vee} \right\rangle \in \mathbb{Z}\_{\geq 1} \right\}.$$

Then, it is easy to see that Δ∨*re λ* forms a subroot system of the coroot system Δ∨. We denote by Π∨ *λ* a base of Δ∨ *re λ* , and put *Wλ* = *rα*|*α* ∈ Π∨ *λ* .

An admissible weight *λ* is called a *principal admissible weight* if Π∨ *λ* is isomorphic to Π∨. In general, the level of a principal admissible weight is a rational number. In fact, it is known from [7] that a rational number *m* = *v u* (*u* ∈ Z≥1, *v* ∈ Z, *gcd*(*u*, *v*) = 1) is the level of principal admissible weights if and only if it satisfies

1. *gcd* (*u*,*r*∨) = 1,

2. *u* (*m* + *h*∨) ≥ *h*∨,

where *r*∨ is the tier number of the transposed generalized Cartan matrix *At* and *h*∨ denotes the dual Coxeter number of g.

Henceforth, we assume that g is an affine Lie algebra with a simple coroot system Π∨ = *α*∨ 0 , ··· , *α*∨ *l* .

Given *u* ∈ Z≥1, put *γ*0 = (*u* − 1)*c* + *α*∨ 0 and *γi* = *α*∨ *i* (1 ≤ *i* ≤ *l*). Define *S*(*u*) = {*γi* | 0 ≤ *i* ≤ *l*}. Then, *S*(*u*) becomes a simple coroot system of Δ∨ ∩ - ∑*l i*=0 Z*γi* if *gcd* (*u*,*r*∨) = 1 (see [13], Lemma 3.2.1). Moreover, the following theorems are known.

**Theorem 1.** *Let m* = *v u with u* ∈ Z≥1*, v* ∈ Z *and gcd*(*u*, *v*) = 1*. Assume that y* ∈ *W*" *satisfies y* -*S*(*u*) ⊂ Δ∨ +*. Write Pm u*,*y for the set of all principal admissible weights λ of level m with* Π∨ *λ* = *y* - *S*(*u*) *. Then, we have*

$$P\_{u,y}^{\mathfrak{m}} = \left\{ y \left( \lambda^0 - (u-1) \left( m+h^\vee \right) \Lambda\_0 + \rho \right) - \rho \mid \lambda^0 \in P\_+^{u(m+h^\vee)-h^\vee} \right\}$$

.

**Proof.** See ([7], Theorem 2.1) or ([9], Proposition 1.5).

**Theorem 2.** *Let m* = *v u with u* ∈ Z≥1*, v* ∈ Z *and gcd*(*u*, *v*) = 1*. Let Pm* + *be the set of all principal admissible weights of level m (we use the same notation as the case of dominant integral weights). Then, Pm* + = % *y Pm u*,*y, where y runs over* # *y* ∈ *W*" |*y* - *S*(*u*) ⊂ Δ∨ + \$ *.*

**Proof.** See ([9], Proposition 1.5).

Let us now review branching functions and their connections with the Virasoro algebra.

Recall the Virasoro algebra is an infinite dimensional Lie algebra *Vir* = ( & *n*∈Z C*n*) ⊕ C*c* with brackets

$$[\ell\_{m}, c] = 0 \text{ for all } m \in \mathbb{Z}\_2$$

and

$$[\ell\_{m}, \ell\_{n}] = (m - n)\ell\_{m + n} + \frac{m^3 - m}{12} \delta\_{m + n, 0} \text{ c for all } m, n \in \mathbb{Z}.$$

Let g be a finite dimensional simple Lie algebra, and g = C ' *t*, *t* −1 ( ⊗ g ⊕ C*K* ⊕ C*d* the non-twisted affine Lie algebra over g. Let *V* be the highest weight g-module of level *m* such that *m* + *h*∨ = 0. Define the operators *L*g *n* (*n* ∈ Z) via

$$L^{\mathfrak{g}}(z) = \sum\_{n \in \mathbb{Z}} L\_n^{\mathfrak{g}} z^{-n-2} = \frac{1}{2 \left( m + h^{\vee} \right)} \sum\_{i=1}^{\dim \mathfrak{g}} : u^i(z) u\_i(z) \ :\_{\prime} \tag{1}$$

where *ui* and {*ui*} are bases of g satisfying *ui*|*uj* = *δij*. It is well-known that *V* becomes a *Vir*-module by letting

$$\ell\_{\mathfrak{H}} \longmapsto L\_n^{\mathfrak{g}} \ (n \in \mathbb{Z}) \text{ and } c \longmapsto \frac{m \dim \widetilde{\mathfrak{g}}}{m + h^{\vee}}.\tag{2}$$

The Virasoro action (2) satisfies the following properties:

$$\mathbb{E}\left[\ell\_{\mathfrak{n}}, t^{j} \otimes X\right] = -jt^{j+n} \otimes X \text{ ( $X \in \mathfrak{g}$ ,  $n \in \mathbb{Z}$  - \{0\},  $j \in \mathbb{Z}$ ),\tag{3}$$

$$\ell\_0 = \frac{\left(\Lambda + 2\rho|\Lambda)}{2\left(m + h^\vee\right)} \text{Id} - d.\tag{4}$$

Let p be a reductive subalgebra of g. Then, p is decomposed as p = p0 ⊕ p1 ⊕···⊕ p*s*, where p0 is the center of p and each p*i* (*i* = 1, ··· ,*s*) is a simple Lie algebra. Assume that

$$
\overline{\mathfrak{p}}\_0 \oplus \left(\sum\_{i=1}^s \overline{\mathfrak{h}}\_i\right) \subset \overline{\mathfrak{h}}\_1
$$

and *s*

$$\sum\_{i=1}^{s} \overline{\mathfrak{p}}\_{i+} \subset \overline{\mathfrak{g}}\_{+\prime}$$

where h*i* (resp. h) is a Cartan subalgebra of p*i* (resp. g) and p*i*+ (resp. g+) is the sum of the positive root spaces of p*i* (resp. g). Consider the affinization p = C[*t*, *t* −1] ⊗ p ⊕ C*K*˙ ⊕ C*d* of p. Since *V* is the highest weight g-module, *V* is also the highest weight p-module. However, the action of the central element *K*˙ on *V* is somewhat complicated. We refer to ([11], Chapter 12) for the details of the action of the central element *K*˙ . Let *m*˙ be the level of *V* as a p-module, and write ( | ) for the standard bilinear form on p. Set

$$L^{\mathfrak{p}} = \sum\_{n \in \mathbb{Z}} L^{\mathfrak{p}}\_n z^{-n-2} = \frac{1}{2 \left( \dot{m} + \dot{h}^\vee \right)} \sum\_{i=1}^{\dim \overline{\mathfrak{p}}} : \dot{u}^i(z) \dot{u}\_i(z) :,\tag{5}$$

where *u*˙ *i* and {*u*˙*i*} are bases of p satisfying *u*˙ *i* |*u*˙ *j* = *δij* and ˙ *h*∨ is the dual Coxeter number of p. Using (1) and (5), define

$$L^{\mathfrak{g};\mathfrak{p}}(z) = L^{\mathfrak{g}} - L^{\mathfrak{p}} = \sum\_{n \in \mathbb{Z}} L\_n^{\mathfrak{g};\mathfrak{p}} z^{-n-2}.$$

Due to (3), it follows that

$$\mathbb{E}\left[L\_n^{\oplus p}, t^j \otimes X\right] = 0 \text{ for all } X \in \mathfrak{F}, \ n \in \mathbb{Z} - \{0\} \text{ and } j \in \mathbb{Z}.\tag{6}$$

Applying the operator product expansions, we can verify that *L*g;p(*z*) is, in fact, a Virasoro field with the central charge *c*g;p = *mdim*g *m*+*h*∨ − *mdim* ˙ p *m*˙ +˙ *h*∨ (see [13,14] for the details). We call the Virasoro field *L*g;p(*z*) the *coset Virasoro field*.

In the remaining part of this section, we assume that *V* = *L* (Λ) for a dominant integral weight Λ of level *m*. Let ˙ h be a Cartan subalgebra of p, and p+ the positive part of p. For *ν* ∈ -˙ h ⊕ C*K*˙ ∗ , set

$$V\_{\nu}^{\mathfrak{g};\mathfrak{p}} = \left\{ \upsilon \in L\left(\Lambda\right) \,|\, X\upsilon = 0 \,\left(\forall \, X \in \mathfrak{p}\_{+}\right), \, H\upsilon = \upsilon\left(H\right)\upsilon \,\left(\forall \, H \in \mathfrak{f} \oplus \mathbb{C}\mathbb{X}\right) \right\}.$$

Due to (4) and (6), *V*g;p *ν* is stable under the actions of *L*g;p *n* (*n* ∈ Z). So, *V*g;p *ν* becomes a *Vir*-module. We call this module the *coset Virasoro module*. Notice that *L* (Λ) is decomposed as a *Vir* ⊕ [p, p]-module into

$$L\left(\Lambda\right) = \bigoplus\_{\nu \in \mathfrak{h}^\* \oplus \mathbb{C}\delta \bmod \mathbb{C}\delta} \left(V\_{\nu}^{\mathbb{R};\mathfrak{p}} \otimes \dot{L}(\nu)\right),\tag{7}$$

where *L*˙ (*ν*) is the irreducible [p, p]-module with highest weight *ν* and ˙ *δ* is identified with *K*˙ via the non-degenerate bilinear form on p. From (7), we define a function

$$c\_{\nu}^{\Lambda}(q) = Tr\_{V\_{\nu}^{\mathsf{g}\mathfrak{p}}} q^{-d} = \sum\_{j \in \mathbb{Z}\_{\geq 0}} \mathtt{mult}\_{\Lambda} \left( \nu - j \delta; \mathfrak{p} \right) q^{j} \left( q = e^{-\delta} \right), \tag{8}$$

where the multiplicity is defined as in ([6], Section 1.6). The function (8) is called the *string function*. Using the string function (8), the decomposition (7) yields the following formula for the character of *L* (Λ):

$$chL\left(\Lambda\right) = \sum\_{\substack{\nu \in \mathfrak{F}^\* \ \ominus \subset \mathcal{S} \ \mathbf{m} \mathbf{d} \subset \mathcal{S}}} c\_{\nu}^{\Lambda}(q) chL\left(\nu\right). \tag{9}$$

Let us now introduce the following numbers:

• *m*Λ = |Λ+*ρ*| 2 2(*m*+*h*∨) − |*ρ*| 2 2*h*∨ , • *m*˙ *ν* = |*ν*+*ρ*˙| 2 2(*m*˙ +˙ *h*∨) − |*ρ*˙| 2 2 ˙ *h*∨ ,

where *ρ*˙ is the Weyl vector associated with p.

Then, we define the branching function as *b*Λ *ν* (*τ*) = *qm*Λ−*m*˙ *ν c*Λ *ν* (*q*) for *q* = *e*2*πiτ*. By the strange formula and (4), we see that the branching function also can be written as *b*Λ *ν* (*τ*) = *q*− 1 24 *c*g;p *TrV*g;p *ν q*-0 (see [11] (Chapter 12) for the strange formula).

Recall that the normalized character *ch L* (Λ) is defined as

$$ch^{'}L\left(\Lambda\right) = \varepsilon^{-m\star\delta}chL\left(\Lambda\right).$$

Introducing the coordinate (*τ*, *z*, *t*) for *h* = 2*πi*(−*τd* + *z* + *tK*) ∈ h, we obtain that *ch L*(Λ) (*τ*, *z*, *t*) = *qm*Λ *chL*(Λ) (*τ*, *z*, *t*). So, the Formula (9) can be rewritten as

$$ch\_{L(\Lambda)}^{'}\left(\tau, z, t\right) = \sum\_{\substack{\nu \in \mathfrak{F}^\* \oplus \mathbb{C}\delta \ \mathbf{mod} \subset \mathcal{S}}} b\_{\nu}^{\Lambda}\left(\tau\right) ch\_{L(\nu)}^{'}\left(\tau, z, t\right).$$

### **4. Tensor Product Decompositions**

In this section, we fix an affine Lie algebra g = C ' *t*, *t* −1 ( ⊗ g ⊕ C*K* ⊕ C*d* over a finite dimensional simple Lie algebra g. We also fix a Cartan subalgebra h of g. For *λ*, *μ* ∈ h∗, let *L* (*λ*) and *L* (*μ*) be irreducible highest weight modules over g. We denote by *πλ* and *πμ* the representations of g on *L* (*λ*) and *L* (*μ*), respectively. Put *m* = *λ* (*K*) and *m* = *μ* (*K*). Assume that *m* + *h*∨ = 0, *m* + *h*∨ = 0 and *m* + *m* + *h*∨ = 0. It follows from (2) that the Virasoro algebra *Vir* acts on *L* (*λ*) and *L* (*μ*). The corresponding Virasoro fields are

$$L^{\lambda}(z) = \frac{1}{2\left(m + h^{\vee}\right)} \sum\_{i=1}^{\dim \overline{\mathfrak{g}}} \colon \pi\_{\lambda}\left(\mu\_{i}(z)\right)\pi\_{\lambda}\left(u^{i}(z)\right) :$$

and

$$L^{\mu}(z) = \frac{1}{2\left(m^{\prime} + h^{\vee}\right)} \sum\_{i=1}^{\dim \overline{\mathfrak{g}}} \colon \pi\_{\mu}\left(\mu\_{i}(z)\right)\pi\_{\mu}\left(\mu^{i}(z)\right) : \dots$$

Notice that the Virasoro algebra *Vir* acts on *L* (*λ*) ⊗ *L* (*μ*) via the tensor product action

$$L^{\lambda,\mu}(z) = L^{\lambda}(z) \otimes \mathrm{Id}\_{L(\mu)} + \mathrm{Id}\_{L(\lambda)} \otimes L^{\mu}(z)$$

with the central charge *mdim*g *m*+*h*∨ + *m dim*g *m* +*h*∨ . *Axioms* **2019**, *8*, 82

On the other hand, we may consider the whole tensor product *L* (*λ*) ⊗ *L* (*μ*) as the highest weight g-module. Applying (2) to the highest weight g-module *L* (*λ*) ⊗ *L* (*μ*), we get the associated Virasoro field

$$L^{\lambda \otimes \mu}(z) = \frac{1}{2\left(m + m' + h^\vee\right)} \sum\_{i=1}^{\dim \overline{\mathfrak{g}}} \left(\pi\_\lambda \otimes \pi\_\mu\right) \left(u\_i(z)\right) \left(\pi\_\lambda \otimes \pi\_\mu\right) \left(u^i(z)\right),$$

with the central charge - *m*+*m dim*g *m*+*m* +*h*∨ . Using (3), we have

$$\begin{cases} \boldsymbol{L}\_{m}^{\lambda,\mu}, \boldsymbol{t}^{n} \otimes \mathcal{X} \end{cases} = -\boldsymbol{n}\boldsymbol{t}^{n+m} \otimes \mathcal{X} \text{ ( $\boldsymbol{X} \in \mathfrak{g}$ ,  $m \in \mathbb{Z}$  - \{0\} $, $ n \in \mathbb{Z} $),$
$$\begin{bmatrix} \boldsymbol{L}\_{m}^{\lambda \otimes \mu}, \boldsymbol{t}^{n} \otimes \mathcal{X} \end{bmatrix} = -\boldsymbol{n}\boldsymbol{t}^{n+m} \otimes \mathcal{X} \text{ ($$
\boldsymbol{X} \in \mathfrak{g} $, $ m \in \mathbb{Z} $ - \{0\}$ ,  $n \in \mathbb{Z}$ )}\,. \tag{10}$$

Set " *L*(*z*) = *Lλ*,*μ*(*z*) − *Lλ*⊗*μ*(*z*) = ∑*n*∈Z " *Lnz*−*n*−2. According to ([15], Proposition 10.3), the field " *L*(*z*) yields the coset Virasoro field on *L* (*λ*) ⊗ *L* (*μ*) with central charge *mdim*g *m*+*h*∨ + *m dim*g *m* +*h*∨ − - *m*+*m dim*g *m*+*m* +*h*∨ .

For *μ* ∈ h ∗ ⊕ C*δ*, we define

$$\mathcal{V}^{\lambda,\mu}\_{\mu} = \left\{ \boldsymbol{\upsilon} \in L\left(\lambda\right) \otimes L\left(\mu\right) \,|\, X\boldsymbol{\upsilon} = 0 \,\left(\forall \,\mathbf{x} \in \mathfrak{g}\_{+}\right), \, H\boldsymbol{\upsilon} = \mu\left(\boldsymbol{H}\right)\boldsymbol{\upsilon} \,\left(\forall \,\boldsymbol{H} \in \overline{\mathfrak{h}} \oplus \mathbb{C}\mathcal{K}\right) \right\}.$$

It follows from (10) that the space *Vλ*,*μ μ* becomes a *Vir*-module via the coset Virasoro field " *L*(*z*). Notice that *L* (*λ*) ⊗ *L* (*μ*) is decomposed as a *Vir* ⊕ [g, g]-module into

$$L\left(\lambda\right) \otimes L\left(\mu\right) = \sum\_{\substack{\mu \in \mathfrak{h}^\* \oplus \mathbb{C}\delta \ \mathsf{mod}\ \mathsf{C}\delta}} V\_{\mu}^{\lambda,\mu} \otimes L\left(\mu\right). \tag{11}$$

We obtain from (11) a string function

$$\mathcal{L}\_{\nu}^{\lambda \odot \mu}(q) = \operatorname{Tr}\_{V\_{\mu}^{\lambda,\mu}} q^{-d} = \sum\_{j \in \mathbb{Z}\_{\geq 0}} \operatorname{mult}\_{\lambda \odot \mu} \left( \mu - j \delta; \mathfrak{g} \right) q^{j}. \tag{12}$$

Using (11) and (12), we get

$$ch\left(\lambda\right)ch\left(\mu\right) = \sum\_{\substack{\mu \in \mathfrak{F}^\* \oplus \mathbb{C}\delta \ \text{mod } \mathbb{C}\delta}} c\_{\nu}^{\lambda \otimes \mu}(q) chL\left(\nu\right). \tag{13}$$

If we define the normalized branching function by

$$b\_{\nu}^{\lambda \odot \mu}(\tau) = q^{m\_{\lambda} + m\_{\mu} - m\_{\nu}} c\_{\nu}^{\lambda \odot \mu}(q) \, .$$

then the Formula (13) yields

$$\left(\operatorname{ch}\_{L(\lambda)}^{'}\left(\pi, z, t\right)\operatorname{ch}\_{L(\mu)}^{'}\left(\pi, z, t\right) = \sum\_{\nu} b\_{\nu}^{\lambda \odot \mu}\left(\pi\right)\operatorname{ch}\_{L(\nu)}^{'}\left(\pi, z, t\right). \tag{14}$$

Let Λ be a dominant integral weight and *μ* a principal admissible weight of the affine Lie algebra g. Then, the branching function of the tensor product *L* (Λ) ⊗ *L* (*μ*) can be expressed in terms of the string functions of *L* (Λ) as follows.

**Theorem 3.** *Let* g *be any affine Lie algebra and m* ∈ Z≥0*. Let m* = *v u with u* ∈ Z≥1, *v* ∈ Z *and gcd*(*u*, *v*) = 1*. Assume that* Λ *and μ*0 *are dominant integral weights of level m and u* - *m* + *h*∨ − *h*∨*,* *respectively. Write c*"Λ *ξ* (*q*) *for the modified string function qm*Λ− |*ξ*| 2 2*m c*Λ *ξ*,(g;g) (*q*) *for ξ* ∈ h ∗ ⊕ C*δ, where c*Λ *ξ*,(g;g) (*q*) *is the string function defined with respect to the pair* (g; g) *(i.e.,* p = g *in (8)). Then, for a principal admissible weight μ* = *y* - *μ*0 − (*u* − 1) - *m* + *h*∨ Λ0 + *ρ* − *ρ* ∈ *Pm u*,*y, the following formula holds:*

$$\left(\operatorname{ch}\_{L(\Lambda)}^{'}\left(\tau,z,t\right)\operatorname{ch}\_{L(\mu)}^{'}\left(\tau,z,t\right) = \sum\_{\substack{\nu \in P\_{\mathfrak{u},\mathfrak{y}}^{m+m'} \text{ s.t. } \nu \equiv \Lambda + \mu \text{ mod } Q}} b\_{\nu}^{\Lambda \otimes \mu}\left(\tau\right) \operatorname{ch}\_{L(\nu)}^{'}\left(\tau,z,t\right),$$

*where*

$$b\_{\nu}^{\Lambda \odot \mu}(\tau) = \sum\_{w \in \mathcal{W}} \epsilon(w) q^{\frac{\binom{\nu'}{m'+h^{\vee}} \binom{\nu}{m+m'+h^{\vee}}}{2m} \left| \frac{w\binom{0}{\nu+\rho} - \frac{\mu^{0}+\rho}{m'+h^{\vee}}}{w+m'+h^{\vee}} \right|^2\_{\tilde{\mathcal{G}}^{\Lambda}\_{\tilde{\mathcal{Y}}(w\backslash \nu^0 + \rho)-(\mu^0+\rho)-(\mu-1)m\Lambda\_0)}(q) \;. $$

### **Proof.** See ([9], Theorem 3.1).

In the next section, we simply write *c*"Λ *λ* for *c*"Λ *λ* (*q*) if no confusion seems likely to arise, and will calculate explicitly the branching functions for some specific cases.

### **5. Explicit Calculations of Branching Functions**

Let Λ0 and Λ1 be the fundamental weights of sl2, and *λ* a principal admissible weight of sl2. In this section, we explicitly calculate the branching functions arising from the tensor product decompositions of (*L* (2Λ0) ⊕ *L* (2Λ1)) ⊗ *L* (*λ*) and *L* (*ρ*) ⊗ *L* (*λ*).

Let us write Π = {*α*} for the simple root system of sl2. Then it is easy to check

$$h^\vee = 2, \,\rho = \Lambda\_0 + \Lambda\_1 = \overline{\rho} + h^\vee \Lambda\_0 \text{ and } \Lambda\_1 = \Lambda\_0 + \frac{1}{2}a \tag{15}$$

for sl\*2. Let *m* = *v u* (*u* ∈ 2Z≥1, *v* ∈ 2Z + 1), and choose a principal admissible weight *λ* of level *m* satisfying *λ* = *λ*0 − (*u* − 1)(*m* + 2)Λ0 ∈ *Pm u*,1 for *λ*0 ∈ *Pu*(*m*+2)−2 + (see Theorems 1 and 2).

Applying Theorem 3 to the tensor product representations *L* (2Λ0) ⊗ *L* (*λ*) and *L* (2Λ1) ⊗ *L* (*λ*), we obtain

$$\begin{array}{ll} \operatorname{ch}'\_{L(2\Lambda\_0)}\left(\tau,z,t\right) \operatorname{ch}'\_{L(\lambda)}\left(\tau,z,t\right) \\ = \sum\_{\nu \in P\_{\nu,l}^{m+2} \text{ s.t. } \nu \equiv 2\Lambda\_0 + \lambda \bmod Q} b\_{\nu}^{2\Lambda\_0 \otimes \lambda}\left(\tau\right) \operatorname{ch}'\_{L(\nu)}\left(\tau,z,t\right), \end{array} \tag{16}$$

where

$$b\_{\nu}^{2\Lambda\_0 \otimes \lambda}(\tau) = \sum\_{w \in W} \epsilon(w) q^{\frac{(m+2)(m+4)}{4} \left| \frac{w\left(\nu^0 + \rho\right)}{m+4} - \frac{\lambda^0 + \rho}{m+2} \right|^2} \tilde{c}\_{w(\nu^0 + \rho) - (\lambda^0 + \rho) - 2(u-1)\Lambda\_0}^{2\Lambda\_0}$$

and

$$\begin{array}{ll} \operatorname{ch}\_{L(2\Lambda\_{1})}^{'}\left(\mathfrak{r},z,t\right)\operatorname{ch}\_{L(\lambda)}^{'}\left(\mathfrak{r},z,t\right) \\ = \sum\_{\substack{\overline{\nu}\in P\_{u,1}^{m+2}\text{ s.t. }\overline{\nu}\equiv 2\Lambda\_{1}+\lambda\text{ mod }Q}} b\_{\overline{\nu}}^{2\Lambda\_{1}\otimes\lambda}\left(\mathfrak{r}\right)\operatorname{ch}\_{L(\overline{\nu})}^{'}\left(\mathfrak{r},z,t\right), \end{array} \tag{17}$$

where

$$b\_{\overline{\nu}}^{2\Lambda\_1 \otimes \lambda}(\tau) = \sum\_{w \in W} \epsilon(w) q^{\frac{(m+2)(m+4)}{4} \left| \frac{w\left(\overline{\nu}^0 + \rho\right)}{m+4} - \frac{\lambda^0 + \rho}{m+2} \right|^2} c\_{w(\overline{\nu}^0 + \rho) - (\lambda^0 + \rho) - 2(\nu - 1)\Lambda\_0}^{2\Lambda\_1}$$

Similarly, if we apply Theorem 3 to the tensor product representation *L* (*ρ*) ⊗ *L* (*λ*) then we have

$$\begin{array}{ll} \operatorname{ch}\_{\mathcal{L}(\rho)}^{'} \left( \mathbf{r}, z, t \right) \operatorname{ch}\_{\mathcal{L}(\lambda)}^{'} \left( \mathbf{r}, z, t \right) \\ = \sum\_{\substack{\nu \in P\_{\mathsf{u}, 1}^{m+2} \ \mathsf{s}.\mathsf{f}.\ \nu \equiv \rho + \lambda \bmod Q}} b\_{\nu}^{\rho \odot \lambda} \left( \mathbf{r} \right) \operatorname{ch}\_{\mathcal{L}(\nu)}^{'} \left( \mathbf{r}, z, t \right), \end{array} \tag{18}$$

where

$$b\_{\nu}^{\rho \odot \lambda}(\tau) = \sum\_{w \in \mathcal{W}} \varepsilon(w) q^{\frac{(m+2)(m+4)}{4} \left| \frac{w\left(\nu^{0} + \rho\right)}{m+4} - \frac{\lambda^{0} + \rho}{m+2} \right|^{2}} \tilde{c}^{\rho}\_{w\left(\nu^{0} + \rho\right) - (\lambda^{0} + \rho) - 2(u-1)\lambda\_{0}}.$$

For *λ*0 ∈ *Pu*(*m*+2)−2 + and *ν*0 ∈ *Pu*(*m*+4)−2 + , let us write

$$\begin{aligned} \lambda^0 &= \left(\mu(m+2) - 2 - n\right)\Lambda\_0 + n\Lambda\_1, \\ \nu^0 &= \left(\mu(m+4) - 2 - n'\right)\Lambda\_0 + n'\Lambda\_1 \end{aligned} \tag{19}$$

for some *n* ∈ Z and *n* ∈ Z. Then, we can rewrite *λ* and *ν* in (16) as

$$
\lambda = \lambda^0 - (\mu - 1)(m + 2)\Lambda\_0 = (m - n)\Lambda\_0 + n\Lambda\_1,
$$

and

$$\nu = \nu^0 - (\mu - 1)(m + 4)\Lambda\_0 = \left(m - n' + 2\right)\Lambda\_0 + n'\Lambda\_1.$$

Since 2Λ0 − (*ν* − *λ*) ∈ *Q* and 2Λ1 − 2Λ0 = *α*, we should have *n* ≡ *n* (mod 2).

Similarly, for *ν*"0 = - *u*(*m* + 4) − 2 − *n* Λ0 + *n* Λ1 ∈ *Pu*(*m*+4)−2 + , we obtain *ν*" = - *m* − *n* + 2 Λ0 + *n* Λ1 - *n* ∈ Z . From the condition 2Λ1 − (*ν*"− *λ*) ∈ Z*α*, we have the same condition *n* ≡ *n* (mod 2) as the case of *ν*. For this reason, we shall identify *ν*" with *ν* in the following Theorem 4. The same argument yields that the condition *ν* ≡ *ρ* + *λ* mod *Q* in (18) is equivalent to the condition *n* ≡ *n* + 1 (mod 2) in (19).

**Theorem 4.** *Let m* = *v u for u* ∈ 2Z≥1 *and v* ∈ 2Z + 1*, and let p* = *u*(*m* + 4) *and p* = *u*(*m* + 2)*.*

*1. Suppose that*

$$
\lambda^0 = \left(\mu(m+2) - 2 - n\right)\Lambda\_0 + n\Lambda\_1,
$$

*and*

$$\nu^0 = \left(\mu(m+2) - 2 - n'\right)\Lambda\_0 + n'\Lambda\_1$$

*for some n* ∈ 4Z *and n* ∈ Z *satisfying n* ≡ *n* (*mod* 2)*. Then, the branching functions in (16) and (17) are explicitly given by*

$$\begin{aligned} \left(b\_{\nu}^{2\Lambda\_{0}\otimes\Lambda}\left(\tau\right)\right)^{2} &= \sum\_{j\in\mathbb{Z}} q^{\frac{1}{8\nu p'}\left(2pp'j + \binom{n'+1}{n'}p' - (n+1)p\right)^{2}} A\\ &- \sum\_{j\in\mathbb{Z}} q^{\frac{1}{8\nu p'}\left(2pp'j - \binom{n'+1}{n'}p' - (n+1)p\right)^{2}} B \end{aligned} \tag{20}$$

*and*

$$\begin{split} \left| b\_{\nu}^{2\Lambda\_{1}\odot\lambda} \left( \pi \right) \right| &= \sum\_{j \in \mathbb{Z}} q^{\frac{1}{8pp'} \left( 2pp'j + \binom{n'+1}{n'}p' - (n+1)p \right)^2} B \\ &- \sum\_{j \in \mathbb{Z}} q^{\frac{1}{8pp'} \left( 2pp'j - \binom{n'+1}{n'}p' - (n+1)p \right)^2} A\_{\nu} \end{split} \tag{21}$$

*where A* = *c*" 2Λ0 2Λ0 , *B* = *c*" 2Λ0 2Λ1 *if n* ≡ 0 (*mod* 4) *A* = *c*" 2Λ0 2Λ1 , *B* = *c*" 2Λ0 2Λ0 *if n* ≡ 2 (*mod* 4).

*2. Assume that*

$$
\lambda^0 = \left(\mu(m+2) - 2 - n\right)\Lambda\_0 + n\Lambda\_1,
$$

*and*

$$\nu^0 = \left(\mu(m+2) - 2 - n'\right)\Lambda\_0 + n'\Lambda\_1$$

*for some n* ∈ 4Z *and n* ∈ 4Z + 1*. Then, the branching function in (18) is explicitly given by*

$$\begin{split} & \mathcal{B}\_{\nu}^{\rho\odot\lambda} \left( \text{tr} \right) \\ &= \sum\_{j\in\mathbb{Z}} \left( q^{\frac{1}{8\eta p^{j}} \left( 2p p' \cdot j + \binom{n'+1}{n'+1} p' - (n+1)p \right)^{2}} - q^{\frac{1}{8\eta p^{j}} \left( 2p p' \cdot j - \binom{n'+1}{n'+1} p' - (n+1)p \right)^{2}} \right) \tilde{c}\_{p}^{\rho} . \end{split} \tag{22}$$

**Proof.** We first prove (20) and (21).

Recall that the Weyl group *W* of sl\*2 is given by *tjα*, *tjαrα*|*j* ∈ Z . By (15) and (19), we have

$$\nu^0 + \rho = \mu(m+4)\Lambda\_0 + \frac{n'+1}{2}\alpha^2$$

and

$$
\lambda^0 + \rho = \mu(m+2)\Lambda\_0 + \frac{n+1}{2}\kappa\_{-1}
$$

So, we get

$$\begin{aligned} t\_{j\mu} \left( \nu^0 + \rho \right) - \left( \lambda^0 + \rho \right) - 2(\mu - 1)\Lambda\_0 \\ \lambda = 2\Lambda\_0 + \left( \mu(m + 4)j + \frac{n' - n}{2} \right) \alpha - \left( \mu(m + 4)j^2 + \binom{n'}{n'}j \right) \delta \end{aligned} \tag{23}$$

and

$$\begin{aligned} t\_{\rm ja}r\_{\rm a}\left(\nu^{0}+\rho\right) - \left(\lambda^{0}+\rho\right) - 2(u-1)\Lambda\_{0} \\ t\_{\rm a} = 2\Lambda\_{0} + \left(\mu(m+4)j - \frac{n'+n+2}{2}\right)a + \left(\mu(m+4)j^{2} - \left(n'+1\right)j\right)\delta. \end{aligned} \tag{24}$$

Notice from ([11], (12.7.9)) that we have

$$
\hat{\mathcal{E}}\_{w\left(\lambda'\right)+2\gamma+a\delta}^{2\Lambda\_0} = \hat{\mathcal{E}}\_{\lambda'}^{2\Lambda\_0} \tag{25}
$$

for *λ* ∈ h∗, *w* ∈ *W*, *γ* ∈ Z*α* and *a* ∈ C. Since *W* = {1,*rα*}, we see from (25) that

$$
\hat{c}^{2\Lambda\_0}\_{\lambda+(2n+1)\kappa+n\delta} = \hat{c}^{2\Lambda\_0}\_{(\lambda+\kappa)+2n\kappa+n\delta} = \hat{c}^{2\Lambda\_0}\_{\lambda+\kappa}
$$

*Axioms* **2019**, *8*, 82

and

$$
\widehat{\mathfrak{c}}\_{r\_a(\lambda)+(2n+1)a+a\delta}^{2\Lambda\_0} = \widehat{\mathfrak{c}}\_{r\_a(\lambda+a)+(2n+2)a+a\delta}^{2\Lambda\_0} = \widehat{\mathfrak{c}}\_{\lambda+a}^{2\Lambda\_0}.
$$

Hence, in any case we obtain

$$
\hat{\mathcal{E}}\_{w(\lambda)+(2u+1)a+a\delta}^{2\Delta\_0} = \hat{\mathfrak{e}}\_{\lambda+u}^{2\Delta\_0} \tag{26}
$$

for *w* ∈ *W*. Since *u* is even, we have

$$
\mu(m+4)j + \frac{n'-n}{2} \equiv \frac{n'-n}{2} \pmod{2}
$$

and

$$
\mu(m+4)j - \frac{n'+n+2}{2} \equiv -\frac{n'+n+2}{2} \pmod{2} \dots
$$

Since *n* ∈ 4Z and *n* ≡ *n* (mod 2), we obtain *n* ≡ 0 (mod 4) or *n* ≡ 2 (mod 4). If *n* ≡ 0 (mod 4), then *n* −*n* 2 ≡ 0 (mod 2) and −*n* +*n*+2 2 ≡ 1 (mod 2). Thus, by (23), (24), (25) and (26) we get

$$\hat{\mathcal{E}}\_{t\_{\hat{\mu}}}^{2\Lambda\_0}(\nu^0+\rho) - (\lambda^0+\rho) - 2(u-1)\Lambda\_0 = \hat{\mathcal{E}}\_{2\Lambda\_0}^{2\Lambda\_0} \tag{27}$$

and

$$
\tilde{\mathfrak{c}}\_{t\_{\mu}r\_{\mathfrak{a}}(\nu^0+\rho)-(\lambda^0+\rho)-2(u-1)\Lambda\_0}^{2\Lambda\_0} = \tilde{\mathfrak{c}}\_{2\Lambda\_0+u}^{2\Lambda\_0} = \tilde{\mathfrak{c}}\_{2\Lambda\_1}^{2\Lambda\_0}.\tag{28}
$$

Similarly, if *n* ≡ 2 (mod 4), then *n* −*n* 2 ≡ 1 (mod 2) and −*n* +*n*+2 2 ≡ 0 (mod 2). So, in this case we have

$$
\hat{\mathfrak{c}}\_{t\_{ja}(\lambda^0+\rho)-(\mu^0+\rho)-2(u-1)\Lambda\_0}^{2\Lambda\_0} = \hat{\mathfrak{c}}\_{2\Lambda\_0+\mathfrak{a}}^{2\Lambda\_0} = \hat{\mathfrak{c}}\_{2\Lambda\_1}^{2\Lambda\_0} \tag{29}
$$

and

$$
\tilde{c}\_{t\_{ja}r\_{\theta}}^{2\Lambda\_0}(\lambda^0+\rho) - (\mu^0+\rho) - 2(u-1)\Lambda\_0 = \tilde{c}\_{2\Lambda\_0}^{2\Lambda\_0}.\tag{30}
$$

We now compute the exponent (*m*+2)(*m*+4) 4 *w*(*ν*0+*ρ*) *m*+4 − *λ*0+*ρ m*+2 2 of *q* in (16) and (17). Since *p* = *u*(*m* + 4) in assumption, we see that

$$t\_{\rm jx} \left( \nu^0 + \rho \right) = p\Lambda \rho + \left( pj + \frac{n'+1}{2} \right) \alpha - \left( pj^2 + \left( n'+1 \right) j \right) \delta. \tag{31}$$

and

$$t\_{\rm ju} r\_a \left( \nu^0 + \rho \right) = p \Lambda\_0 + \left( pj - \frac{n'+1}{2} \right) a - \left( pj^2 - \left( n'+1 \right) j \right) \delta. \tag{32}$$

It also follows from the assumption *p* = *u*(*m* + 2) that

$$\frac{(m+2)(m+4)}{4} \left| \frac{w\left(\nu^{0} + \rho\right)}{m+4} - \frac{\lambda^{0} + \rho}{m+2} \right|^{2}$$

$$= \frac{u^{2}(m+2)(m+4)}{4} \left| \frac{w\left(\nu^{0} + \rho\right)}{u(m+4)} - \frac{\lambda^{0} + \rho}{u(m+2)} \right|^{2} \tag{33}$$

$$= \frac{pp'}{4} \left| \frac{w\left(\nu^{0} + \rho\right)}{p} - \frac{\lambda^{0} + \rho}{p'} \right|^{2}.$$

Notice from (31) and (32) that

$$\begin{aligned} &\frac{t\_{j\alpha}\left(\nu^{0}+\rho\right)}{p}-\frac{\lambda^{0}+\rho}{p'}\\ &=\quad \left(\Lambda\_{0}+\left(j+\frac{n'+1}{2p}\right)\alpha\right)-\frac{1}{p'}\left(p'\Lambda\_{0}+\frac{n+1}{2}\alpha\right)\bmod\mathbb{C}\delta\\ &=\quad \left(j+\frac{n'+1}{2p}-\frac{n+1}{2p'}\right)\alpha\bmod\mathbb{C}\delta\end{aligned}$$

and

$$\begin{aligned} &\frac{\mathfrak{f}\_{j\mathfrak{a}}r\_{\mathfrak{a}}\left(\upsilon^{0}+\rho\right)}{p}-\frac{\lambda^{0}+\rho}{p'}\\ &=\ \left(\Lambda\_{0}+\left(j-\frac{n'+1}{2p}\right)a\right)-\frac{1}{p'}\left(p'\Lambda\_{0}+\frac{n+1}{2}a\right)\bmod\mathbb{C}\delta\\ &=\ \left(j-\frac{n'+1}{2p}-\frac{n+1}{2p'}\right)a\bmod\mathbb{C}\delta.\end{aligned}$$

Thus, we obtain

$$\left|\frac{t\_{ja}\left(\nu^0+\rho\right)}{p}-\frac{\lambda^0+\rho}{p'}\right|^2 = \frac{1}{2\left(pp'\right)^2}\left(2pp'j+\left(n'+1\right)p'-(n+1)p\right)^2,$$

$$\left|\frac{t\_{ja}r\_a\left(\nu^0+\rho\right)}{p}-\frac{\lambda^0+\rho}{p'}\right|^2 = \frac{1}{2\left(pp'\right)^2}\left(2pp'j-\left(n'+1\right)p'-(n+1)p\right)^2. \tag{34}$$

Hence, if *n* ≡ 0 (mod 4), then we obtain from (27), (28), (33) and (34) that

$$\begin{split} b^{2\Lambda\_0 \otimes \lambda}\_{\nu} \left( \tau \right) &= \sum\_{j \in \mathbb{Z}} q^{\frac{1}{8pp'} \left( 2pp'j + \binom{n'+1}{n'+1}p' - (n+1)p \right)^2} \hat{c}^{2\Lambda\_0}\_{2\Lambda\_0} \\ &- \sum\_{j \in \mathbb{Z}} q^{\frac{1}{8pp'} \left( 2pp'j - \binom{n'+1}{n'+1}p' - (n+1)p \right)^2} \hat{c}^{2\Lambda\_0}\_{2\Lambda\_1}. \end{split}$$

If *n* ≡ 2 (mod 4) then we also obtain from (29), (30), (33) and (34)

$$\begin{split} b\_{\nu}^{2\Lambda\_{0}\odot\lambda} \left( \tau \right) &= \sum\_{j\in\mathbb{Z}} q^{\frac{1}{8pp'} \left( 2pp'j + \binom{n'+1}{n'+1}p' - (n+1)p \right)^2} \hat{c}\_{2\Lambda\_{1}}^{2\Lambda\_{0}} \\ &- \sum\_{j\in\mathbb{Z}} q^{\frac{1}{8pp'} \left( 2pp'j - \binom{n'+1}{n'+1}p' - (n+1)p \right)^2} \hat{c}\_{2\Lambda\_{0}}^{2\Lambda\_{0}} .\end{split}$$

The Formula (20) now follows.

Applying the same argument as above to the case of *b* 2Λ1⊗*λ ν* (*τ*), we obtain

$$\begin{cases} \hat{c}\_{t\_{j\bar{\mu}}(\nu^0+\rho)-(\lambda^0+\rho)-2(u-1)\Lambda\_0}^{2\Delta\_1} = \hat{c}\_{2\Lambda\_0}^{2\Lambda\_1} & \text{if } n' \equiv 0 \pmod{4} \\ \hat{c}\_{t\_{j\bar{\mu}}x\_0(\nu^0+\rho)-(\lambda^0+\rho)-2(u-1)\Lambda\_0}^{2\Delta\_1} = \hat{c}\_{2\Lambda\_1}^{2\Lambda\_1} & \text{if } n' \equiv 0 \pmod{4} \end{cases} \tag{35}$$

and

$$\begin{cases} \hat{c}\_{t\_{ja}(\nu^0+\rho)-(\lambda^0+\rho)-2(u-1)\Lambda\_0}^{2\Delta\_1} = \hat{c}\_{2\Lambda\_1}^{2\Lambda\_1} & \text{if } n' \equiv 2 \pmod{4} \\ \hat{c}\_{t\_{ja}r\_a(\nu^0+\rho)-(\lambda^0+\rho)-2(u-1)\Lambda\_0}^{2\Lambda\_1} = \hat{c}\_{2\Lambda\_0}^{2\Lambda\_1} & \text{if } n' \equiv 2 \pmod{4} \end{cases} \tag{36}$$

Notice that we have *c*" *M*Λ0+*N*Λ1 *m*Λ0+*n*Λ1 = *c*" *N*Λ0+*M*Λ1 *n*Λ0+*m*Λ1 due to the outer automorphism of sl2. Hence, we obtain that

$$
\hat{c}\_{2\Lambda\_0}^{2\Lambda\_1} = \hat{c}\_{2\Lambda\_1}^{2\Lambda\_0} \text{ and } \hat{c}\_{2\Lambda\_1}^{2\Lambda\_1} = \hat{c}\_{2\Lambda\_0}^{2\Lambda\_0} \tag{37}
$$

Therefore, if *n* ≡ 0 (mod 4) then we get from (35), (36), (33), (34) and (37) that

$$\begin{split} b\_{\nu}^{2\Lambda\_{1}\otimes\lambda}(\boldsymbol{\pi}) &= \sum\_{\boldsymbol{j}\in\mathbb{Z}} q^{\frac{1}{8pp'} \left(2pp'\boldsymbol{j} + \binom{\boldsymbol{n'}+1}{\boldsymbol{n'}}p' - (\boldsymbol{n}+1)p\right)^{2}} \widehat{\mathfrak{E}}\_{2\Lambda\_{1}}^{2\Lambda\_{0}} \\ &- \sum\_{\boldsymbol{j}\in\mathbb{Z}} q^{\frac{1}{8pp'} \left(2pp'\boldsymbol{j} - \binom{\boldsymbol{n'}+1}{\boldsymbol{n'}}p' - (\boldsymbol{n}+1)p\right)^{2}} \widehat{\mathfrak{E}}\_{2\Lambda\_{0}}^{2\Lambda\_{0}} .\end{split}$$

Similarly, if *n* ≡ 2 (mod 4) then we obtain that

$$\begin{split} b\_{\nu}^{2\Lambda\_{1}\otimes\lambda}(\boldsymbol{\pi}) &= \sum\_{\boldsymbol{j}\in\mathbb{Z}} q^{\frac{1}{8pp'} \left(2pp'\boldsymbol{j} + \binom{\boldsymbol{n}'+1}{\boldsymbol{n}}p' - (\boldsymbol{n}+1)p\right)^{2}} \widehat{c}\_{2\Lambda\_{0}}^{2\Lambda\_{0}} \\ &- \sum\_{\boldsymbol{j}\in\mathbb{Z}} q^{\frac{1}{8pp'} \left(2pp'\boldsymbol{j} - \binom{\boldsymbol{n}'+1}{\boldsymbol{n}}p' - (\boldsymbol{n}+1)p\right)^{2}} \widehat{c}\_{2\Lambda\_{1}}^{2\Lambda\_{0}} .\end{split}$$

The Formula (21) now follows.

Let us now prove (22).

The proof is exactly the same as those of (20) and (21) except for calculations of the string function *c*" *ρ w*(*ν*0+*ρ*)−(*λ*0+*ρ*)−2(*u*−1)Λ0 . Recall from the assumption that *n* ∈ 4Z and *n* ∈ 4Z + 1. Then, by (23)–(25) we obtain

$$\widetilde{\mathcal{E}}\_{t\_{j\mathfrak{a}}(\nu^0+\rho)-(\lambda^0+\rho)-2(u-1)\Lambda\_0}^{\rho} = \widetilde{\mathcal{E}}\_{2\Lambda\_0+\frac{u'-u}{2}\mathfrak{a}}^{\rho} = \widetilde{\mathcal{E}}\_{2\Lambda\_0+\frac{1}{2}\mathfrak{a}}^{\rho} = \widetilde{\mathcal{E}}\_{\rho}^{\rho}$$

and

$$\widetilde{\mathcal{E}}\_{t\_{ja}r\_{\mathfrak{a}}(\nu^0+\rho)-(\lambda^0+\rho)-2(\mathfrak{u}-1)\Lambda\_0}^{\rho} = \widetilde{\mathcal{E}}\_{2\Lambda\_0-\frac{\mathfrak{a}'+\mathfrak{a}+2}{2}\mathfrak{a}}^{\rho} = \widetilde{\mathcal{E}}\_{2\Lambda\_0-2\mathfrak{a}+\frac{1}{2}\mathfrak{a}}^{\rho} = \widetilde{\mathcal{E}}\_{\rho}^{\rho}.$$

The result now follows.

It is immediate from Theorem 4 that the branching function of (*L* (2Λ0) ⊕ *L* (2Λ1)) ⊗ *L* (*λ*) for sl2 is given by

$$\begin{split} & \quad b\_{\nu}^{2\Lambda\_{0}\otimes\lambda} \left( \tau \right) + b\_{\nu}^{2\Lambda\_{1}\otimes\lambda} \left( \tau \right) \\ &= \quad \sum\_{j\in\mathbb{Z}} q^{\frac{1}{8pp'} \left( 2pp'j + \binom{\nu'+1}{\nu'+1}p' - (n+1)p \right)^{2}} \left( \hat{c}\_{2\Lambda\_{0}}^{2\Lambda\_{0}} + \hat{c}\_{2\Lambda\_{1}}^{2\Lambda\_{0}} \right) \\ & \quad \quad - \sum\_{j\in\mathbb{Z}} q^{\frac{1}{8pp'} \left( 2pp'j - \binom{\nu'+1}{\nu'+1}p' - (n+1)p \right)^{2}} \left( \hat{c}\_{2\Lambda\_{0}}^{2\Lambda\_{0}} + \hat{c}\_{2\Lambda\_{1}}^{2\Lambda\_{0}} \right) . \end{split}$$

In the following theorem, we explicitly calculate *c*" 2Λ0 2Λ0 + *c*" 2Λ0 2Λ1 and *c*" *ρ ρ* in terms of the Dedekind eta function.

$$\textbf{Theorem 5. } \hat{c}\_{2\Lambda\_0}^{2\Lambda\_0} + \hat{c}\_{2\Lambda\_1}^{2\Lambda\_0} = \frac{\eta(\tau)}{\eta\left(\frac{\tau}{2}\right)\eta(2\tau)} \text{ and } \hat{c}\_{\rho}^{\rho} = \frac{\eta(2\tau)}{\eta(\tau)^2}, \text{ where } \eta\left(\tau\right) = q^{\frac{1}{24}}\Pi\_{n=1}^{\infty}\left(1 - q^n\right).$$

*Axioms* **2019**, *8*, 82

**Proof.** It follows from the Weyl-Kac character formula that

$$chL\left(2\Lambda\_0\right) = \frac{1}{e^{\rho}R} \sum\_{w \in \mathcal{W}} \epsilon(w)e^{w\left(2\Lambda\_0 + \rho\right)}\,,\tag{38}$$

where

$$\mathcal{W} = \left\{ t\_{j\alpha\prime} \,|\, t\_{j\alpha} r\_{\alpha} \,|\, j \in \mathbb{Z} \right\}$$

and

$$R = \Pi\_{n=1}^{\infty} \left( 1 - q^n \right) \left( 1 - e^{-a} q^{n-1} \right) \left( 1 - e^a q^n \right) \dots$$

Calculating *w* (2Λ0 + *ρ*) for *w* ∈ *W*, we obtain from (38)

$$chL\left(2\Lambda\_0\right) = \frac{1}{e^{\rho}R} \left(\sum\_{j \in \mathbb{Z}} e^{2\Lambda\_0 + \rho + 4ja - \frac{(4j)^2 + 4j}{4}\delta} - \sum\_{j \in \mathbb{Z}} e^{2\Lambda\_0 + \rho + (-4j - 1)a - \frac{(-4j - 1)^2 + (-4j - 1)}{4}\delta}\right). \tag{39}$$

Similarly, we can evaluate *chL* (2Λ1) as follows:

$$\frac{1}{e^{\rho}R}q^{-\frac{1}{2}}\left(\sum\_{j\in\mathbb{Z}}e^{2\Lambda\_{0}+\rho+(4j+1)a-\frac{\left(4j+1\right)^{2}+\left(4j+1\right)}{4}\delta}-\sum\_{j\in\mathbb{Z}}e^{2\Lambda\_{0}+\rho+\left(-4j-2\right)a-\frac{\left(-4j-2\right)^{2}+\left(-4j-2\right)}{4}\delta}}\right).\tag{40}$$

Using (39), (40) and the Jacobi triple product identity, we have

$$\begin{split} & \quad \text{ch}\, \text{l} (2\lambda\_0) - q^{\frac{1}{2}} \text{ch}\, (2\lambda\_1) \\ & \quad = \frac{1}{\sigma^q R} \sum\_{j \in \mathbb{Z}} (-1)^j e^{2\lambda\_0 + p + j\alpha - \frac{\alpha}{4}j\epsilon} \\ & \quad = \frac{e^{2\lambda\_0}}{R} \sum\_{j \in \mathbb{Z}} (-1)^j e^{j\epsilon} q^{\frac{\beta^2 + j}{4}} \\ & \quad = \frac{e^{2\lambda\_0}}{R} \prod\_{n=1}^{\infty} \left( 1 - q^{\frac{n}{2}} \right) \left( 1 - e^{\epsilon} q^{\frac{n}{2}} \right) \left( 1 - e^{-\epsilon} q^{\frac{n-1}{2}} \right) \\ & \quad = \epsilon^{2\lambda\_0} \prod\_{n=1}^{\infty} \left( 1 - q^{\frac{2n-1}{2}} \right) \left( 1 - e^{\epsilon} q^{\frac{2n-1}{2}} \right) \left( 1 - e^{-\epsilon} q^{\frac{2n-1}{2}} \right) \\ & \quad = \epsilon^{2\lambda\_0} \prod\_{n=1}^{\infty} \frac{1 - q^{n-\frac{1}{2}}}{1 - q^n} \prod\_{n=1}^{\infty} \left( 1 - q^n \right) \left( 1 - e^{\epsilon} q^{\frac{2n-1}{2}} \right) \left( 1 - e^{-\epsilon} q^{\frac{2n-1}{2}} \right) \\ & \quad = \epsilon^{2\lambda\_0} \prod\_{n=1}^{\infty} \frac{1 - q^{n-\frac{1}{2}}}{1 - q^n} \sum\_{j \in \mathbb{Z}} (-1)^j e^{\epsilon} q^{\frac{2^j}{2}}. \end{split}$$

Recall from ([11], (12.7.1)) that

$$chL\left(2\Lambda\_0\right) = \sum\_{\lambda \in \max(2\Lambda\_0)} c\_{\lambda}^{2\Lambda\_0} c^{\lambda}.\tag{42}$$

and

$$chL\left(2\Lambda\_1\right) = \sum\_{\lambda \in \mathbf{max}(2\Lambda\_1)} c\_{\lambda}^{2\Lambda\_1} c^{\lambda}.\tag{43}$$

From (42) and (43), the coefficient of *e*2Λ0 in *chL* (2Λ0) − *q* 1 2 *chL* (2Λ1) should be equal to

$$c\_{2\Lambda\_0}^{2\Lambda\_0} - q^{\frac{1}{2}}c\_{2\Lambda\_0}^{2\Lambda\_1}.\tag{44}$$

*Axioms* **2019**, *8*, 82

Comparing (44) with the coefficient of *e*2Λ0 in (41), we obtain

$$c\_{2\Lambda\_0}^{2\Lambda\_0} - q^{\frac{1}{2}}c\_{2\Lambda\_0}^{2\Lambda\_1} = \prod\_{n=1}^{\infty} \frac{1 - q^{n - \frac{1}{2}}}{1 - q^n}. \tag{45}$$

By substituting *x* = *q* 1 2 , we obtain from (45)

$$c\_{2\Lambda\_0}^{2\Lambda\_0} - x c\_{2\Lambda\_0}^{2\Lambda\_1} = \prod\_{n=1}^{\infty} \frac{1 - x^{2n-1}}{1 - x^{2n}}.$$

By letting *x* −→ −*x*, we get

$$c\_{2\Lambda\_0}^{2\Lambda\_0} + xc\_{2\Lambda\_0}^{2\Lambda\_1} = \prod\_{n=1}^{\infty} \frac{1 + x^{2n-1}}{1 - x^{2n}}.$$

and this implies

$$c\_{2\Lambda\_0}^{2\Lambda\_0} + q^{\frac{1}{2}}c\_{2\Lambda\_0}^{2\Lambda\_1} = \prod\_{n=1}^{\infty} \frac{1 + q^{n - \frac{1}{2}}}{1 - q^n}. \tag{46}$$

On the other hand, it is easy to check that *m*2Λ0 − |2Λ0| 2 4 = − 1 16 and *m*2Λ1 − |2Λ0| 2 4 = 7 16 , and these yield that *c*" 2Λ0 2Λ0 = *q*− 1 16 *c* 2Λ0 2Λ0 and *c*" 2Λ1 2Λ0 = *q* 7 16 *c* 2Λ1 2Λ0 . So, (46) gives rise to

$$q^{\frac{1}{16}} \left( \hat{c}\_{2\Lambda\_0}^{2\Lambda\_0} + \hat{c}\_{2\Lambda\_0}^{2\Lambda\_1} \right) = \prod\_{n=1}^{\infty} \frac{1 + q^{n - \frac{1}{2}}}{1 - q^n}. \tag{47}$$

Thus,

$$\begin{array}{c} \frac{\eta\left(\tau\right)}{\eta\left(\frac{\tau}{2}\right)\eta\left(2\tau\right)}\\ = \frac{q^{-\frac{1}{16}}}{\prod\_{n=1}^{\infty}\left(1-q^{\frac{n}{2}}\right)\prod\_{n=1}^{\infty}\left(1+q^{n}\right)}\\ = \frac{q^{-\frac{1}{16}}\prod\_{n=1}^{\infty}\left(1+q^{\frac{n}{2}}\right)}{\prod\_{n=1}^{\infty}\left(1-q^{n}\right)\prod\_{n=1}^{\infty}\left(1+q^{n}\right)}\\ = \frac{q^{-\frac{1}{16}}\prod\_{n=1}^{\infty}\left(1+q^{n-\frac{1}{2}}\right)}{\prod\_{n=1}^{\infty}\left(1-q^{n}\right)}\\ = \frac{\hat{\varsigma}\_{2\lambda\_{0}}^{2\lambda\_{0}}+\hat{\varsigma}\_{2\lambda\_{0}}^{2\lambda\_{1}}\left(\text{see } (47)\right)}. \end{array}$$

Next, we compute *c*" *ρ ρ*.

Replacing all positive roots *α* by *kα* (*k* ∈ Z≥1), we obtain from the denominator identity that

$$e^{k\rho} \prod\_{\alpha \in \Lambda\_+} \left( 1 - e^{-k\alpha} \right)^{\text{mult}(\alpha)} = \sum\_{w \in W} \epsilon(w) e^{w(k\rho)}.$$

Thus, it follows from the Jacobi triple identity that

$$\begin{split} &h^{\dagger}(\rho) \\ &= \frac{1}{e^{\rho}\lambda} \sum\_{w \in W} \epsilon(w) e^{w(2\rho)} \\ &= \frac{1}{e^{\rho}\lambda} e^{2\rho} \prod\_{a \in A\_{+}} \left(1 - e^{-2a}\right)^{\text{mult}(a)} \\ &= e^{\rho} \prod\_{j=1}^{\infty} \frac{\left(1 - q^{2j}\right) \left(1 - e^{-2a} q^{2(j-1)}\right) \left(1 - e^{2a} q^{2j}\right)}{\left(1 - q^{j}\right) \left(1 - e^{-a} q^{(j-1)}\right) \left(1 - e^{a} q^{j}\right)} \left(q = e^{-\delta}\right) \\ &= e^{\rho} \prod\_{j=1}^{\infty} \frac{\left(1 + q^{j}\right)}{\left(1 - q^{j}\right)} \prod\_{j=1}^{\infty} \left(1 - q^{j}\right) \left(1 + e^{-a} q^{(j-1)}\right) \left(1 + e^{a} q^{j}\right) \\ &= \prod\_{j=1}^{\infty} \frac{\left(1 + q^{j}\right)}{\left(1 - q^{j}\right)} \sum\_{j \in \mathbb{Z}} e^{\rho - ja} q^{\frac{j^{2}}{2} - j} \\ &= \prod\_{j=1}^{\infty} \frac{\left(1 + q^{j}\right)}{\left(1 - q^{j}\right)} \sum\_{j \in \mathbb{Z}} e^{\rho - ja - \frac{j^{2}}{2} - j}. \end{split} \tag{48}$$

On the other hand, we get from ([11], (12.7.1)) that

$$\text{chL}\left(\rho\right) = \sum\_{\lambda \in \text{max}\left(\rho\right)} c\_{\lambda}^{\rho} e^{\lambda}. \tag{49}$$

Comparing the coefficients of *eρ* in (48) and (49), we have

$$c\_{\rho}^{\rho} = \prod\_{j=1}^{\infty} \frac{1 + q^j}{1 - q^j}.$$

Moreover, it is easy to check *mρ* − |*ρ*| 2 4 = 0 which implies *c*" *ρ ρ* = *c ρ ρ*. The result now follows.

### **6. Super-Virasoro algebras**

In this section, we shall investigate relationships between our results on branching functions and the representation theory of super-Virasoro algebras. As by-products, we generalize the tensor product decomposition formulas ([6], (4.1.2a) and (4.1.2b)) to the case of principal admissible weights.

Let us first review the theta functions associated to an affine Lie algebra g = C ' *t*, *t* −1 ( ⊗ g ⊕ C*K* ⊕ C*d* and its Cartan subalgebra h.

For *λ* ∈ *Pm* (*m* ∈ Z≥0), the theta function *θλ* is defined as

$$\theta\_{\lambda} = e^{-\frac{|\lambda|^2}{2m}\delta} \sum\_{\alpha \in \overline{Q}} e^{t\_{\alpha}\lambda} \rho$$

where *Q* is the root lattice of g. Using the coordinate (*τ*, *z*, *t*) for the Cartan subalgebra h, we get

$$\theta\_{\lambda} \left( \tau, z, t \right) = e^{2\pi i m t} \sum\_{\gamma \in \overline{\mathbb{Q}} + \frac{\mathsf{T}}{m}} q^{\frac{m}{2} \left| \gamma \right|^2} e^{2\pi i m \left( \gamma \left| z \right| \right)\_{\rho}}$$

where *λ* is the projection of *λ* onto h.

*Axioms* **2019**, *8*, 82

In particular, if we take *λ* = *md* + 1 2*nα* + *rK* ∈ *Pm* for g = sl2 then the corresponding theta function is

$$\theta\_{\lambda} \left( \tau, z, t \right) = e^{2\pi i m t} \sum\_{k \in \mathbb{Z} + \frac{n}{m}} q^{mk^2} e^{2\pi i m \left( ka \middle| z \right)}. \tag{50}$$

Evaluating (50) at (*τ*, 0, 0), we have

$$\theta\left(\tau,0,0\right) = \sum\_{j\in\mathbb{Z}} q^{\mathfrak{m}\left(j+\frac{\mu}{2\pi}\right)^2} \left(q = e^{2\pi i\tau}\right). \tag{51}$$

For convenience, we shall simply write *θn*,*m* for (51) in the remaining part of this section.

Next, we review the super-Virasoro algebras *Vir-* - *-* = 0, 1 2 . (For *-* = 0 or 1 2 , *Vir* is called the *Ramond* and *Neveu-Schwarz* superalgebra, respectively.)

The super-Virasoro algebra *Vir* is the complex superalgebra with a basis *c*, *j*, *gm*|*j* ∈ Z and *m* ∈ *-* + Z , and it satisfies commutation relations

where { , } denotes an anti-commutator bracket between two odd elements.

Recall that every minimal series irreducible module of *Vir* corresponds to the pair of numbers *z* - *p*,*p* , *h* - *p*,*p r*,*s*;*-* . Here, *z* - *p*,*p* is the central charge equals *z* - *p*,*p* = 3 2 1 − 2 - *p*−*p* 2 *pp* , and *h* - *p*,*p r*,*s*;*-* is the minimal eigenvalue of -0 equals *h* - *p*,*p r*,*s*;*-* = - *pr*−*p s* 2 − - *p*−*p* 2 8*pp* + 1 16 (1 − 2*-*) for *p*, *p* ,*r*,*s* ∈ Z, 2 ≤ *p* < *p*, *p* − *p* ∈ 2Z, *gcd p*−*p* 2 , *p* = 1, 1 ≤ *r* ≤ *p* − 1, 1 ≤ *s* ≤ *p* − 1 and *r* − *s* ∈ 2Z (we refer to ([16], Theorem 5.2) for the details).

Write *V- z* - *p*,*p* , *h p*,*p r*,*s*;*-* for the minimal series module over *Vir* corresponding to *z* - *p*,*p* , *h* - *p*,*p r*,*s*;*-* . According to [17,18], it follows that

$$chV\_{\mathfrak{c}}\left(z^{\left(p,p'\right)},h^{\left(p,p'\right)}\_{r,s;x}\right) = q^{\frac{1}{2}z^{\left(p,p'\right)}}\eta\_{\mathfrak{c}}\left(\tau\right)\left(\theta\_{\frac{pr-p's}{2},\frac{pp'}{2}} - \theta\_{\frac{pr+p's}{2},\frac{pp'}{2}}\right)\tau$$

where *η-* (*τ*) = ⎧ ⎨ ⎩ *η*(2*τ*) *η*(*τ*) 2 if *-* = 0 *η*(*τ*) *η*( *τ* 2 )*η*(2*τ*) if *-* = 1 2 .

By (51), we see that

$$\begin{array}{rcl} \theta\_{\frac{pr-p's}{2},\frac{pp'}{2}} & = & \sum\_{j\in\mathbb{Z}} q^{\frac{1}{8pp'} \left(2pp'j+pr-p's\right)^2} \\ \theta\_{\frac{pr+p's}{2},\frac{pp'}{2}} & = & \sum\_{j\in\mathbb{Z}} q^{\frac{1}{8pp'} \left(2pp'j+pr+p's\right)^2} \end{array}$$

*Axioms* **2019**, *8*, 82

So, the normalized character of *V- z* - *p*,*p* , *h* - *p*,*p r*,*s*;*-* is

$$\begin{aligned} &\chi\_{r,s;x}^{\left(p,p'\right)}\left(\tau\right) \\ &=\quad \eta\_{\mathcal{E}}\left(\tau\right)\left(\sum\_{j\in\mathbb{Z}}q^{\frac{1}{8pp'}\left(2pp'j+pr-p's\right)^{2}}-\sum\_{j\in\mathbb{Z}}q^{\frac{1}{8pp'}\left(2pp'j+pr+p's\right)^{2}}\right),\end{aligned} \tag{52}$$

where *χ* - *p*,*p r*,*s*;*-* (*τ*) = *q*− 1 24 *z p*,*p chV- z* - *p*,*p* , *h* - *p*,*p r*,*s*;*-* .

Let *r* = −(*n* + 1) and *s* = *n* + 1 in (52). Then, by Theorem 4, Theorem 5 and (52), we obtain the following result.

**Proposition 1.** *Let m* = *v u* (*u* ∈ 2Z≥1, *v* ∈ 2Z + 1)*. Suppose that λ is a principal admissible weight of* sl\*2 *such that λ* = *λ*0 − (*u* − 1)(*m* + 2)Λ0 ∈ *Pm u*,1 *for λ*0 ∈ *Pu*(*m*+2)−2 + *. Then, the branching function b* 2Λ0⊗*λ ν* (*τ*) + *b* 2Λ1⊗*λ ν* (*τ*) *(resp. b ρ ν* (*τ*)*) of* (*L* (2Λ0) ⊕ *L* (2Λ1)) ⊗ *L* (*λ*) *(resp. L* (*ρ*) ⊗ *L* (*λ*)*) is the same as the normalized character χ* - *p*,*p* −(*n*+1),*n* +1; 1 2 (*τ*) *(resp. χ* - *p*,*p* −(*n*+1),*n* +1;0 (*τ*)*) of the Neveu-Schwarz (resp. Ramond) superalgebra.*

It follows from Section 4 that

$$L\left(L\left(2\Lambda\_0\right)\oplus L\left(2\Lambda\_1\right)\right)\otimes L\left(\lambda\right) = \sum\_{\nu} \left(V\_{\nu}^{2\Lambda\_0\lambda} \oplus V\_{\nu}^{2\Lambda\_1\lambda}\right)\otimes L\left(\nu\right) \tag{53}$$

and

$$L\left(\rho\right) \otimes L\left(\lambda\right) = \sum\_{\nu'} V\_{\nu'}^{\rho,\lambda} \otimes L\left(\nu'\right),\tag{54}$$

where *ν* and *ν* are taken over *Pm*+2 *u*,1 such that *ν* ≡ 2Λ0 + *λ* mod *Q* and *ν* ≡ *ρ* + *λ* mod *Q*, respectively.

According to [17] the coset Virasoro action introduced in Section 4 can be extended to the action of super-Virasoro algebras, and (53) and (54) can be considered as decompositions of *V-* ⊕ [g, g]-module. Thus, (14) and Proposition 1 imply that *V*2Λ0 *ν* ⊕ *V*2Λ1 *ν* (resp. *Vρ ν* ) should be isomorphic to the minimal series module *V*1 2 *zp*,*p* , *hp*,*p* −(*n*+1),*n* +1; 1 2 (resp. *V*0 *zp*,*p* , *hp*,*p* −(*n*+1),*n* +1;0 ) as *Vir* 1 2 -modules (resp. *Vir*0-modules). Hence, we obtain the following theorem.

**Theorem 6.** *Let m and λ be the same as Proposition 1. Then, we have*

$$L\left(L\left(2\Lambda\_0\right)\oplus L\left(2\Lambda\_1\right)\right)\otimes L\left(\lambda\right) = \sum\_{\nu} V\_{\frac{1}{2}}\left(z^{p,p'}, h^{p,p'}\_{-\left(n+1\right),n'+1;\frac{1}{2}}\right)\otimes L\left(\nu\right)$$

*and*

$$L\left(\rho\right) \otimes L\left(\lambda\right) = \sum\_{\nu'} V\_0\left(z^{p,p'}, h^{p,p'}\_{-(n+1),n'+1;0}\right) \otimes L\left(\nu'\right),$$

*where ν and ν are taken over Pm*+2 *u*,1 *such that ν* ≡ 2Λ0 + *λ mod Q and ν* ≡ *ρ* + *λ mod Q, respectively.*

**Funding:** This research was supported by the Daegu University Research Grant, 2016.

**Conflicts of Interest:** The author declares no conflict of interest.

### **References**

c 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Cohomology Theory of Nonassociative Algebras with Metagroup Relations**

### **Sergey V. Ludkowski**

Department of Applied Mathematics, MIREA—Russian Technological University, av. Vernadsky 78, 119454 Moscow, Russia<EMAIL_ADDRESS>
Received: 30 April 2019; Accepted: 24 June 2019; Published: 4 July 2019

**Abstract:** Nonassociative algebras with metagroup relations and their modules are studied. Their cohomology theory is scrutinized. Extensions and cleftings of these algebras are studied. Broad families of such algebras and their acyclic complexes are described. For this purpose, different types of products of metagroups are investigated. Necessary structural properties of metagroups are studied. Examples are given. It is shown that a class of nonassociative algebras with metagroup relations contains a subclass of generalized Cayley–Dickson algebras.

**Keywords:** nonassociative algebra; cohomology; extension; metagroup

**MSC:** 16E40; 16D70; 18G60; 17A60; 03C60; 03C90

### **1. Introduction**

Nonassociative algebras comprise a large area of algebra. Among them, Lie algebras and their modifications are widely used in different branches of mathematics and its applications including PDEs, physics, quantum mechanics, informatics, and biology (see, for example, [1–5] and references therein). There are other classes of nonassociative algebras which are less investigated. For example, octonions and generalized Cayley–Dickson algebras play very important roles in mathematics and quantum field theory [6–11]. Their structures and identities have attracted great attention. They are used not only in algebra and noncommutative geometry, but also in noncommutative analysis and PDEs, particle physics, mathematical physics, in the theory of Lie groups and algebras and their generalizations, mathematical analysis, operator theory, and in applications in natural sciences including physics and quantum field theory (see [2,7–9,12–25] and references therein).

A multiplicative law of their canonical generators is nonassociative and leads to a more general notion of a metagroup instead of a group [26]. The preposition meta is used to emphasize that such an algebraic object has milder properties than a group. Their axiomatic metagroups satisfy Conditions (1)–(3) with the weak relation (9), as shown in Definition 1 in Section 2. They were used in [26] to investigate automorphisms and derivations of nonassociative algebras.

An extensive area of investigation of PDEs intersects with cohomologies and deformed cohomologies [27]. Therefore, it is important to develop this area using octonions, Cayley–Dickson algebras, and more general metagroup algebras.

It appears that generators of Cayley–Dickson algebras form objects, which are nonassociative generalizations of groups. They are called metagroups. This means that metagroup algebras include the Cayley–Dickson algebras . This article is devoted to algebras generated by metagroups. Note that a class of metagroups differs substantially from a class of groups. Indeed, a metagroup may be nonassociative, power non-associative, or nonalternative. Moreover, left or right inverse elements in the metagroup may not exist or it may contain elements for which left and right inverse elements do not coincide (see Definition 1 in Section 2).

On the other hand, algebras are frequently studied using cohomology theory. However, the already developed cohomology theory operates with associative algebras. It has been investigated by Hochschild and other authors [1,28–30], but it is not applicable to nonassociative algebras. In some particular cases of nonassociative algebras, such as Lie algebras, pre-Lie algebras, flexible algebras, and alternative algebras, homology theory was developed for the needs of studies of their structures (see, for example, [2,31–33] and references therein). It is necessary to note that classes of these algebras are quite different from classes of generalized Cayley–Dickson algebras and nonassociative algebras with metagroup relations. This work is devoted to the development of cohomology theory for nonassociative algebras, namely for its subclass of algebras with metagroup relations.

Previously, cohomologies of loop spaces on quaternion and octonion manifolds were studied in [17]. They have specific features in comparison with the case of complex manifolds. This is especially caused by the noncommutativity of the quaternion skew field and the nonassociativity of the octonion algebra.

In this article, nonassociative algebras with metagroup relations are studied. Their modules and acyclic complexes are investigated. Their cohomology theory is scrutinized in Section 2. This requires the development of a specific axiomatic model of such algebras and their modules. Necessary structural properties of metagroups are studied in Lemmas 1 and 2. Acyclic complexes and co-chain complexes are described in Proposition 1 and Theorem 1. A relation of the cohomologies with quotient modules is given by Theorem 2. Extensions and cleftings of these algebras are studied in Theorems 3–5 under the framework of cohomology theory. Broad families of such algebras are described. In Theorem 6, inner derivations of nonassociative algebras are investigated. A semisimplicity of nonassociative algebras is investigated in Theorem 7 and Corollary 1.

Different types of products of metagroups are investigated in Theorems 8 and 9 in Section 3. Examples are given. It is shown that a class of nonassociative algebras with metagroup relations contains a subclass of generalized Cayley–Dickson algebras.

All of the key results of this paper are obtained for the first time. They can be used for further studies of nonassociative algebras cohomologies, the structure of nonassociative algebras, operator theory, and the spectral theory of Cayley–Dickson algebras, PDEs, noncommutative analysis, noncommutative geometry, mathematical physics, and their applications in the sciences.

### **2. Cohomology Theory of Nonassociative Algebras**

To avoid misunderstandings we give the necessary definitions.

**Definition 1.** *Let G be a set with a single-valued binary operation (multiplication) G*2 (*a*, *b*) → *ab* ∈ *G, where G satisfies the following conditions:*

*The set of all elements h* ∈ *G commuting and associating with G is*

C(*G*) := *Com*(*G*) ∩ *N*(*G*) *is called the center* C(*G*) *of G.*

*We call G a metagroup if a set G possesses a single-valued binary operation and satisfies Conditions (1)–(3) and*

*(9)* (*ab*)*c* = t3(*a*, *b*, *c*)*a*(*bc*) *for each a, b, and c in G, where* t3(*a*, *b*, *c*) ∈ **Ψ***,* **Ψ** ⊂ C(*G*)*; where* t3 *shortens a notation* t3,*G, where* **Ψ** *denotes a (proper or improper) subgroup of* C(*G*)*.*

*Then G will be called a central metagroup if, in addition to* (9)*, it satisfies the condition*

*(10) ab* = t2(*a*, *b*)*ba for each a and b in G, where* t2(*a*, *b*) ∈ **Ψ***.*

> Particularly, *Invl*(*a*) = *Divl*(*a*,*e*) is a left inversion, and *Invr*(*a*) = *Divr*(*a*,*e*) is a right inversion. In view of the nonassociativity of *G*, in general, a product of several elements of *G* is usually

specified by opening "(" and closing ")" parentheses. We denote the product of elements *a*1,...,*an* in *G* by {*a*1, ..., *an*}*q*(*n*), where a vector *q*(*n*) indicates an order of pairwise multiplications of elements in the row *a*1, ..., *an* in braces in the following manner. The enumerate positions are as follows: before *a*1 by 1, between *a*1 and *a*2 by 2,..., by *n* between *an*−1 and *an*, and by *n* + 1 after *an*. Then, we put *qj*(*n*)=(*k*, *m*) if there are *k* opening "(" and *m* closing ")" parentheses in the ordered product at the *j*-th position of the type )...)(...(, where *k* and *m* are nonnegative integers, *q*(*n*)=(*q*1(*n*), ...., *qn*+1(*n*)) with *q*1(*n*)=(*k*, 0) and *qn*+1(*n*)=(0, *m*).

Traditionally, *Sn* denotes the symmetric group of the set {1, 2, ..., *n*}. Henceforth, maps and functions on metagroups are assumed to be single-valued unless otherwise specified .

Let *ψ* : *G* → *G* be a bijective surjective map satisfying the following condition: *ψ*(*ab*) = *ψ*(*a*)*ψ*(*b*) for each *a* and *b* in *G*. Then, *ψ* is called an automorphism of the metagroup *G*.

**Lemma 1.** (*i*)*. Let G be a central metagroup. Then, for every a*1*,...,an in G, v* ∈ *Sn and vectors q*(*n*) *and u*(*n*) *indicating an order of pairwise multiplications and n* ∈ **N***, there exists an element tn* = *tn*(*a*1, ..., *an*; *q*(*n*), *u*(*n*)|*v*) ∈ **Ψ** *such that*

$$\iota(1) \quad \{a\_1, \ldots, a\_n\}\_{\eta(n)} = t\_n \{a\_{\upsilon(1)}, \ldots, a\_{\upsilon(n)}\}\_{\mathfrak{u}(n)}\cdots$$

(*ii*)*. If G is a metagroup and if v is the neutral element v* = *id in Sn, then property* (1) *is satisfied.*

**Proof.** From Conditions (1)–(8) in Definition 1, it follows that C(*G*) itself is a commutative group.

(*i*). For *n* = 1, evidently *t*1 = 1, since *a* = 1*a* for each *a* ∈ *G*. For *n* = 2, Formula (1) is a direct consequence of condition (10) in Definition 1. Consider *n* = 3. When *u* is the identity element of *S*3, the statement follows from condition (9) in Definition 1. For any transposition *u* of two elements of the set {1, 2, 3}, the statement follows from (9) and (10) in Definition 1. Elements of *S*3 can be obtained by multiplication of pairwise transpositions. Therefore, from the condition **Ψ** ⊂ C(*G*), it follows that formula (1) is valid.

Now, let *n* ≥ 4 and suppose that this lemma is proved for any products consisting of less than *n* elements. In view of Properties (1) and (2) in Definition 1, it is sufficient to verify Formula (1) of this lemma for {*a*1, ..., *an*}*q*(*n*) = (...((*a*1*a*2)*a*3)...)*an* =: {*a*1, ..., *an*}*l*(*n*) since **Ψ** ⊂ C(*G*). In this particular case, {*av*(1), ..., *av*(*n*)}*u*(*n*) = {*av*(1), ..., *av*(*n*−1)}*u*(*n*−1)*an*. Formula (1) follows from the induction hypothesis, since (...((*a*1*a*2)*a*3)...)*an*−1 = *tn*−1{*av*(1), ..., *av*(*n*−1)}*u*(*n*−1) and hence ((...((*a*1*a*2)*a*3)...)*an*−1)*an* = *tn*−1({*av*(1), ..., *av*(*n*−1)}*u*(*n*−1)*an*) and putting *tn* = *tn*−1, where *tn*−1 = *tn*−1(*a*1, ..., *an*−1; *q*(*n* − 1), *u*(*n* − 1)|*w*) with *w* = *v*|{1,...,*n*−1}, *v*(*n*) = *n*.

In the general case, {*av*(1), ..., *av*(*n*)}*u*(*n*) = {*b*1, ..., *bj*, ..., *bk*}*p*(*k*), where *j* is such that either *bj* = *cjan* with *cj* = {*av*(*j*), ..., *av*(*j*+*m*−1)}*r*(*m*) and with *v*(*j* + *m*) = *n* or *bj* = *ancj* with *cj* = {*av*(*j*+1), ..., *av*(*j*+*m*)}*r*(*m*) and with *v*(*j*) = *n*. Also, *b*1 = *av*(1),...,*bj*−1 = *av*(*j*−1), *bj*+1 = *av*(*j*+1),..., *bk* = *av*(*n*) with suitable vectors *p*(*k*) and *r*(*m*). If *m* > 1, then *k* < *n* and using the induction hypothesis for {*b*1, ..., *bj*, ..., *bk*}*p*(*k*) and *bj*, we get that elements *s* and *t* in **Ψ** exist so that {*b*1, ..., *bj*, ..., *bk*}*p*(*k*) = *s*{*b*1, ..., *bj*−1, *bj*+1, , ..., *bk*}*p*(*k*−1)*bj* = *st*({*b*1, ..., *bj*−1, *bj*+1, , ..., *bk*}*p*(*k*−1)*cj*)*an*, where *p*(*k* − 1) is a corresponding vector prescribing an order of multiplications.

Again, applying the induction hypothesis to the product of *n* − 1 elements {*b*1, ..., *bj*−1, *bj*+1, , ..., *bk*}*p*(*k*−1)*cj*, we deduce that there exists *w* ∈ **Ψ**, such that

{*av*(1), ..., *av*(*n*)}*u*(*n*) = *stw*((...((*a*1*a*2)*a*3)...)*an*−1)*an*.

Therefore, a case remains when *m* = 1. Let the first multiplication in {*av*(1), ..., *av*(*n*)}*u*(*n*) containing *an* be (*av*(*k*)*av*(*k*+1)) =: *bk*. We put *bj* = *av*(*j*) for each 1 ≤ *j* ≤ *k* − 1. We also put *bj*−1 = *av*(*j*) for each *k* + 1 < *j* ≤ *n*, where either *an* = *av*(*k*) or *an* = *av*(*k*+1). Therefore, using previous identities, we rewrite the considered product as {*av*(1), ..., *av*(*n*)}*u*(*n*) = {*by*(1), ..., *by*(*n*−1)}*w*(*n*−1) with an element *y* ∈ *Sn*−1 of the symmetric group and a vector *w*(*n* − 1), indicating an order of pairwise products (see Definition 1). From the induction hypothesis, we deduce that there exists *tn*−1 ∈ **Ψ**, so that *tn*−1{*by*(1), ..., *by*(*n*−1)}*w*(*n*−1) = *pbk* with *p* = {*b*1, ..., *bk*−1, *bk*+1, ..., *bn*−1}*w*(*n*−1), because *G* is the central metagroup. Applying the induction hypothesis for *n* = 3, we infer that *t*3 ∈ **Ψ** exists, such that *tn*−1*t*3{*by*(1), ..., *by*(*n*−1)}*w*(*n*−1) = (*pa*)*an*, where either *a* = *av*(*k*+1) or *a* = *av*(*k*), correspondingly. From the induction hypothesis for *n* − 1, it follows that ˜*tn*−1 ∈ **Ψ** exists, so that ˜*tn*−1 *pa* = (...((*a*1*a*2)*a*3)...)*an*−1, and hence, {*a*1, ..., *an*}*l*(*n*) = *tn*{*av*(1), ..., *av*(*n*)}*u*(*n*), where *tn* = ˜*tn*−1*tn*−1*t*3.

(*ii*). Now, let *G* be a metagroup and *v* = *id* be the neutral element of the symmetric group *Sn*, where *id*(*k*) = *k* for each *k* ∈ **N**. Then, using condition (10) of Definition 1 is unnecessary, because transpositions are already not utilized. For *n* = 1 and *n* = 2, we get *t*1 = 1 and *t*2 = 1, since *a* = 1*a* and *ab* = 1*ab* for each *a* and *b* in *G*. For *n* = 3, Formula (1) of this lemma follows on from condition (9) in Definition 1. Then, the proof in case (*ii*) by induction is a simplification of that of case (*i*).

**Lemma 2.** *If G is a metagroup, then for each a and b* ∈ *G, the following identities are fulfilled:*

**Proof.** Conditions (1)–(3) in Definition 1 imply that

for each *a* and *b* in *G*. Using Condition (9) in Definition 1 and Identities (4) and (5), we deduce that *e*/*b* = (*e*/*b*)(*b*(*b* \ *e*)) = (*b* \ *e*)/t3(*e*/*b*, *b*, *b* \ *e*) which leads to (1).

Let *c* = *a* \ *b*. Then, from Identities (1) and (4), it follows that (*a* \ *e*)*b* = (*e*/*a*)t3(*e*/*a*, *a*, *a* \ *e*)(*ac*) = ((*e*/*a*)*a*)(*a* \ *b*)t3(*e*/*a*, *a*, *a* \ *e*)/t3(*e*/*a*, *a*, *a* \ *b*) which provides (2).

Now, let *d* = *b*/*a*. Then, Identities (1) and (5) imply that *b*(*e*/*a*)=(*da*)(*a* \ *e*)/t3(*e*/*a*, *a*, *a* \ *e*) = (*b*/*a*)t3(*b*/*a*, *a*, *a* \ *e*)/t3(*e*/*a*, *a*, *a* \ *e*) which demonstrates (3).

**Definition 2.** *Let A be an algebra over an associative unital ring* T *, such that A has a natural structure of a* (T , T )*-bimodule with a multiplication map A* × *A* → *A, which is right and left distributive, a*(*b* + *c*) = *ab* + *ac,* (*b* + *c*)*a* = *ba* + *ca, and also satisfies the identities r*(*ab*)=(*ra*)*b,* (*ar*)*b* = *a*(*rb*)*,* (*ab*)*r* = *a*(*br*)*, s*(*ra*)=(*sr*)*a, and* (*ar*)*s* = *a*(*rs*) *for any a, b, and c in A, r and s in* T *. Let G be a metagroup and* T *be an associative unital ring.*

*Henceforth, the ring* T *is assumed to be commutative, unless otherwise specified.*

*Then, by* T [*G*]*, a metagroup algebra is denoted over* T *for all formal sums s*1*a*1 + ... + *snan satisfying Conditions* (1 − 3) *below, where n is a positive integer, s*1*,...,sn are in* T *, and a*1*,...,an belong to G:*

**Note 1.** *Let M be an additive commutative group such that M is a two-sided G-module, where G is a metagroup. We remind the reader that this means that automorphisms p*(*g*) *and s*(*g*) *of M correspond to each g* ∈ *G. For short, we use gx* = *p*(*g*)*x and xg* = *xs*(*g*) *for each g* ∈ *G.*

*Note that, usually, M has a natural structure of a two-sided* **Z***-module, because M is the additive commutative group, where* **Z** *denotes the ring of all integers. Therefore, M is a two-sided G-module if and only if it is a two-sided* **Z**[*G*]*-module according to the formulas* (∑*g*∈*G n*(*g*)*g*)*x* = ∑*g*∈*G n*(*g*)(*gx*) *and x*(∑*g*∈*G n*(*g*)*g*) = ∑*g*∈*G*(*xg*)*n*(*g*)*, where n*(*g*) ∈ **Z** *for each g* ∈ *G.*

*One can consider the additive group of integers* **Z** *as the trivial two-sided G-module putting gn* = *ng* = *n for each g* ∈ *G and n* ∈ **Z***, where G is a metagroup.*

**Example 1.** *I. Recall the following: Let A be a unital algebra over a commutative associative unital ring F supplied with a scalar involution a* → *a*¯ *so that its norm N and trace T maps have values in F and fulfill conditions:*

*for each a and b in A.*

*If a scalar f* ∈ *F satisfies the condition* ∀*a* ∈ *A fa* = 0 ⇒ *a* = 0*, then such element f is called cancelable. For a cancelable scalar f , the Cayley–Dickson doubling procedure provides new algebra C*(*A*, *f*) *over F such that*

*for each a and b in A. Then, l is called a doubling generator. From the definitions of T and N, it follows that* ∀*a* ∈ *A*, ∀*b* ∈ *A T*(*a*) = *T*(*a* + *bl*) *and N*(*a* + *bl*) = *N*(*a*) + *f N*(*b*)*. The algebra A is embedded into C*(*A*, *f*) *as A a* → (*a*, 0)*, where* (*a*, *b*) = *a* + *bl. This is put by induction An*(*f*(*n*)) = *C*(*An*−1, *fn*)*, where A*0 = *A, f*1 = *f , n* = 1, 2, ...*, f*(*n*) = (*f*1, ..., *fn*)*. Then, An*(*f*(*n*)) *are generalized Cayley–Dickson algebras when F is not a field or Cayley–Dickson algebras when F is a field.*

It is natural to put *A*∞(*f*) := %∞ *n*=1 *An*(*f*(*n*)), where *f* = (*fn* : *n* ∈ **N**). If *char*(*F*) = 2, let *Im*(*z*) = *z* − *T*(*z*)/2 be the imaginary part of a Cayley–Dickson number *z*, and hence *N*(*a*) := *N*2(*a*, *a*¯)/2, where *N*2(*a*, *b*) := *T*(*a*¯ *b*).

If the doubling procedure starts from *A* = *F*1 =: *A*0, then *A*1 = *C*(*A*, *f*1) is a ∗-extension of *F*. If *A*1 has a basis {1, *u*} over *F* with the multiplication table *u*2 = *u* + *w*, where *w* ∈ *F* and 4*w* + 1 = 0 with the involution 1¯ = 1, *u*¯ = 1 − *u*, then *A*2 is the generalized quaternion algebra, and *A*3 is the generalized octonion (Cayley–Dickson) algebra.

When *F* = **R** and *fn* = 1, each *n* by A*r* will denote the real Cayley–Dickson algebra with generators *i*0, ..., *i*2*r*−1, such that *i*0 = 1, *i* 2 *j* = −1 for each *j* ≥ 1, *ijik* = −*ikij* for each *j* = *k* ≥ 1. Note that the Cayley–Dickson algebra A*r* for each *r* ≥ 3 is nonassociative, for example, (*i*1*i*2)*i*4 = −*i*1(*i*2*i*4), etc. Moreover, for each *r* ≥ 4, the Cayley–Dickson algebra A*r* is nonalternative (see [7–9]). Frequently, *a*¯ is also denoted by *a*∗ or *a*˜. Then, *Gr* = {*ij*, − *ij* : *j* = 0, 1, ..., 2*r* − 1} is a finite metagroup for each 3 ≤ *r* < ∞.

Let *An* be a Cayley–Dickson algebra over a commutative associative unital ring R that is characteristically different from two, such that *A*0 = R, *n* ≥ 2. Take its basic generators *i*0, *i*1, ..., *i*2*n*−1, where *i*0 = 1. Choose **Ψ** as a multiplicative subgroup contained in the ring R, such that *fj* ∈ **Ψ** for each *j* = 0, ..., *n*. Put *Gn* = {*i*0, *i*1, ..., *i*2*n*−1} × **Ψ**. Then, *Gn* is a central metagroup.

II. More generally, let *H* be a group such that **Ψ** ⊂ *H* with relations *hik* = *ikh* and (*hg*)*ik* = *h*(*gik*) for each *k* = 0, 1, ..., 2*n* − 1 and each *h* and *g* in *H*. Then, *Gn* = {*i*0, *i*1, ..., *i*2*n*−1} × *H* is also a metagroup. If the group *H* is noncommutative, then the latter metagroup can be noncentral (see Condition (10) in

Definition 1). Using the notation of Example 1. I, we get that the Cayley–Dickson algebra A∞ over the real field **R** with *fn* = 1 for each *n* provides an example of a metagroup *G*∞ = {*ij*, − *ij* : 0 ≤ *j* ∈ **Z**}, where **Z** denotes the ring of integers.

III. Generally metagroups need not be central. From given metagroups, new metagroups can be constructed using their direct or semidirect products. Certainly, each group is a metagroup also. Therefore, there are abundant families of noncentral metagroups and also of central metagroups different from groups.

In another way, smashed products of groups and of metagroups can be considered by providing other examples of metagroups (for more detail, see Section 3).

**Definition 3.** *Let* R *be a ring, which may be nonassociative relative to the multiplication. If the mapping* R × *M* → *M,* R × *M* (*a*, *m*) → *am* ∈ *M exists, such that a*(*m* + *k*) = *am* + *ak and* (*a* + *b*)*m* = *am* + *bm for each a and b in* R*, m and k in M, then M will be called a generalized left* R*-module or, for short, a left* R*-module or left module over* R*.*

If R is a unital ring and 1*m* = *m* for each *m* ∈ *M*, then *M* is called a left unital module over R, where 1 denotes the unit element in the ring R. Symmetrically, a right R-module is defined.

If *M* is a left and right R-module, then it is called a two-sided R-module or a (R, R)-bimodule. If *M* is a left R-module and a right S-module, then it is called a (R, S)-bimodule.

A two-sided module *M* over R is called cyclic if an element *y* ∈ *M* exists such that *M* = R(*y*R) and *M* = (R*y*)R, where R(*y*R) = {*s*(*yp*) : *s*, *p* ∈ R} and (R*y*)R = {(*sy*)*p* : *s*, *p* ∈ R}.

Let *G* be a metagroup. Take a metagroup algebra *A* = T [*G*] and a two-sided *A*-module *M*, where T is an associative unital ring (see Definition 2). Let *Mg* be a two-sided T -module for each *g* ∈ *G*, where *G* is the metagroup. Let *M* have the decomposition *M* = ∑*g*∈*G Mg* as a two-sided T -module. Let *M* also satisfy the following conditions:

(1) *hMg* = *Mhg* and *Mgh* = *Mgh*,

(2) (*bh*)*xg* = *b*(*hxg*) and *xg*(*bh*)=(*xgh*)*b* and *bxg* = *xgb*,

(3) (*hs*)*xg* = t3(*h*,*s*, *g*)*h*(*sxg*) and (*hxg*)*s* = t3(*h*, *g*,*s*)*h*(*xgs*) and (*xgh*)*s* = t3(*g*, *h*,*s*)*xg*(*hs*)

for every *h*, *g*,*s* in *G* and *b* ∈ T and *xg* ∈ *Mg*. Then, a two-sided *A*-module *M* satisfying conditions (1)–(3) is called smashly *G*-graded. For short, it is also called "*G*-graded" instead of "smashly *G*-graded". In particular, if the module *M* is *G*-graded and splits into a direct sum *M* = & *g*∈*G Mg* of two-sided T -submodules *Mg*, then we say that that *M* is directly *G*-graded. For a nontrivial (nonzero) *G*-graded module *X* with the nontrivial metagroup *G*, it is supposed that *g* ∈ *G* exists such that *Xg* = *Xe* if something else is not outlined.

Similarly, *G*-graded left and right *A*-modules are defined. Henceforward, speaking about *A*-modules (left, right, or two-sided), it is supposed that they are *G*-graded and, for short, "an *A*-module" is written instead of "a *G*-graded *A*-module", unless otherwise specified.

If *P* and *N* are left *A*-modules and a homomorphism *γ* : *P* → *N* is such that *γ*(*ax*) = *aγ*(*x*) for each *a* ∈ *A* and *x* ∈ *P*, then *γ* is called a left *A*-homomorphism. Analogously, right *A*-homomorphisms are defined for right *A*-modules. For two-sided *A* modules, a left and right *A*-homomorphism is called an *A*-homomorphism.

For left T -modules *M* and *N* by *Hom*T (*M*, *N*), a family of all left T -homomorphisms is defined from *M* into *N*. A similar notation is used for a family of all T -homomorphisms (or right T -homomorphisms) of two-sided T -modules (or right T -modules correspondingly). If an algebra *A* is specified, a homomorphism may be written for short, instead of an *A*-homomorphism.