††institutetext: Department of Physics and Astronomy, Rutgers University ,
126 Frelinghuysen Rd., Piscataway NJ 08855, USA

# Dyson’s Classification And Real Division Superalgebras

Roman Geiko and Gregory W. Moore<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

It is well-known that unitary irreducible representations of groups can be
usefully classified in a 3-fold classification scheme: Real, Complex,
Quaternionic. In 1962 Freeman Dyson pointed out that there is an analogous
10-fold classification of irreducible representations of groups involving both
unitary and antiunitary operators. More recently, it was realized that there
is also a 10-fold classification scheme involving superdivision algebras. Here
we give a careful proof of the equivalence of these two 10-fold ways.

###### Keywords:

antiunitary symmetries, time-reversal

††arxiv: 2010.01675

##  1 Introduction And Conclusion

A fundamental theorem of E. Wigner Wigner1931 ; Bargmann1964 ; Freed:2011 ;
Weinberg:1995 relates the theory of symmetry in quantum mechanical systems to
group representation theory. Complex linear representations can be
characterized as real, complex, and quaternionic (a.k.a. pseudoreal). These
distinctions can play an important role in physical applications of
representation theory. One example is Kramers’ theorem: From a mathematical
viewpoint, it is simply the observation that the complex dimension of a
quaternionic representation is even. A second example arises when studying
index theory and the closely related subject of anomalies in quantum field
theories with fermions. In this case, the nature of the fermionic
representation of the gauge group plays an important role. Wigner’s theorem
actually allows for symmetry transformations which are both unitary and
antiunitary. In 1962, when studying symmetry properties of Hamiltonians in
nuclear physics F. Dyson pointed out that the three-fold classification of
complex linear representations should be generalized to a ten-fold
classification of representations that contain both unitary and antiunitary
symmetry transformations Dyson_1962 .

In more recent times, several other ten-fold classifications of aspects of
symmetry in quantum mechanics have appeared. One of these, described in
Fidkowski:2009dba ; Freed:2013 , is based on the 10-fold classification of
real superalgebras established by C.T.C. Wall Wall1964 . It was suggested in
Freed:2013  that the Dyson classification and the Wall classification are
“essentially the same,” although the precise relation between the two schemes
was imperfectly understood. It is the purpose of this paper to fill in this
gap and clarify the relation. We will explain, rather thoroughly and
carefully, that they are, indeed, essentially the same. We must stop and savor
the delicious irony that the English physicist Dyson and the English
topologist Wall were establishing their results at the same time, but the
connection between their work only became apparent half a century later.

We should mention in passing some of the other 10-fold classification schemes
which have appeared in the literature on symmetry and quantum mechanics.
Foremost among these is the renowned “Altland-Zirnbauer” scheme for
classification of free Fermion Hamiltonians Atland1997 ; Freed:2013 ;
Heinzner:2004xj ; Zirnbauer2013 ; Zirnbauer:2020nab . Another classification
scheme called a “relativistic ten-fold way” was put forward by Freed and
Hopkins in Freed:2016rqq . In quantum systems that come with a graded Hilbert
space, the symmetry groups actually have a natural
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ grading as stressed in Freed:2013 ;
Moore:2014 . One ${\mathbb{Z}}_{2}$ is provided by the unitary/antiunitary
dichotomy and the other by the grading. In this case there is a similar
10-fold classification scheme based on superalgebras similar to that discussed
here. The AZ classification scheme is a special case of this more general
statement. The generalization of the AZ classification scheme in principle
applies to bosons as well as fermions, interacting or not, as emphasized in
Moore:2014 . The ensembles in which the Hamiltonian is an odd element will
have the special property that the energy spectrum has an inversion symmetry
around zero.

Section two of this paper reviews the representation theory of groups
involving both unitary and antiunitary operators. Two categories of
representations are studied and we discuss how the Schur lemma leads to 3-fold
and 10-fold classifications of representations in the two cases. In section
three we review Dyson’s 10-fold classification. In section four we formulate
and prove our main theorem, Theorem 4.1, that relates Dyson’s classification
to superalgebras. The main result is summarized by the one-one correspondence
given by the first two columns of Table 1. Section five comments on an analog
of the Frobenius-Schur indicator for the 10-fold way. It is closely related to
recent work of Ichikawa and Tachikawa ichikawa2020super . Section six
demonstrates how, starting from complex representations of a group $G$, one
can construct all 10 Dyson Types. Appendix A summarizes basic facts about
superalgebras. Appendix B summarizes our notation for Clifford algebras.
Finally, for convenience, we give a reprise of the statement and proof of
Wall’s classification, following a treatment by P. Deligne.

We would like to call attention to two other closely related papers. In
neeb2017 $\phi$-representations were studied in relation to modular theory of
operator algebras. While completing our draft we learned of a very closely
related paper by D. Rumynin and J. Taylor rumynin2020real , which contains a
modern review of Dyson’s classification.

## Acknowledgements

We thank Dan Freed, Alexei Kitaev, Yuji Tachikawa, and Martin Zirnbauer for
helpful conversations and correspondence. R.G. thanks Andrei Grekov for
helpful discussions. G.M. had the great good fortune to have known Freeman
Dyson. It was a pleasure discussing with him the problem which is solved in
this paper. The authors are supported by the DOE under grant DOE-SC0010008 to
Rutgers.

##  2 Review Of $\phi$-representation Theory

We wish to define a complex semilinear action of a compact group $G$ on a
complex vector space. By semi-linearity we mean that the group elements act
either $\mathbb{C}$-linearly or $\mathbb{C}$-antilinearly. In order to do that
conveniently, we “decomplexify” our complex vector spaces by presenting them
as real vector spaces equipped with a complex structure
$V:=(V_{{\mathbb{R}}},I)$. Below, we give basic definitions of
$\phi$-representations (also known as antiunitary representations, semilinear
representations, and co-representations), their subrepresentations etc.

######  Definition 2.1.

Let $G$ be a compact $\mathbb{Z}_{2}$-graded group, where the grading is given
by a group homomorphism

$\phi:G\to\mathbb{Z}_{2}\;,$ (1)

which we assume to be surjective. Here ${\mathbb{Z}}_{2}$ denotes the
multiplicative group of square roots of unity.

1) A $\phi$-representation $(\rho,V_{{\mathbb{R}}},I)$ of a
$\mathbb{Z}_{2}$-graded group $(G,\phi)$ is a real vector space with a complex
structure $(V_{{\mathbb{R}}},I)$ and a homomorphism

$\rho:G\to\mbox{End}_{\;{\mathbb{R}}}(V_{{\mathbb{R}}})\;$ (2)

such that

$\rho(g)I=\phi(g)I\rho(g)\;,\quad\mbox{for}\quad g\in G\;.$ (3)

2) A $\phi$-subrepresentation $(\rho^{\prime},W_{{\mathbb{R}}},I_{W})$ of the
$\phi$-representation $(\rho,V_{{\mathbb{R}}},I_{V})$ is a
$\phi$-representation such that $W_{{\mathbb{R}}}$ is a real vector subspace
of $V_{{\mathbb{R}}}$ with $I_{V}|_{W}=I_{W}$ and $\rho|_{W}=\rho^{\prime}$.

3) A $\phi$-representation is called irreducible if it has no non-trivial
$\phi$-subrepresentations.

The set of $\phi$-reps of the graded group $G$ can be seen as objects of some
category. In the next two sections, we define two different categories
consisting of $\phi$-reps with different types of morphisms among them.

###  2.1 The Category Rep${}_{\mathbb{C}}(G,\phi)$

Here, we define the category Rep${}_{\mathbb{C}}(G,\phi)$ of
$\phi$-representations of the compact graded group $G$ with
$\mathbb{C}$-linear $G$-equivariant morphisms.

######  Definition 2.2.

A complex-linear morphism (intertwiner) between two $\phi$-representations
$(\rho^{\prime},V_{{\mathbb{R}}}^{\prime},I^{\prime})$ and
$(\rho^{\prime\prime},V_{{\mathbb{R}}}^{\prime\prime},I^{\prime\prime})$ is a
complex-linear map $f\in\mbox{Hom}_{\mathbb{C}}(V^{\prime},V^{\prime\prime})$
commuting with the $G$-action

$\displaystyle f\rho^{\prime}(g)$
$\displaystyle=\rho^{\prime\prime}(g)\;f\;,\quad\forall g\in G\;,$ (4)
$\displaystyle fI^{\prime}$ $\displaystyle=I^{\prime\prime}f\;.$ (5)

We denote the set of such morphisms as
$\mbox{Hom}_{\mathbb{C}}^{G}(V^{\prime},V^{\prime\prime})$.

Since the group $G$ under consideration is compact, all representations are
fully reducible. Indeed, the compactness of $G$ allows us to make all the
$\rho(g)$ unitary or antiunitary. In other words, there always exists a
$\phi$-equivariant inner product, which allows us to define orthogonal
complements to irreducible subrepresentations. Thus, we can always decompose
any $\phi$-representation into a direct sum of irreducible ones. In other
words, we can say that Rep${}_{\mathbb{C}}(G,\phi)$ is a semisimple category.

The morphisms between any pair of simple objects are described by Schur’s
lemma.

######  Theorem 2.1.

(Schur’s lemma for $\phi$\- representations)

1\. Let $f$ be a complex-linear intertwiner between two $\phi$-irreps. Then,
$f$ is either the zero map or an isomorphism.

2\. The set of complex-linear self-intertwiners of a $\phi$-irrep,
$\mbox{End}\;^{G}_{\mathbb{C}}(V)$, has the structure of a real division
algebra.

###### Proof.

1\. Let us notice that $\mbox{ker}(f)$ and $\mbox{Im}(f)$ are
$\phi$-subrepresentations. Since we are looking for a map between irreducible
representations, the map can be either an isomorphism or the zero-map.

2\. In general, $\mbox{Hom}^{G}_{\mathbb{C}}(V^{\prime},V^{\prime\prime})$ is
a real vector space. Indeed, assume we found $f$ satisfying (4) and (5). Our
$\phi$-irreps always contain at least one $\mathbb{C}$-antilinear $\rho(g)$.
For such $g$:

$f\rho^{\prime}(g)=\rho^{\prime\prime}(g)\;f\quad\centernot\implies
fI^{\prime}\rho^{\prime}(g)=\rho^{\prime\prime}(g)I^{\prime\prime}f\;.$ (6)

So, $fI^{\prime}$ and $I^{\prime\prime}f$ do not necessarily belong to
$\mbox{Hom}^{G}_{\mathbb{C}}(V^{\prime},V^{\prime\prime})$.

It is clear that $\mbox{End}_{\mathbb{C}}^{G}(V)$ has the structure of a
division algebra. This is a vector space with a multiplication - composition
of maps. Due to the first point, every non-zero element of this algebra is
invertible. ∎

A theorem due to Frobenius C.2 lists all real (unital associative) division
algebras: they are isomorphic to ${\mathbb{R}}$, $\mathbb{C}$ or $\mathbb{H}$.
Then, a self-intertwiner of an irreducible $\phi$-representation acts as one
of the real division algebras on the underlying representation space. The fact
that there are precisely three real associative unital division algebras
underlies the number three in Dyson’s famous Three-Fold Way.

###  2.2 The Category Rep${}_{{\mathbb{R}}}(G,\phi)$

Here, we define a category with morphisms being general real-linear maps, and
discuss its basic properties. In analogy with morphisms among graded rings,
which might shift the grading, we introduce the representation category with
morphisms being semilinear maps.

As defined, $\phi$-representations always refer to real vector spaces endowed
with complex structures. The most general morphisms between real vector spaces
are real linear maps. Using the complex structures on both source and target
spaces $V^{\prime}=(V_{{\mathbb{R}}}^{\prime},I^{\prime})$ and
$V^{\prime\prime}=(V_{{\mathbb{R}}}^{\prime\prime},I^{\prime\prime})$, we can
define an involution on the space of ${\mathbb{R}}$-linear maps:

$f\mapsto-I^{\prime\prime}fI^{\prime}\;,\quad
f\in\mbox{Hom}_{{\mathbb{R}}}(V^{\prime}_{{\mathbb{R}}},V^{\prime\prime}_{{\mathbb{R}}}).$
(7)

There exists a homogeneous basis, such that even morphisms are
$\mathbb{C}$-linear and odd morphisms are $\mathbb{C}$-antilinear maps
correspondingly. This defines a super vector space structure on
$\mbox{Hom}_{{\mathbb{R}}}(V^{\prime},V^{\prime\prime})$. Since the
composition of morphisms respects the ${\mathbb{Z}}_{2}$-grading, it follows
that
$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}}):=\mbox{Hom}_{{\mathbb{R}}}(V_{{\mathbb{R}}},V_{{\mathbb{R}}})$
has the structure of a real superalgebra. For any real vector space
$V_{{\mathbb{R}}}$ with the complex structure $I$, we denote the superalgebra
of ${\mathbb{R}}$-linear maps graded by the adjoint action of $I$ as
$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I)$.

Having this observation at hand, we can define morphisms between $\phi$-reps
to be ${\mathbb{R}}$-linear super G-equivariant maps. Homogeneous
$f^{0},f^{1}\in\mbox{Hom}_{{\mathbb{R}}}^{G}(V^{\prime},V^{\prime\prime})$
satisfy:

$\displaystyle f^{0}I^{\prime}$ $\displaystyle=+I^{\prime\prime}f^{0}\;,$ (8)
$\displaystyle f^{0}\rho^{\prime}(g)$
$\displaystyle=+\rho^{\prime\prime}(g)f^{0}\;,$ (9) $\displaystyle
f^{1}I^{\prime}$ $\displaystyle=-I^{\prime\prime}f^{1}\;,$ (10) $\displaystyle
f^{1}\rho^{\prime}(g)$ $\displaystyle=\phi(g)\rho^{\prime\prime}(g)f^{1}\;.$
(11)

By $\textbf{Rep}_{{\mathbb{R}}}^{G}(G,\phi)$ we denote the category of
$\phi$-representations with ${\mathbb{R}}$-linear super $G$-equivariant maps
as morphisms.

######  Theorem 2.2.

(super-Schur’s lemma for $\phi$\- representations)

1\. Let $f$ be a homogeneous ${\mathbb{R}}$-linear morphism between
$\phi$-irreps. Then, $f$ is either the zero map or an isomorphism.

2\. The set of ${\mathbb{R}}$-linear endomorphisms
$\mbox{End}^{G}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I)$ of a $\phi$-irrep
$(V_{{\mathbb{R}}},I)$, has the structure of a real division superalgebra.

###### Proof.

The proof of the first point is literally the same as for the Schur’s lemma
2.1. Point two follows from what we said above. ∎

Due to a theorem by Wall Wall1964 (see Appendix C for a review), there are 10
real division superalgebras, so that any simple object of
$\textbf{Rep}_{{\mathbb{R}}}(G,\phi)$ has one of the ten types of
endomorphisms. If we endow the category of $\phi$-representations with real-
linear morphisms, we get the 10-fold classification of non-trivial morphisms
among irreducible $\phi$-reps. Let us denote an endomorphism superalgebra of
the given $\phi$-irrep as ${\cal W}$. The set of real division superalgebras
can be identified with the graded Brauer monoid of reals $sBR({\mathbb{R}})$,
considered as a set. The latter is as a set,

$sBR({\mathbb{R}})=\\{C\ell_{-3},\;C\ell_{-2},\;C\ell_{-1},\;C\ell_{1},\;C\ell_{2},\;C\ell_{3},\;\;{\mathbb{R}}\;,\mathbb{H},\;\mathbb{C},\;\mathbb{C}\ell_{1}\\}\;.$
(12)

See Appendix B for the notation. Note we consider superalgebras isomorphic if
they are isomorphic in the graded sense: an isomorphism must be grading-
preserving. Thus, we have a map from $\phi$-irreps to $sBR({\mathbb{R}})$

$(\rho,V_{{\mathbb{R}}},I)\mapsto[{\cal W}]\;.$ (13)

We denote this map by $\delta$.

##  3 Review of Dyson’s Classification of $\phi$-irreps

Here, we review some aspects of theory of $\phi$-representations discussed by
Dyson in his seminal paper. In order to obtain a coarse classification of
irreducible $\phi$-representations of $(G,\phi)$, we shall compress the
initial data $(\rho,V_{{\mathbb{R}}},I)$ into a triple of real associative
algebras (${\cal D}$, ${\cal B}$, ${\cal A}$).

Following Dyson, we define triples of algebras $({\cal D},{\cal B},{\cal A})$,
and list all possible triples potentially arising from an arbitrary
$\phi$-irrep. Since $V_{{\mathbb{R}}}$ has a complex structure, it has even
real dimension; let us denote $2n=\mbox{dim}(V_{{\mathbb{R}}})$.

######  Definition 3.1.

Let $(\rho,V_{{\mathbb{R}}},I)$ be a $\phi$-irrep of a graded group
$(G,\phi)$.

1) We denote the real algebra generated by $\rho(g)$ for $g\in G$ such that
$\phi(g)=1$ as ${\cal A}\subset\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}})$,
and by $\mathcal{X}$ we denote its commutant within
$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}})$.

2) The real algebra obtained by joining $I$ to ${\cal A}$ is denoted by ${\cal
B}$, and ${\cal Y}$ is its commutant within
$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}})$.

3) The real algebra generated by $\rho(g)$ for all $g\in G$ and $I$ is denoted
by ${\cal D}$, and ${\cal Z}$ is its commutant within
$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}})$.

First of all, we notice that irreducibility of $\rho$ is equivalent to
simplicity of ${\cal D}$. Then, ${\cal D}$ is a real simple algebra and, due
to a theorem by Wedderburn, is isomorphic to the full matrix algebra with
real, complex, or quaternionic coefficients. It is convenient to introduce the
following notation: we denote the simple real matrix algebra with coefficients
in a real division algebra
$\mbox{End}_{{\mathbb{R}}}({\mathbb{R}}^{n})\otimes_{{\mathbb{R}}}\mathbb{K}\cong\mbox{End}_{{\mathbb{R}}}(\mathbb{K}^{n})$
by $\mathbb{K}(n)$. Also, we will face algebras made of $m$ copies of
$\mathbb{K}(n)$, we denote them by
$m\mathbb{K}(n)\cong\boldsymbol{1}_{m}\otimes\mathbb{K}(n)$. Explicitly,
$m\mathbb{K}(n)$ consists of matrices of the form

$\begin{pmatrix}\mathbf{M}&&&&\\\ &&\ddots&&&\\\
&&&&\mathbf{M}\end{pmatrix}\;,\quad\mbox{with}\;\mathbf{M}\in\mathbb{K}(n)\;,$
(14)

and as an algebra, $m\mathbb{K}(n)\cong\mathbb{K}(n)$. Using this notation, we
present three alternatives

${\cal D}\cong{\mathbb{R}}(2n)\;,\quad{\cal D}\cong\mathbb{C}(n)\;,\quad{\cal
D}=\mathbb{H}(m)\;,$ (15)

where $n=2m$. Using the double commutant theorem, we can find the commutants
of those algebras within ${\mathbb{R}}(2n)$. With those three alternatives,
the commutants respectively are

${\cal Z}\cong 2n{\mathbb{R}}\;,\quad{\cal Z}\cong n\mathbb{C}\;,\quad{\cal
Z}=m\mathbb{H}\;.$ (16)

Clearly, these three algebras coincide with the algebras appearing in Schur’s
lemma for irreps from $\textbf{Rep}_{\mathbb{C}}(G,\phi)$.

Now we proceed to the possible types of ${\cal B}$ with having ${\cal D}$
fixed. The real dimension of ${\cal D}$ is twice that of ${\cal B}$, since $G$
has non-trivial $\mathbb{Z}_{2}$-grading. In other words, multiplication by
$\rho(g)$ for any $g$ with $\phi(g)=-1$ defines an isomorphism from ${\cal B}$
to a complement of ${\cal B}$ within ${\cal D}$. While $V$ need not be an
irrep of ${\cal B}$, it will be completely reducible.

Let us analyze the general structure of a unitary subrepresentation within the
given $\phi$-irrep. Denote the map $\rho$ restricted to $G^{0}=\\{g\in
G\;|\phi(g)=1\\}$ as $\rho^{\mathbb{C}}$, since we refer to a standard
$\mathbb{C}$-linear representation. So, over the complex numbers,
$\rho^{\mathbb{C}}$ can be reduced to $N$ blocks:

$\rho^{\mathbb{C}}(u)=\rho^{\mathbb{C}}_{1}(u)\oplus\dots\oplus\rho^{\mathbb{C}}_{N}(u)\;,\quad\mbox{where}\quad
u\in G^{0}\;.$ (17)

For any odd element $a$ of the group $G$, we can define a map

$V_{a}(u)=a^{-1}\;u\;a\;,$ (18)

which is an automorphism of $G^{0}$ and consequently

$\rho^{\mathbb{C}}\mapsto\rho(a)^{-1}\rho^{\mathbb{C}}\;\rho(a)$ (19)

is an automorphism of $\rho^{\mathbb{C}}$. The map (19) has to exchange the
components of (17) in order for $\rho$ to be irreducible. In other words, (19)
acts as a non-trivial permutation on the components of (17). In order to
proceed further, let us apply (19) twice

$\rho^{\mathbb{C}}\mapsto\rho(a^{2})^{-1}\;\rho^{\mathbb{C}}\;\rho(a^{2})=\rho^{\mathbb{C}}(w)^{-1}\;\rho^{\mathbb{C}}\;\rho^{\mathbb{C}}(w)\;,\quad\mbox{where}\quad
w\in G^{0}\;,$ (20)

which gives us an automorphism of the unitary subrepresentation, and leaves
the components of (17) invariant. This shows that $\eqref{Vaction}$ defines an
involution on $\rho^{\mathbb{C}}$. This involution either acts trivially on a
component of (17), or transposes a pair of components, so that irreducible
unitary subrepresentations form either singlets or doublets under the action
of $\eqref{Vaction}$. A crucial observation is that each singlet, as well as
each doublet, corresponds to precisely one irreducible $\phi$-representation.

######  Proposition 3.1.

Unitary representation $\rho^{\mathbb{C}}$ contains either one or two
irreducible components; if there are two components, then (19) exchanges them.

We can now show that, given knowledge of ${\cal D}$, one can deduce the
algebra ${\cal B}$. Note that ${\cal B}$ is a complex semisimple algebra
containing up to two blocks of the same dimension (since (19) is an invertible
map). Also, as follows from the consideration above, the simple components
have to be isomorphic algebras. Using what we said above and the fact that the
real dimension of ${\cal D}$ is twice that of ${\cal B}$, we can fix the
complex algebra ${\cal B}$ uniquely (up to isomorphism):

$\displaystyle{\cal D}$
$\displaystyle\cong\mathbb{R}(2n)\quad\Rightarrow\quad{\cal
B}\cong\mathbb{C}(n)\;,$ (21) $\displaystyle{\cal D}$
$\displaystyle\cong\mathbb{C}(n)\;\;\quad\Rightarrow\quad{\cal
B}\cong\mathbb{C}(m)\oplus\mathbb{C}(m)\;,\quad\quad m=n/2\;,$
$\displaystyle{\cal D}$
$\displaystyle\cong\mathbb{H}(m)\;\quad\Rightarrow\quad{\cal B}\cong
2\mathbb{C}(m)\;.$

We now proceed to ${\cal A}$: this is an algebra such that adjoining $I$ to it
gives ${\cal B}$. The simple components are simple real algebras. If $I$
belongs to ${\cal A}$, then its order is $2n^{2}$, otherwise it is $n^{2}$.
Assuming ${\cal B}$ has one simple component:

${\cal B}\cong\mathbb{C}(n)\quad\Rightarrow\quad\begin{cases}\quad{\cal
A}\cong 2{\mathbb{R}}(n)\;,\\\ \quad{\cal A}\cong\mathbb{C}(n)\;,\\\
\quad{\cal A}\cong\mathbb{H}(m)\;.\end{cases}$ (22)

This result can be directly applied to the first and third line in (21). For
the second line, the result (22) should apply to the two components
independently. As follows from the derivation of Proposition 3.1, there are
automorphisms of the full group algebra permuting its simple components.
Rephrased, ${\cal B}$ might have up to two simple components being isomorphic
as algebras. Matching the dimensions, we get the following possibilities:

${\cal
B}=\mathbb{C}(m)\oplus\mathbb{C}(m)\quad\Rightarrow\quad\begin{cases}\quad{\cal
A}=2{\mathbb{R}}(m)\oplus 2{\mathbb{R}}(m)\;,\\\ \quad{\cal
A}=\mathbb{C}(m)\oplus\mathbb{C}(m)\;,\\\ \quad{\cal A}=2\mathbb{C}(m)\;,\\\
\quad{\cal A}=\mathbb{H}(p)\oplus\mathbb{H}(p)\;,\\\ \end{cases}$ (23)

where $p=n/4$. According to the standard terminology, we say that the complex
subrepresentation $\rho^{\mathbb{C}}$ is potentially real, truly complex, or
potentially quaternionic if ${\cal A}$ has real, complex, or quaternionic type
respectively.

This finishes the list of possible triples $({\cal D},{\cal B},{\cal A})$
arising as Dyson’s algebras 3.1 of the $\phi$-irrep, and any triple of
algebras from this table may correspond to some $\phi$-irrep. Let us call a
triple of algebras from this table a Dyson triple. It is clear that we can
introduce an equivalence relation on the Dyson triples: we say that two
triples are similar if the constituent algebras differ by a common tensor
factor ${\mathbb{R}}(p)$ for some $p$. Then, we obtain ten similarity classes
of the Dyson triples. Similarity classes of the Dyson triples are called Dyson
Types. Recall, ${\cal B}$ is not independent, so that $({\cal D},{\cal
B},{\cal A})$ contains the same data as $({\cal D},{\cal A})$. According to
that, any Dyson Type can be encoded by the similarity classes of algebras
${\cal D}$ and ${\cal A}$. Following Dyson, we denote the similarity class of
$({\cal D},{\cal A})$ as $[{\cal D}][{\cal A}]$, and for brevity
$[\mathbb{C}][\mathbb{C}\oplus\mathbb{C}]\equiv\mathbb{C}\mathbb{C}_{1}$, and
$[\mathbb{C}][\mathbb{C}]\equiv\mathbb{C}\mathbb{C}_{2}$, where square
brackets stand for the equivalence class defined above.

$\mathbb{D}$ $Type$ ${\cal D}$ ${\cal B}$ ${\cal A}$ ${\cal X}$ ${\cal Y}$
${\cal Z}$ $C\ell_{1}$ $\mathbb{RR}$ $\mathbb{R}(2n)$ $\mathbb{C}(n)$
$2\mathbb{R}(n)$ $n{\mathbb{R}}(2)$ $n\mathbb{C}$ $2n{\mathbb{R}}$
${\mathbb{R}}$ $\mathbb{RC}$ ${\mathbb{R}}(2n)$ $\mathbb{C}(n)$
$\mathbb{C}(n)$ $n\mathbb{C}$ $n\mathbb{C}$ $2n{\mathbb{R}}$ $C\ell_{-1}$
$\mathbb{RH}$ ${\mathbb{R}}(4m)$ $\mathbb{C}(2m)$ $\mathbb{H}(m)$
$m\mathbb{H}^{\scriptstyle{opp}}$ $2m\mathbb{C}$ $4m{\mathbb{R}}$ $C\ell_{2}$
$\mathbb{CR}$ $\mathbb{C}(2m)$ $\mathbb{C}(m)\oplus\mathbb{C}(m)$
$2{\mathbb{R}}(m)\oplus 2{\mathbb{R}}(m)$ $m{\mathbb{R}}(2)\oplus
m{\mathbb{R}}(2)$ $m\mathbb{C}\oplus m\mathbb{C}$ $2m\mathbb{C}$ $\mathbb{C}$
$\mathbb{CC}_{1}$ $\mathbb{C}(2m)$ $\mathbb{C}(m)\oplus\mathbb{C}(m)$
$\mathbb{C}(m)\oplus\mathbb{C}(m)$ $m\mathbb{C}\oplus m\mathbb{C}$
$m\mathbb{C}\oplus m\mathbb{C}$ $2m\mathbb{C}$ $\mathbb{C}\ell_{1}$
$\mathbb{CC}_{2}$ $\mathbb{C}(2m)$ $\mathbb{C}(m)\oplus\mathbb{C}(m)$
$2\mathbb{C}(m)$ $m\mathbb{C}(2)$ $m\mathbb{C}\oplus m\mathbb{C}$
$2m\mathbb{C}$ $C\ell_{-2}$ $\mathbb{CH}$ $\mathbb{C}(4p)$
$\mathbb{C}(2p)\oplus\mathbb{C}(2p)$ $\mathbb{H}(p)\oplus\mathbb{H}(p)$
$p\mathbb{H}^{\scriptstyle{opp}}\oplus p\mathbb{H}^{\scriptstyle{opp}}$
$2p\mathbb{C}\oplus 2p\mathbb{C}$ $4p\mathbb{C}$ $C\ell_{3}$ $\mathbb{HR}$
$\mathbb{H}(m)$ $2\mathbb{C}(m)$ $4{\mathbb{R}}(m)$ $m\mathbb{R}(4)$
$m\mathbb{C}(2)$ $m\mathbb{H}^{\scriptstyle{opp}}$ $\mathbb{H}$ $\mathbb{HC}$
$\mathbb{H}(m)$ $2\mathbb{C}(m)$ $2\mathbb{C}(m)$ $m\mathbb{C}(2)$
$m\mathbb{C}(2)$ $m\mathbb{H}^{\scriptstyle{opp}}$ $C\ell_{-3}$ $\mathbb{HH}$
$\mathbb{H}(2p)$ $2\mathbb{C}(2p)$ $2\mathbb{H}(p)$
$p\mathbb{H}^{\scriptstyle{opp}}$ $2p\mathbb{C}(2)$
$2p\mathbb{H}^{\scriptstyle{opp}}$

Table 1: The real division superalgebras matched with the Dyson Types and
their Dyson triples. The first column will be explained below - it is one the
main points of this paper.

Examples of $\phi$-irreps can be found in the original paper Dyson_1962 . In
particular, Dyson shows that all ten possibilities indeed occur. In section 6
below, we discuss how $\phi$-irreps arise from ordinary complex-linear irreps.

We have defined a map from $\phi$-irreps to the Dyson triples.

$(\rho,V_{{\mathbb{R}}},I)\to({\cal D},{\cal B},{\cal A})\;.$ (24)

Further, taking the similarity class of the Dyson triple defines a map to the
Dyson Type:

$\displaystyle\alpha:\quad\mbox{Irreps}(G,\phi)$ $\displaystyle\to
Dyson\;Type\;,$ (25) $\displaystyle(\rho,V_{{\mathbb{R}}},I)$
$\displaystyle\mapsto[{\cal D}][{\cal A}]\;.$ (26)

##  4 Real Division Superalgebras And Dyson Types

We have seen that there is a correspondence between the three possible
similarity types of ${\cal D}$ and three real division algebras. It is natural
to ask - is there a correspondence between the ten Dyson Types and ten real
division superalgebras? This question was first raised in Freed:2013 , and
formulated as a conjecture about a 1-1 correspondence between the two. Some
examples are worked out in Moore:2013 . One of the main goals of this paper is
to show that the classifications by Dyson Types and graded commutants are
essentially the same. We formulate this as a theorem and prove it below.

######  Theorem 4.1.

There is a 1-1 correspondence between the Dyson Types and real division
superalgebras, such that for any given $\phi$-irrep of any
$\mathbb{Z}_{2}$-graded group $(G,\phi)$, the endomorphism superalgebra ${\cal
W}$ for this irrep corresponds to the Dyson Type extracted from this irrep.

For any $\phi$-irrep, we can determine its Dyson Type and the endomorphism
superalgebra ${\cal W}$ separately. We shall prove that, for any $(G,\phi)$
and any $\phi$-irrep the two are governed by the 1-1 correspondence of ten
objects in the first two columns of Table 1.

###  4.1 The Road Map

We prove the theorem in two steps. Firstly, for any $\phi$-irrep we introduce
the superalgebra ${\cal A}_{s}$ as a superalgebra generated by $\rho(g)$ for
all $g\in G$, where the grading is inherited from that of $G$. We will prove
that for any $\phi$-irrep, ${\cal A}_{s}$ and the Dyson triple extracted from
this $\phi$-irrep are in 1-1 correspondence, so that there is a map

$Dyson\;triples\to Superalgebras\;.$ (27)

Where, $Superalgebras$ stands for the set of superalgebras ${\cal A}_{s}$
which can arise from $\phi$-irreps. Note, there are restrictions on ${\cal
A}_{s}$ dictated by the fact they occur from representations of graded groups,
so they are not general superalgebras. In the previous section, we introduced
the equivalence relation of having a common tensor factor ${\mathbb{R}}(n)$;
under this equivalence Dyson triples descend to Dyson Types, and
$Superalgebras$ descend to $GroupSAlg$. There is an isomorphism of sets:

$\displaystyle\beta:\quad Dyson\;Types$ $\displaystyle\to GroupSAlg\;,$ (28)
$\displaystyle Dyson\;Type$ $\displaystyle\mapsto[{\cal A}_{s}]\;.$ (29)

where $GroupSAlg$ stands for the set of equivalence classes of the
superalgebras ${\cal A}_{s}$. It turns out that $GroupSAlg$ contains all the
ten classes from $sBR({\mathbb{R}})$.

The next step will be to calculate the graded commutant of the superalgebra
${\cal A}_{s}$ within $\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I)$. This
graded commutant is nothing but an endomorphism superalgebra of the
$\phi$-irrep. Taking the graded commutant is an invertible operation, which
defines an isomorphism of sets

$\displaystyle\gamma:GroupSAlg$ $\displaystyle\to sBR({\mathbb{R}})\;$ (30)
$\displaystyle[{\cal A}_{s}]$ $\displaystyle\mapsto[{\cal W}]\;.$ (31)

Our goal is to end up with an isomorphism of sets $Dyson\;Types\to
sBR({\mathbb{R}})$ not depending on the specific group and representation. We
achieve it by constructing the isomorphism $\gamma\circ\beta$ without
referring to any specific $\phi$-irrep. We analyze what are the algebraic
conditions the $\phi$-irrep has to satisfy in order to have some fixed Dyson
Type. Those conditions allow us to fix the corresponding superalgebra ${\cal
A}_{s}$ up to equivalence. In its turn, any simple superalgebra has a unique
graded commutant within $\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I)$. In
fact, we can go other way unambiguously, such that $\gamma\circ\beta$ is an
isomorphism of sets.

###  4.2 Group Superalgebras

As we discussed above, the full matrix algebra
$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}})$ admits a
${\mathbb{Z}}_{2}$-grading given by the adjoint action of the complex
structure. Then, the $\phi$-representation $(\rho,V_{{\mathbb{R}}},I)$ can be
seen as a grade-preserving homomorphism
$\rho:G\to\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I)$. This morphism can
always fit into the commutative diagram:

${(G,\phi)}$${\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I)}$${{\mathbb{R}}^{\phi}[G]}$$\scriptstyle{\rho}$$\scriptstyle{\psi}$

where ${\mathbb{R}}^{\phi}$ is the group superalgebra (a super vector space
with the basis given by the elements of $(G,\phi)$), and $\psi$ is an
${\mathbb{R}}$-linear grade-preserving injective map. Let us denote the image
of $\psi$ as ${\cal A}_{s}$. Alternatively, ${\cal A}_{s}$ is an
${\mathbb{R}}$-linear span of $\\{\rho(g)|\;g\in G\\}$. The superalgebra
${\cal A}_{s}$ is a close analog of Dyson’s algebras associated to the
$\phi$-irrep. The even part of ${\cal A}_{s}$ is nothing but the Dyson’s
${\cal A}\equiv{\cal A}_{s}^{0}$, and ${\cal A}_{s}^{1}$ is the image of the
odd elements ${\cal A}_{s}^{1}\equiv\\{\rho(g)|\;g\in G,\;\phi(g)=-1\\}$.

Let us notice that the largest of the Dyson algebras ${\cal D}$ contains
${\cal A}_{s}^{\scriptstyle{ungr}}$ together with an element which commutes
with ${\cal A}_{s}^{0}$ and anticommutes with ${\cal A}_{s}^{1}$: ${\cal
D}=\langle{\cal A}_{s}^{\scriptstyle{ungr}},I\rangle$, where angle brackets
stand for additive and multiplicative closure.

As follows from definitions, ${\cal A}_{s}$ is a simple superalgebra for any
$\phi$-irrep. Since it arises from irreps of non-trivially graded groups,
${\cal A}_{s}$ is not concentrated in the even degree. Moreover, there is an
isomorphism of vector spaces ${\cal A}_{s}^{0}\cong{\cal A}_{s}^{1}$. We can
always present this morphism as a multiplication by an odd element $u\in{\cal
A}_{s}^{0}$. Indeed, $G$ admits a coset decomposition
$G=G^{0}\bigcup\tilde{g}G^{0}$, where $\tilde{g}$ is a coset representative;
we can take $u=\rho(\tilde{g})$, and it has the properties of a coset
representative as well. Summarizing, there is an invertible element $u$ such
that ${\cal A}_{s}^{1}=u{\cal A}_{s}^{0}$.

###  4.3 Aside: $\phi$-Representation As Real-Linear Representations

Let us clarify what we mean by the relation ${\cal D}\equiv\langle{\cal
A}_{s}^{ungr},I\rangle$. Definition 2.1 of a $\phi$-representation of $G$ can
be recast as the definition of an ordinary representation of a group
$\tilde{G}$ related to $G$. Indeed, $\tilde{G}$ will simply be a semidirect
product of $G$ with $U(1)$ (or $\mathbb{Z}_{4}$).

Let us identify $\phi$ with the non-trivial homomorphism
$\phi:G\to\mbox{Aut}(U(1))\cong{\mathbb{Z}}_{2}$. The non-trivial generator of
${\rm Aut}(U(1))$ is the Galois action $\omega\to\omega^{-1}$. Using $\phi$,
we can define a semidirect product $\tilde{G}\equiv U(1)\rtimes_{\phi}G$. It
is clear from the definitions that any $\phi$-representation
$(\rho,V_{{\mathbb{R}}},I)$ of $(G,\phi)$ is equivalent to a real linear
representation $(\rho,V_{{\mathbb{R}}})$ of $\tilde{G}$. Further, we can
introduce the group algebra ${\mathbb{R}}[\tilde{G}]$: its image under $\rho$
will be nothing but ${\cal D}$. This useful point of view was used to describe
Dyson’s classification in rumynin2020real .

###  4.4 Proof Of The Conjecture

Firstly, we wish to convert the Dyson triples into superalgebras. With the
${\cal A}_{s}$ defined above, we claim that the Dyson triples are equivalent
to ${\cal A}_{s}$.

######  Proposition 4.1.

For any $\phi$-irrep, there is a 1-1 correspondence between the Dyson triple
and the superalgebra ${\cal A}_{s}$.

Any triple of algebras $({\cal D},{\cal B},{\cal A})$ coming from a
$\phi$-irrep implicitly contains information about $u$ because we can generate
${\cal D}$ with the elements of ${\cal A}$, $u$, and $I$. Let us recall that
the Dyson Type contains Dyson triples differing only by a tensor factor of
${\mathbb{R}}(n)$. Assume ${\cal D}\cong\langle{\cal A},u,I\rangle$ such that
${\cal A}_{s}\cong{\cal A}\oplus u{\cal A}$, then ${\cal D}^{\prime}\cong{\cal
D}\otimes{\mathbb{R}}(n)\cong\langle{\cal A}\otimes{\mathbb{R}}(n),u,I\rangle$
corresponds to ${\cal A}_{s}^{\prime}\cong{\cal A}_{s}\otimes{\mathbb{R}}(n)$.
From here it follows that it is enough to consider the minimal Dyson triple: a
Dyson triple from the Table 1 having the parameter $n$ (or $m$, or $p$) equal
to 1.

Our proof is by direct computation. Beginning with the minimal Dyson triple
$({\cal D},{\cal B},{\cal A})$, we reconstruct $u$. Having obtained ${\cal
A}_{s}$, we get back to a triple. Within the minimal irrep, we always pick the
standard complex structure. As a real matrix acting on
$V_{{\mathbb{R}}}^{2q}$, where $q$ is the complex dimension of the minimal
$\phi$-irrep, it always looks like $I=i\otimes\boldsymbol{1}_{q}$, where
$i\in\mathbb{C}$ is represented by the matrix
$\bigl{(}\begin{smallmatrix}0&-1\\\ 1&0\end{smallmatrix}\bigr{)}$. We are
looking for an operator $u$, anticommuting with $I$ such that ${\cal
D}\cong\langle{\cal B},u\rangle$. When working with ${\mathbb{R}}(2)$, we
always pick the “quaternionic” basis.

####  $({\mathbb{R}}{\mathbb{R}})\Longrightarrow C\ell_{1}$.

Here, $u$ is the real structure. It is an element of ${\mathbb{R}}(2)$
squaring to $+\boldsymbol{1}$ and anticommuting with $I$. Then, ${\cal
A}_{s}\cong C\ell_{1}$.

####  $({\mathbb{R}}\mathbb{C})\Longrightarrow C\ell_{2}$.

Here, ${\cal A}=2\mathbb{C}$ and ${\cal
B}=\mathbb{C}\oplus\mathbb{C}\cong\mathbb{C}\otimes\langle\boldsymbol{1},P\rangle$,
where $P$ is an abstract element of ${\mathbb{R}}(2)$ squaring to
$\boldsymbol{1}$. In order to complete ${\cal B}$ to ${\cal D}$, we take
$u=P^{\prime}\otimes P^{\prime}$, where $P^{\prime}\in{\mathbb{R}}(2)$ is an
operator squaring to $\boldsymbol{1}$, and anticommuting with $i$ and $P$.
This shows ${\cal A}_{s}=C\ell_{2}$.

####  $({\mathbb{R}}\mathbb{H})\Longrightarrow C\ell_{3}$.

Given ${\cal A}=\mathbb{H}$ and ${\cal B}=\mathbb{C}(2)$, the complex
structure $I=i^{\prime}\in\mathbb{H}^{\scriptstyle{opp}}$. Then,
$u=j^{\prime}\in\mathbb{H}^{\scriptstyle{opp}}$. One can check that if ${\cal
A}_{s}^{1}=j^{\prime}\mathbb{H}$, then ${\cal A}_{s}\cong C\ell_{3}$.

####  $(\mathbb{C}{\mathbb{R}})\Rightarrow C\ell_{1,-1}$.

Here, ${\cal A}=2{\mathbb{R}}\oplus 2{\mathbb{R}}\cong
2{\mathbb{R}}\otimes\langle\boldsymbol{1},P\rangle$. The complex structure
$I=i\otimes\boldsymbol{1}$, and $u$ is an antilinear operator squaring to
$+\boldsymbol{1}$, let us call it $P^{\prime}\otimes P^{\prime}$, where
$P^{\prime}\in{\mathbb{R}}(2)$, is an operator anticommuting with
$P\in{\mathbb{R}}(2)$ and squaring to $+\boldsymbol{1}$. This leads to ${\cal
A}_{s}\cong C\ell_{1,-1}$

####  $(\mathbb{C}\mathbb{C}_{1})\Longrightarrow\mathbb{C}\ell_{2}$.

Here, ${\cal
A}=\mathbb{C}\oplus\mathbb{C}\cong\mathbb{C}\otimes\langle\boldsymbol{1},P\rangle$.
The complex structure $I=i\otimes\boldsymbol{1}$, and $u$ is the same as in
the previous case. If appropriately normalized, $u^{2}=\pm 1$ leads to ${\cal
A}_{s}\cong\mathbb{C}\ell_{2}$.

######  Remark 1.

Even though, the superalgebras $C\ell_{1,-1}$ and $\mathbb{C}\ell_{2}$ are
similar to ${\mathbb{R}}$ and $\mathbb{C}$ correspondingly, we keep writing
them as they are. We are stressing that $\phi$ is a surjective map, and
consequently ${\cal A}_{s}$ is not a purely even superalgebra. Nothing
prevents ${\cal A}_{s}$ to be similar to a purely even superalgebra.

####  $(\mathbb{C}\mathbb{C}_{2})\Longrightarrow\mathbb{C}\ell_{1}$.

Here, ${\cal A}=2\mathbb{C}\cong\boldsymbol{1}\otimes\mathbb{C}$. The complex
structure $I=i\otimes\boldsymbol{1}$, and $u$ is the same as in the previous
two cases. If appropriately normalized, $u^{2}=\pm 1$ leads to ${\cal
A}_{s}\cong\mathbb{C}\ell_{1}$.

####  $(\mathbb{C}\mathbb{H})\Longrightarrow C\ell_{-4}$.

Here, ${\cal A}=\mathbb{H}\otimes\langle\boldsymbol{1},P\rangle$. The complex
structure is $I=i^{\prime}\otimes\boldsymbol{1}$ with
$i^{\prime}\in\mathbb{H}^{\scriptstyle{opp}}$, and $u$ is a complex-linear
combination of $\boldsymbol{1}\otimes P^{\prime}$ and $\boldsymbol{1}\otimes
i$. Then, ${\cal A}_{s}^{1}=u{\cal A}$ leads to ${\cal A}_{s}\cong C\ell_{4}$
(as well as to ${\cal A}_{s}\cong C\ell_{-4}$).

####  $(\mathbb{H}{\mathbb{R}})\Longrightarrow C\ell_{-1}$.

In this case, ${\cal A}=4{\mathbb{R}}$, ${\cal B}=2\mathbb{C}$, such that
$I=i$ and $u$ is any operator anticommuting with $I$ and squaring to
$-\boldsymbol{1}$, so is $u=j\in\mathbb{H}$. Then, ${\cal A}_{s}^{1}=\\{j\\}$,
and ${\cal A}_{s}\cong C\ell_{-1}$.

####  $(\mathbb{H}\mathbb{C})\Longrightarrow C\ell_{-2}$.

In this case, ${\cal A}=2\mathbb{C}$, ${\cal B}=2\mathbb{C}$, such that $I=i$
and $u=j\in\mathbb{H}$. One may identify ${\cal A}_{s}\cong C\ell_{-2}$.

####  $(\mathbb{H}\mathbb{H})\Longrightarrow C\ell_{-3}$.

Let us use the dimension argument. The superalgebra ${\cal A}_{s}$ is a simple
superalgebra of the real dimension $8$ (this dimension is twice the dimension
of ${\cal A}$). The candidates are the Clifford algebras of rank 3:
$C\ell_{3}$, $C\ell_{-3}$, $C\ell_{2,-1}\cong
C\ell_{1}\otimes{\mathbb{R}}(2)$, and $C\ell_{1,-2}$. Matching the even parts
leaves us with $C\ell_{3}$, and $C\ell_{-3}$.

If ${\cal A}\cong\mathbb{H}$, the full algebra ${\cal D}$, as a vector space,
can be decomposed as ${\cal D}=\mathbb{H}\oplus I\mathbb{H}\oplus
u\mathbb{H}\oplus uI\mathbb{H}$. Then, depending on the class of $u^{2}$ in
${\mathbb{R}}^{2}/{\mathbb{R}}^{*}$, we get the two possibilities. If
$u^{2}=-\boldsymbol{1}$, then ${\cal
D}=\mathbb{H}\otimes\mathbb{H}\cong{\mathbb{R}}(4)$. If
$u^{2}=\boldsymbol{1}$, then ${\cal
D}=\mathbb{H}\otimes{\mathbb{R}}(2)\cong\mathbb{H}(2)$. Since we are
interested in ${\cal D}=\mathbb{H}(2)$, the coset representative $u$ should
satisfy $u^{2}=\boldsymbol{1}$. Then, ${\cal A}_{s}^{1}\cong u\mathbb{H}$
implies ${\cal A}_{s}\cong C\ell_{-3}$.

####

Now we wish to obtain a Dyson triple from the superalgebra ${\cal A}_{s}$.
Recall, ${\cal D}=\langle{\cal A}_{s}^{\scriptstyle{ungr}},I\rangle$ such that
$I$ defines the grading on ${\cal A}_{s}$. So, the dimension of ${\cal D}$ is
twice that of ${\cal A}_{s}$ in the case $I\in{\cal A}_{s}^{0}$ and it is four
times the dimension of ${\cal A}_{s}^{0}$ if $I\notin{\cal A}_{s}^{0}$.
Summarizing, we say that ${\cal D}$ is a simple algebra containing ${\cal
A}_{s}^{\scriptstyle{ungr}}$ as a subalgebra or coinciding with the latter.

For any superalgebra, its even subalgebra and the algebra obtained by
forgetting the grading is known. Let us recall some relevant cases in the
following Table 2.

$[{\cal A}_{s}]$ $[{\cal A}_{s}^{0}]$ $[{\cal A}_{s}^{\scriptstyle{ungr}}]$
$C\ell_{-4}$ $\mathbb{H}\oplus\mathbb{H}$ $\mathbb{H}(2)$ $C\ell_{-3}$
$\mathbb{H}$ $\mathbb{H}\oplus\mathbb{H}$ $C\ell_{-2}$ $\mathbb{C}$
$\mathbb{H}$ $C\ell_{-1}$ ${\mathbb{R}}$ $\mathbb{C}$ $C\ell_{1,-1}$
${\mathbb{R}}\oplus{\mathbb{R}}$ ${\mathbb{R}}(2)$ $C\ell_{1}$ ${\mathbb{R}}$
${\mathbb{R}}\oplus{\mathbb{R}}$ $C\ell_{2}$ $\mathbb{C}$ ${\mathbb{R}}(2)$
$C\ell_{3}$ $\mathbb{H}$ $\mathbb{C}(2)$ $\mathbb{C}\ell_{1}$ $\mathbb{C}$
$\mathbb{C}\oplus\mathbb{C}$ $\mathbb{C}\ell_{2}$ $\mathbb{C}\oplus\mathbb{C}$
$\mathbb{C}(2)$

Table 2: The set of distinguished Morita-classes of superalgebras appearing
form $\phi$-irreps.

####  $C\ell_{1}\Longrightarrow({\mathbb{R}}{\mathbb{R}})$.

In this case, ${\cal
A}_{s}^{\scriptstyle{ungr}}={\mathbb{R}}\oplus{\mathbb{R}}$. The order of
${\cal D}$ is $4$, it is either ${\mathbb{R}}(2)$, or $\mathbb{H}$. Obviously,
it cannot be $\mathbb{H}$, as there is not enough elements squaring to
$-\boldsymbol{1}$. So, ${\cal D}={\mathbb{R}}(2)$.

####  $C\ell_{3}\Longrightarrow({\mathbb{R}}\mathbb{H})$.

The order of ${\cal D}$ is 16, so it is either $\mathbb{H}(2)$ or
${\mathbb{R}}(4)$. We are completing $\mathbb{C}(2)$ with an operator $I$
commuting with ${\cal A}_{s}^{0}=\mathbb{H}$. In this case, ${\cal A}_{s}^{1}$
can be identified with $j^{\prime}\in\mathbb{H}^{\scriptstyle{opp}}$, where
$j^{\prime}$ is odd and anticommutes with $I$. Together, $I$ and $j^{\prime}$
generate the full quaternion algebra $\mathbb{H}^{\scriptstyle{opp}}$. So,
${\cal D}=\mathbb{H}\mathbb{H}^{\scriptstyle{opp}}\cong{\mathbb{R}}(4)$.

####  $C\ell_{1,-1}\Longrightarrow(\mathbb{C}{\mathbb{R}})$.

The order of ${\cal D}$ is 8, so this is $\mathbb{C}(2)$.

####  $\mathbb{C}\ell_{1}\Longrightarrow(\mathbb{C}\mathbb{C}_{2})$.

The order of ${\cal D}$ is 8, so this is $\mathbb{C}(2)$.

####  $C\ell_{4}\Longrightarrow(\mathbb{C}\mathbb{H})$.

The order of ${\cal D}$ is 32, so this is $\mathbb{C}(4)$.

####  $C\ell_{-1}\Longrightarrow(\mathbb{H}{\mathbb{R}})$.

The order of ${\cal D}$ is 4, the candidates are ${\mathbb{R}}(2)$ and
$\mathbb{H}$. Clearly, ${\cal D}=\mathbb{H}$.

####  $C\ell_{-3}\Longrightarrow(\mathbb{H}\mathbb{H})$.

The order of ${\cal D}$ is 16. The candidates are $\mathbb{H}(2)$ and
${\mathbb{R}}(4)$. However, ${\mathbb{R}}(4)$ does not contain ${\cal
A}_{s}^{\scriptstyle{ungr}}=\mathbb{H}\oplus\mathbb{H}$ as a subalgebra, while
$\mathbb{H}(2)$ clearly contains it.

This finishes the proof of proposition 4.1.

###  4.5 Graded Commutants

We have transferred data encoded in Dyson’s analysis into the language of
superalgebras. Thanks to that, computation of graded commutants of irreducible
$\phi$-reps defined in 2.2 becomes a conventional computation of graded
commutants of group superalgebras within
$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I)$.

######  Definition 4.1.

We define the graded commutant ${\cal Z}_{s}(A,B)$ of a superalgebra $A$
within a superalgebra $B$ as the subset of $B$ containing elements of the form
$w=w^{0}+w^{1}\in B$ such that

$\displaystyle w^{0}a\;=+a\;w^{0}\;,$ (32) $\displaystyle
w^{1}a^{0}=+a^{0}w^{1}\;,$ (33) $\displaystyle w^{1}a^{1}=-a^{1}w^{1}\;.$ (34)

In this notation, ${\cal Z}_{s}(A)\equiv{\cal Z}_{s}(A,A)$ is the supercenter
of $A$.

Matching this definition with that of ${\cal W}$, we can identify the graded
commutant of a $\phi$-irrep with the graded commutant of its group
superalgebra within $\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I)$:

$\mbox{End}^{G}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I)\equiv{\cal Z}_{s}({\cal
A}_{s},\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I))\;.$

To proceed further, let us analyze the superalgebra
$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I)$. It is a real simple
superalgebra with the supercenter ${\mathbb{R}}$, so that it is similar to one
of the real Clifford algebras. Picking the standard complex structure in
$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}})$, one can directly check the
following isomorphisms:

$\mbox{End}_{{\mathbb{R}}}({\mathbb{R}}^{2},I)\cong
C\ell_{2}\;,\quad\mbox{End}_{{\mathbb{R}}}({\mathbb{R}}^{4},I)\cong
C\ell_{3,-1}\;,\quad\mbox{End}_{{\mathbb{R}}}({\mathbb{R}}^{8},I)\cong
C\ell_{4,-2}\;.$ (35)

There are two more isomorphisms:
$\mbox{End}_{{\mathbb{R}}}({\mathbb{R}}^{4},I)\cong
C\ell_{2}\otimes{\mathbb{R}}(2)$ and
$\mbox{End}_{{\mathbb{R}}}({\mathbb{R}}^{8},I)\cong
C\ell_{2}\otimes{\mathbb{R}}(4)$, which will be useful. In general, we define
a grading on $\mbox{End}_{{\mathbb{R}}}({\mathbb{R}}^{2n},I)$ by the adjoint
action of the standard complex structure $I=i\otimes\boldsymbol{1}_{n}$. Then,
as a super vector space,
$\mbox{End}_{{\mathbb{R}}}({\mathbb{R}}^{2n},I)\cong\langle\boldsymbol{1},i\rangle\otimes{\mathbb{R}}(n)\oplus\\{P,P^{\prime}\\}\otimes{\mathbb{R}}(n)$,
where we used the “quaternionic” basis for ${\mathbb{R}}(2)$. Since we
consider ${\mathbb{R}}(n)$ as an ungraded algebra and the tensor product is
the usual tensor product, $\mbox{End}_{{\mathbb{R}}}({\mathbb{R}}^{2n},I)$ is
isomorphic to $C\ell_{2}\otimes{\mathbb{R}}(n)$, so is similar to $C\ell_{2}$.
By the same argument,

$\mbox{End}_{{\mathbb{R}}}({\mathbb{R}}^{4n},I)\cong
C\ell_{3,-1}\otimes{\mathbb{R}}(n)\;,\;\mbox{End}_{{\mathbb{R}}}({\mathbb{R}}^{8n},I)\cong
C\ell_{4,-2}\otimes{\mathbb{R}}(n)\;.$ (36)

Now, our aim is to compute graded commutants of superalgebras from the second
column of the following Table 3 within the superalgebras from the third
column. The resulting commutant ${\cal W}$ is presented in the fourth column;
we will calculate it shortly.

Type ${\cal A}_{s}$ $\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I)$ ${\cal W}$
${\mathbb{R}}{\mathbb{R}}$ $C\ell_{1}\otimes{\mathbb{R}}(n)$
$C\ell_{2}\otimes{\mathbb{R}}(n)$ $C\ell_{1}$ ${\mathbb{R}}\mathbb{C}$
$C\ell_{2}\otimes{\mathbb{R}}(n)$ $C\ell_{2}\otimes{\mathbb{R}}(n)$
${\mathbb{R}}$ ${\mathbb{R}}\mathbb{H}$ $C\ell_{3}\otimes{\mathbb{R}}(m)$
$C\ell_{3,-1}\otimes{\mathbb{R}}(n)$ $C\ell_{-1}$ $\mathbb{C}{\mathbb{R}}$
$C\ell_{1,-1}\otimes{\mathbb{R}}(m)$ $C\ell_{3,-1}\otimes{\mathbb{R}}(m)$
$C\ell_{2}$ $\mathbb{C}\mathbb{C}_{1}$
$\mathbb{C}\ell_{2}\otimes{\mathbb{R}}(m)$
$C\ell_{3,-1}\otimes{\mathbb{R}}(m)$ $\mathbb{C}$ $\mathbb{C}\mathbb{C}_{2}$
$\mathbb{C}\ell_{1}\otimes{\mathbb{R}}(m)$
$C\ell_{3,-1}\otimes{\mathbb{R}}(m)$ $\mathbb{C}\ell_{1}$
$\mathbb{C}\mathbb{H}$ $C\ell_{4}\otimes{\mathbb{R}}(p)$
$C\ell_{4,-2}\otimes{\mathbb{R}}(p)$ $C\ell_{-2}$ $\mathbb{H}{\mathbb{R}}$
$C\ell_{-1}\otimes{\mathbb{R}}(m)$ $C\ell_{3,-1}\otimes{\mathbb{R}}(m)$
$C\ell_{3}$ $\mathbb{H}\mathbb{C}$ $C\ell_{-2}\otimes{\mathbb{R}}(m)$
$C\ell_{3,-1}\otimes{\mathbb{R}}(m)$ $\mathbb{H}$ $\mathbb{H}\mathbb{H}$
$C\ell_{-3}\otimes{\mathbb{R}}(p)$ $C\ell_{4,-2}\otimes{\mathbb{R}}(p)$
$C\ell_{-3}$

Table 3: The Dyson Types matched with the superalgebras ${\cal A}_{s}$.

First of all, let us notice that ${\cal
Z}_{s}(A\otimes{\mathbb{R}}(n),B\otimes{\mathbb{R}}(n))\cong{\cal
Z}_{s}(A,B)$, for any simple superalgebras $A$ and $B$, what is a consequence
of the known isomorphism ${\cal
Z}({\mathbb{R}}(n),{\mathbb{R}}(n))\cong{\mathbb{R}}$. Then, we are left with
a calculation of graded commutants of some Clifford algebras within other
Clifford algebras. Recall that we define Clifford algebras through a vector
space equipped by a symmetric bilinear form, see Appendix B. The following
proposition relates operations of taking commutants in the context of
superalgebras with operation of taking orthogonal complement in the context of
bilinear forms on vector spaces.

######  Proposition 4.2.

Let $(V,B)$ and $(W,B^{\prime})$ be a pair of a real vector spaces equipped
with symmetric bilinear forms such that $V\subset W$ and $B^{\prime}|_{V}=B$.
Then, ${\cal Z}_{s}(C\ell(V,B),C\ell(W,B^{\prime}))\cong
C\ell(V^{\perp},B^{\perp})$, where $V^{\perp}$ is the orthogonal complement to
$W$ in the sense of the bilinear form $B^{\prime}$, and
$B^{\perp}=B^{\prime}|_{V^{\perp}}$.

The proof follows from matching the conditions of (4.1) and the Clifford
relations (58).

Using this proposition, we immediately obtain the graded commutants
corresponding to the types ${\mathbb{R}}{\mathbb{R}}$,
${\mathbb{R}}\mathbb{C}$, ${\mathbb{R}}\mathbb{H}$, $\mathbb{C}{\mathbb{R}}$,
$\mathbb{C}\mathbb{H}$, $\mathbb{H}{\mathbb{R}}$ in Table 3. The four
remaining cases should be computed separately.

We begin with ${\cal Z}_{s}(C\ell_{-2},C\ell_{3,-1})$. Let $C\ell_{3,-1}$ be
generated by $x_{1},x_{2},x_{3}$, and $y$ such that $x_{i}^{2}=\boldsymbol{1}$
and $y^{2}=-\boldsymbol{1}$. Then, the superalgebra $C\ell_{-2}$ generated by
odd anticommuting elements $a$ and $b$ and squaring to $-\boldsymbol{1}$, can
be embedded into $C\ell_{3,-1}$ in the following way:

$a\to y\;,\quad b\to x_{1}x_{2}x_{3}\;.$ (37)

One can verify that the graded commutant is purely even and contains
$\boldsymbol{1},x_{1}x_{2},x_{1}x_{3},x_{2}x_{3}$, what implies ${\cal
Z}_{s}(C\ell_{-2},C\ell_{3,-1})\cong\mathbb{H}$. We work out the case
$\mathbb{H}\mathbb{H}$ next. Let $C\ell_{4,-2}$ be generated by $x_{i}$ with
$1\leq i\leq 4$ and $y_{1},y_{2}$ such that $x_{i}^{2}=\boldsymbol{1}$ and
$y_{j}^{2}=-\boldsymbol{1}$. We generate $C\ell_{-3}$ with three odd elements
$a,b,c$, and embed it into $C\ell_{4,-2}$ as follows:

$a\to y_{1}\;,\quad b\to y_{2}\;,\quad c\to x_{1}x_{2}x_{3}\;.$ (38)

As a result, the commutant contains
$x_{4},~{}x_{1}x_{2}x_{4},~{}x_{1}x_{3}x_{4},~{}x_{2}x_{3}x_{4}$, what implies
${\cal Z}_{s}(C\ell_{-3},C\ell_{4,-2})\cong C\ell_{-3}$.

We proceed with the case of a $\phi$-irrep of the type
$\mathbb{C}\mathbb{C}_{2}$. Let $C\ell_{3,-1}$ be generated by
$x_{1},x_{2},x_{3},y$ such that $x_{i}^{2}=\boldsymbol{1}$ and
$y^{2}=-\boldsymbol{1}$. In its turn, $\mathbb{C}\ell_{1}$ as a real
superalgebra contains four elements: $\boldsymbol{1}$, $i$, $e$, $ie$ such
that $e$ and $ie$ are odd, commuting, and squaring to $\boldsymbol{1}$ and
$-\boldsymbol{1}$ correspondingly. There is the embedding of real
superalgebras:

$\displaystyle\boldsymbol{1}$
$\displaystyle\to\boldsymbol{1}\;,\quad\quad\;\;\;e\to x_{1}\;,$ (39)
$\displaystyle i$ $\displaystyle\to x_{2}x_{3}\;,\quad ie\to
x_{1}x_{2}x_{3}\;.$ (40)

One can verify that the supercommutant for this embedding ${\cal
Z}_{s}(\mathbb{C}\ell_{1},C\ell_{3,-1})$ contains $\boldsymbol{1}$,
$x_{2}x_{3}$, $y$, $x_{2}x_{3}y$, and can be identified with
$\mathbb{C}\ell_{1}$. The same computation can be performed for the type
$\mathbb{C}\mathbb{C}_{1}$, where ${\cal A}_{s}\sim\mathbb{C}\ell_{2}$ and
obtain ${\cal Z}_{s}(\mathbb{C}\ell_{2},C\ell_{3,-1})$. This finishes the
calculations needed to obtain Table 3. Clearly, we can use this technique in
order to reproduce the similarity type of ${\cal A}_{s}$ from ${\cal W}$. This
proves that $\gamma$ defined in (30) is an isomorphism of sets.

######  Remark 2.

The procedure of taking the graded commutant described above is nothing but
solving the following equation in the graded Brauer monoid of reals (see
Appendix C):

$[{\cal A}_{s}][{\cal W}]=[C\ell_{2}]\;.$ (41)

Taking the graded commutant within
$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}},I)$ can be seen as a reflection in
the graded Brauer monoid across the element 2.

Our proof can be recast into the following diagram, which is true for any
$\phi$-irrep $(\rho,V_{{\mathbb{R}}},I)$:

$(\rho,V_{{\mathbb{R}}},I)$Dyson Type$[{\cal A}_{s}]$$[{\cal
W}]$$\alpha$$\beta$$\gamma$$\delta$

where $\beta$ and $\gamma$ are isomorphisms of sets and the diagram commutes.
We explained how to construct the maps $\alpha$, $\beta$, and $\gamma$, and
showed that $\beta$ and $\gamma$ are bijective. The fact this is true for any
$\phi$-irrep of any ${\mathbb{Z}}_{2}$-graded group and makes the diagram
above commutative.

##  5 A Frobenius-Schur Indicator For $\phi$-irreps

In the original paper Dyson_1962 , Dyson defined classical Frobenius-Schur
indicators for the algebras ${\cal D}$ and ${\cal A}$. Recall, for any unitary
representation $(\rho^{\mathbb{C}},V)$ of the group $G^{0}$, we can define a
Frobenius-Schur indicator

$\Pi(\rho^{\mathbb{C}})=\frac{1}{|G^{0}|}\sum_{u\in
G^{0}}\mbox{Tr}(\rho^{\mathbb{C}}(u^{2}))\;.$ (42)

One can prove that the F-S indicator on any irreducible representation takes
one of the three values: $\\{-1,0,+1\\}$. According to the value of the F-S
indicator, we call an irrep potentially quaternionic, complex, or potentially
real. As we reviewed above, the $\phi$-irrep, restricted to its unitary piece,
might be reducible. However, it contains at most two components of the same
type. Then, we can apply the Frobenius-Schur indicator to any of the two
irreducible unitary subrepresentations with the same result. We always assume
that the F-S indicator is applied to only one component. The three possible
values of the F-S indicator correspond to one of the three possible similarity
types of the simple component of ${\cal A}$.

Further, Dyson (following V. Bargmann) introduced an indicator distinguishing
types of ${\cal D}$. It is defined for any $\phi$-irrep in the following way:

$\Pi^{\prime}(\rho^{\mathbb{C}})=\frac{1}{|G^{1}|}\sum_{a\in
G^{1}}\mbox{Tr}(\rho^{\mathbb{C}}(a^{2}))\;,$ (43)

It was proven by Dyson that this indicator also takes the three possible
values $\\{-1,0,+1\\}$, corresponding to the three possible types of ${\cal
D}$: $\mathbb{H}$, $\mathbb{C}$, and ${\mathbb{R}}$ respectively.

These two indicators can be assembled into one indicator

$\pi=\Pi^{\prime}(\rho^{\mathbb{C}})+i\Pi(\rho^{\mathbb{C}})\;.$ (44)

which has remarkable properties. Set $\omega=0$ if $\pi=0$ and
$\omega\equiv\pi/|\pi|$ when $\pi$ is nonzero. Now, the graded Brauer group of
reals $Br({\mathbb{R}})$ is isomorphic to ${\mathbb{Z}}_{8}$ which we identify
as the multiplicative group of eighth roots of unity. Choosing a suitable
isomorphism, the image of ${\cal W}$ in $Br({\mathbb{R}})$ will be identical
to the value of $\omega$. That is, $\omega$ takes the following values
depending on the Dyson Type of the $\phi$-irrep:

$\displaystyle\omega=\begin{cases}0\;,\quad\quad\quad\quad\quad\quad\mathbb{C}\mathbb{C}_{1},\;\mathbb{C}\mathbb{C}_{2}\;,\\\
\exp(2\pi in/8)\;,\quad\mbox{others}\;.\end{cases}$ (45)

Put differently, we have the following table:

${\cal W}$ $Type$ $\Pi^{\prime}$ $\Pi$ $\omega$ $C\ell_{1}$ $\mathbb{RR}$ $+1$
$+1$ $\exp(\scriptstyle{+1\cdot\frac{2\pi i}{8}})$ ${\mathbb{R}}$
$\mathbb{RC}$ $+1$ $0$ $\exp(\scriptstyle{0\cdot\frac{2\pi i}{8}})$
$C\ell_{-1}$ $\mathbb{RH}$ $+1$ $-1$ $\exp(\scriptstyle{-1\cdot\frac{2\pi
i}{8}})$ $C\ell_{2}$ $\mathbb{CR}$ $0$ $+1$
$\exp(\scriptstyle{+2\cdot\frac{2\pi i}{8}})$ $C\ell_{-2}$ $\mathbb{CH}$ $0$
$-1$ $\exp(\scriptstyle{-2\cdot\frac{2\pi i}{8}})$ $C\ell_{3}$ $\mathbb{HR}$
$-1$ $+1$ $\exp(\scriptstyle{+3\cdot\frac{2\pi i}{8}})$ $\mathbb{H}$
$\mathbb{HC}$ $-1$ $0$ $\exp(\scriptstyle{+4\cdot\frac{2\pi i}{8}})$
$C\ell_{-3}$ $\mathbb{HH}$ $-1$ $-1$ $\exp(\scriptstyle{-3\cdot\frac{2\pi
i}{8}})$

Table 4: The Dyson Types having real central graded commutants matched with
their Frobenius-Schur indicators.

A similar indicator was introduced in Gow1979 , and rediscovered recently in a
study of crossed product group superalgebras by Ichikawa and Tachikawa in
ichikawa2020super .

##  6 How To Construct Examples of Different Dyson Types of $\phi$-irreps From
Unitary Representations

Here, we discuss one particular way of obtaining $\phi$-irreps of a given type
from a complex representation.

The simplest example comes from a graded group being a direct product
${\mathbb{Z}}_{2}\times G^{0}$. Let $(\rho^{\mathbb{C}},V)$ be a complex
unitary irrep of the group $G^{0}$. In the following five cases, we map the
non-trivial element of ${\mathbb{Z}}_{2}$ to an element of ${\cal X}$ of the
given unitary irrep, or that of its double-copy. We begin with the type
${\mathbb{R}}{\mathbb{R}}$. If $\rho^{\mathbb{C}}$ is a potentially real
irrep, then the commutant ${\cal X}$ contains an antilinear operator squaring
to $+\boldsymbol{1}$. We simply include this operator to our irrep and obtain
a $\phi$-irrep of the type ${\mathbb{R}}{\mathbb{R}}$. In order to obtain the
type $\mathbb{H}{\mathbb{R}}$ from a potentially real irrep, we have to take
its double-copy
$(\rho^{\mathbb{C}}\otimes\boldsymbol{1}_{2},V\otimes{\mathbb{R}}^{2})$. This
way we enlarge the commutant and it now includes an antiunitary operator
squaring to $-\boldsymbol{1}$, which we use in order to get a $\phi$-irrep of
the type $\mathbb{H}{\mathbb{R}}$.

Assume $\rho^{\mathbb{C}}$ is a potentially quaternionic irrep. Then, we can
add the antilinear element of ${\cal X}$ squaring to $-\boldsymbol{1}$ and
obtain a $\phi$-irrep of the type ${\mathbb{R}}\mathbb{H}$. Analogously, we
can take a double-copy, such that the commutant contains an element squaring
to $+\boldsymbol{1}$. Adjoining this gives us the type $\mathbb{H}\mathbb{H}$.

Finally, the commutant of a “complexification” of a truly complex irrep always
contains an antilinear operator exchanging the two irreducible components,
that allows us to obtain a $\phi$-irrep of the type
$\mathbb{C}\mathbb{C}_{2}$. One can check that a complex structure still
belongs to the commutant ${\cal Z}$.

There is a universal way of constructing $\phi$-irreps from a complex-linear
irrep $(\rho^{\mathbb{C}},V)$ of a group $H$. We define $G^{0}\equiv H\times
H$ and its representation $(\rho^{\mathbb{C}}\oplus\rho^{\mathbb{C}},V\oplus
V)$. There is an automorphism of $G^{0}$ permuting the factors of $H$. Let us
denote the non-trivial generator of ${\mathbb{Z}}_{2}$ by $p$, then

$\displaystyle{\mathbb{Z}}_{2}$ $\displaystyle\to Aut(G^{0})$ (46)
$\displaystyle p:(h_{1},h_{2})$
$\displaystyle\mapsto(h_{2},h_{1})\;,\;\;\mbox{where}\;h_{1},h_{2}\in H\;,$
(47)

what allows us to define a semidirect product $G\equiv{\mathbb{Z}}_{2}\ltimes
G^{0}$. The group $G$ contains pairs $[t,(h_{1},h_{2})]$ with
$t\in{\mathbb{Z}}_{2}$ and is clearly ${\mathbb{Z}}_{2}$-graded. We define a
$\phi$-irrep of $G$ by the following rule:

$\rho([p,(h_{1},h_{2})]=\rho^{\mathbb{C}}(h_{2})J\oplus\rho^{\mathbb{C}}(h_{1})J\;,$
(48)

where $J$ is a complex-conjugation operator acting on $V$. If $I$ is the
complex structure for $V$, then the commutant ${\cal Z}$ contains precisely
one non-trivial operator $I\oplus(-I)$. On the other hand, the type of
$\rho^{\mathbb{C}}$ does not matter for this construction and we obtain any of
the types $\mathbb{C}{\mathbb{R}}$, $\mathbb{C}\mathbb{H}$, or
$\mathbb{C}\mathbb{C}_{1}$.

There are two remaining cases where $G$ should be taken as the general
semidirect product group. Given a truly-complex irreducible representation of
$G^{0}$, we can map the non-trivial generator of ${\mathbb{Z}}_{2}$ to a real
or quaternionic structure and obtain a $\phi$-irrep of the type
${\mathbb{R}}\mathbb{C}$ or $\mathbb{H}\mathbb{C}$ correspondingly.

###  6.1 A Special Case: Groups Containing “Time-Reversal”

In this section, we review Dyson’s treatment of $\phi$-irreps of
“factorizable” groups. It refers to groups of symmetries containing a
distinguished time-reversal operation commuting with all unitary
transformations. We note that our discussion of antiunitary symmetries is not
limited to operations of time-reversal, and time-reversal symmetries are not
limited to those commuting with unitary symmetries.

Until this point we made an assumption that the group $G$ is non-trivially
graded. We assumed that map 1 is surjective, or equivalently, the following
sequence is exact

${1}$${G^{0}}$${G}$${\mathbb{Z}_{2}}$${1}$$\scriptstyle{\phi}$

Let us consider a special case when $G$ is a semidirect products of $G^{0}$
with ${\mathbb{Z}}_{2}$, so that the sequence is split. In other words, there
exists a group homomorphism

$s:{\mathbb{Z}}_{2}\to G\;$ (49)

such that $\phi\circ s=id_{{\mathbb{Z}}_{2}}$. Then, $G$ is a split extension;
they are known to be classified by group homomorphisms

$\psi:{\mathbb{Z}}_{2}\to\mbox{Aut}(G^{0})\;,$ (50)

allowing as to define the semidirect product $G\cong
G^{0}\rtimes_{\psi}{\mathbb{Z}}_{2}$.

######  Remark 3.

The first example of a group admitting ${\mathbb{Z}}_{2}$-grading and not
being a semidirect product is the quaternionic group ${\cal Q}_{8}$. The
normal subgroup $G^{0}$ here is generated by $\\{\pm 1,\pm
i\\}\cong{\mathbb{Z}}_{4}$. Accordingly, the only order-2 subgroup of ${\cal
Q}_{8}$ is the one generated by $\\{\pm 1\\}$. Those two subgroups intersect
non-trivially and cannot form a semidirect product. Then, ${\cal Q}_{8}$ is
not isomorphic to ${\mathbb{Z}}_{4}\rtimes{\mathbb{Z}}_{2}$.

Furthermore, we make an assumption that every automorphism of $G^{0}$ is
inner, or equivalently, the group of outer automorphisms is trivial. Then,
there always exists $\tilde{g}\in G^{0}$ such that

$\psi(P)(g)=\tilde{g}^{-1}g\tilde{g}\;,\quad\mbox{for}\;g\in G^{0}\;.$ (51)

and $P$ is the non-trivial element of ${\mathbb{Z}}_{2}$. On the other hand,
we can use the splitting homomorphisms in order to construct the map (50)

$\psi(P)(g)=s(P)^{-1}g\;s(P)\;.$ (52)

For any semidirect product $G\cong G^{0}\rtimes_{\psi}{\mathbb{Z}}_{2}$, with
a group $G^{0}$ having only inner automorphisms, specified by the homomorphism
$s$ and $\tilde{g}$, we can define an operator $T=s(P)\tilde{g}^{-1}\in
G^{1}$. Matching 51 and 52, we see that $T$ commutes with $G^{0}$.

######  Remark 4.

The smallest example of a group which is a semidirect product and not every
automorphism is inner is the dihedral group of 8 elements $D_{4}$. Here,
$G^{0}={\mathbb{Z}}_{4}$ and it has one non-trivial outer automorphism, the
one used to build $D_{4}$. Seen geometrically, this is the group of symmetries
of the square. If we represent the elements involving reflections antiunitary,
there will be no distinguished “time-reversal”.

Let us note that such $T$ is well-defined in the case when both $G^{0}$ and
${\mathbb{Z}}_{2}$ are normal subgroups of $G$. In this case, $G\cong
G^{0}\times{\mathbb{Z}}_{2}$ and $T$ is simply the non-trivial generator of
${\mathbb{Z}}_{2}$. Then, we impose no restrictions on $G^{0}$. Another
situation, if $G^{0}$ has trivial group of outer automorphisms and trivial
center ($G^{0}$ is complete), $G$ is automatically a direct product of $G^{0}$
and ${\mathbb{Z}}_{2}$.

Let us call groups satisfying any of two properties above factorizable. For a
factorizable symmetry group, there always exists an operator $T$ commuting
with all unitary symmetries. This operator is canonically associated with the
operation of time-reversal in physics. If the symmetry group $G$ is a
semidirect product and not factorizable, one may voluntarily call any
antiunitary symmetry as the time-reversal, but there is no canonical choice.

Let us consider irreducible $\phi$-representation of a factorizable group. It
immediately follows for such $\phi$-irrep that ${\cal X}$ is not a one-
dimensional algebra. Indeed, the element $\rho(T)$ commutes with all ${\cal
A}$. However, this element does not belong to ${\cal Y}$ as it is
$\mathbb{C}$-antilinear. So, a $\phi$-irrep of a factorizable group cannot
have a purely even graded commutant.

We proceed further turning to group superalgebras. Here, $\rho(T)$ can be
associated with the element $u$ such that ${\cal A}_{s}^{1}=u{\cal A}^{0}$ and
$u\in{\cal X}$. Matching with Table 2 and Table 3, we imply that $\phi$-irreps
of factorizable groups are always of the following Dyson Types:
${\mathbb{R}}{\mathbb{R}}$, ${\mathbb{R}}\mathbb{H}$,
$\mathbb{C}\mathbb{C}_{2}$, $\mathbb{H}{\mathbb{R}}$, $\mathbb{H}\mathbb{H}$.
In conclusion, the group superalgebra of graded factorizable groups must be
similar to one of the following: $C\ell_{\pm 1},C\ell_{\pm
3},C\ell_{1,-1},C\ell_{4}$.

Finally, we can discuss the sign of $\rho(T)^{2}$ when $A_{s}^{0}$ is a
central simple algebra over ${\mathbb{R}}$. According to that,
$\rho(T)^{2}=+\boldsymbol{1}$ for the Dyson Types ${\mathbb{R}}{\mathbb{R}}$
and $\mathbb{H}\mathbb{H}$, and $\rho(T)^{2}=-\boldsymbol{1}$ for the types
${\mathbb{R}}\mathbb{H}$ and $\mathbb{H}{\mathbb{R}}$. This way, the sign of
$\rho(T)^{2}$ is fixed by the full Dyson Type of the $\phi$-irrep and not
solely by the type of ${\cal D}$ or ${\cal A}$.

##  Appendix A General Facts About Superalgebras

Here, we give basic definitions concerning superalgebras. Taking even
components everywhere, would give us definitions concerning ordinary algebras.
Vectors spaces below are defined over a field $\kappa$, for our needs it will
stand for ${\mathbb{R}}$ or $\mathbb{C}$. Throughout the paper, all
superalgebras are assumed to be unital and associative.

######  Definition A.1.

a) A $\kappa$-superalgebra $A$ is a vector superspace over $\kappa$ together
with a product $a\otimes a^{\prime}\to aa^{\prime}$. For homogeneous elements
of $A$,

$\mbox{deg}(aa^{\prime})=\mbox{deg}(a)+\mbox{deg}(a^{\prime})\;.$ (53)

b) Two homogeneous elements of a superalgebra are said to graded-commute, or
super-commute provided

$aa^{\prime}=(-1)^{\scriptstyle{deg}(a)\scriptstyle{deg}(a^{\prime})}a^{\prime}a$

If every pair of elements $a$ and $a^{\prime}$ in $A$ graded-commute, then the
superalgebra $A$ is called supercommutative.

######  Definition A.2.

Algebra $A^{\scriptstyle{opp}}$ is said to be opposite to $A$ if it consist of
the same elements, but multiplication is performed in opposite order.

######  Definition A.3.

The center ${\cal Z}(A)$ of a superalgebra $A$ is the maximal commutative
ungraded subalgebra of $A$:

$z\in{\cal Z}(A)\Leftrightarrow za=az\quad\mbox{for any $a\in A$}\;.$ (54)

The supercenter ${\cal Z}_{s}(A)$ of a superalgebra $A$ is the maximal
supercommutative graded subalgebra of $A$. Homogeneous elements of the
supercenter $z^{0},z^{1}\in A$ satisfy:

$\displaystyle z^{0}\in{\cal Z}_{s}^{0}(A)\Leftrightarrow
z^{0}a^{0}=a^{0}z^{0}\;,\quad\&\quad z^{0}a^{1}=+a^{1}z^{0}\;,$ (55)
$\displaystyle z^{1}\in{\cal Z}_{s}^{1}(A)\Leftrightarrow
z^{1}a^{0}=a^{0}z^{1}\;,\quad\&\quad z^{1}a^{1}=-a^{1}z^{1}\;.$ (56)

A superalgebra is called supercentral if ${\cal Z}_{s}(A)$ coincides with
$\kappa$.

######  Definition A.4.

Let $A$ and $B$ be superalgebras. We define the graded tensor product
$A\hat{\otimes}B$ as the superalgebra, which is the graded tensor product as a
vector space and the multiplication of homogeneous elements satisfy

$(a_{1}\hat{\otimes}b_{1})(a_{2}\hat{\otimes}b_{2})=(-1)^{\scriptstyle{deg}(a_{2})\scriptstyle{deg}(b_{1})}(a_{1}a_{2})\hat{\otimes}(b_{1}b_{2})\;.$
(57)

######  Definition A.5.

An ideal $I$ of superalgebra $A$ is called homogenious if it has the form
$I=I^{0}\oplus I^{1}$ such that $I^{0}\subset A^{0}$ and $I^{1}\subset A^{1}$.
A superalgebra $A$ is called simple if its only homogeneous two-sided ideals
are ${0}$ and $A$ itself.

Simple superalgebras are objects of our main interest. In their turn, Clifford
algebras are main examples of simple superalgebras.

##  Appendix B Clifford Algebras

######  Definition B.1.

Let $V$ be a vector space over a field $\kappa$ and let $B$ be a non-
degenerate symmetric bilinear form on $V$ valued in $\kappa$. The Clifford
algebra $C\ell(V_{\kappa},B)$ is the algebra over field $\kappa$ generated by
the basis ${e_{i}}$ for $V_{\kappa}$ and defined by relations

$e_{i}e_{j}+e_{j}e_{i}=2B(e_{i},e_{j})\;.$ (58)

Even subalgebra of $C\ell(V_{\kappa},B)$ consist of products of even numbers
of generators, including the base field, while odd part consists of products
of odd number of generators. The grading is consistent with multiplication
because the Clifford structure is given by the quadratic relations. For
Clifford algebras, it is easy to find the opposites using the sign rule
(remember that the multiplication is a bilinear map
$A\hat{\otimes}_{\kappa}A\to A$).

$C\ell(V_{\kappa},B)^{\scriptstyle{opp}}\cong C\ell(V_{\kappa},-B)\;.$ (59)

The most interesting examples for us are real and complex Clifford algebras.
Assume we fixed the basis of $V_{\kappa}$ such that $B$ is diagonal. Then,
over the reals this form has the signature $(r,s)$, where $r$ is the number of
positive elements on the diagonal and $s$ is the number of negative ones. We
denote the real Clifford algebra corresponding to the bilinear form of
signature $(r,s)$ by $C\ell_{r,-s}$. If the form is positive-definite
(negative-definite), then we use the following one-index notation:
$C\ell_{r,0}\equiv C\ell_{r}$ and $C\ell_{0,-s}\equiv C\ell_{-s}$.

Without loss of generality, we can diagonalize the form $B$ over the complex
numbers in such a way that it consists of $+1$’s on the diagonal. We denote
the complex Clifford algebra corresponding to a rank-$n$ bilinear form by
$\mathbb{C}\ell_{n}$.

As follows from the Clifford relations (58), the graded tensor product of two
Clifford algebras is a Clifford algebra:

$C\ell(V_{\kappa}^{\prime},B^{\prime})\hat{\otimes}_{\kappa}C\ell(V_{\kappa}^{\prime\prime},B^{\prime\prime})\cong
C\ell(V_{\kappa}^{\prime}\oplus V_{\kappa}^{\prime\prime},B^{\prime}\oplus
B^{\prime\prime})\;.$ (60)

##  Appendix C (Graded)-Brauer Groups

A theorem by Wedderburn provide us with a structure of simple
(super)-algebras. In order to formulate that theorem, we need the notion of a
division superalgebra.

######  Definition C.1.

A division superalgebra $A$ is a superalgebra such that any non-zero
homogeneous element is invertible.

######  Theorem C.1.

(super-Wedderburn) Any simple superalgebra over the field $\kappa$ is a matrix
superalgebra with coefficients in a division superalgebra over the field
$\kappa$.

A very good source to look up the proof is the book by V.S. Varadarajan
varadarajansupersymmetry . For our applications, we invoke the theorems by
Frobenius and Wall, listing all real division algebras and superalgebras
respectively.

######  Theorem C.2.

(Frobenius) There are three associative real division algebras:
$\mathbb{R},\mathbb{C}$, and $\mathbb{H}$.

######  Theorem C.3.

(Wall, Deligne) There are ten associative real division superalgebras: three
purely even superalgebras $\mathbb{R},\mathbb{C}$, and $\mathbb{H}$ and seven
Clifford superalgebras $\mathbb{C}\ell_{1},C\ell_{\pm 1},C\ell_{\pm
2},C\ell_{\pm 3}$.

We sketch a proof of the Deligne-Wall theorem below. A consequence of those
theorems is that any real simple algebra has the form
$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}})\otimes_{{\mathbb{R}}}\mathbb{K}$,
where $\mathbb{K}$ is one of the three algebras: ${\mathbb{R}}$, $\mathbb{C}$,
and $\mathbb{H}$. Any real simple superalgebra is isomorphic to
$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}}^{m|n})\hat{\otimes}_{{\mathbb{R}}}\mathbb{D}$,
where $\mathbb{D}$ is one of the ten real division superalgebras above.

The following relation is an important consequence of the Wedderburn theorem
(see e.g. varadarajansupersymmetry ):

$\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}}^{m|n})\;\hat{\otimes}\;\mathbb{D}\;\hat{\otimes}\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}}^{p|q})\;\hat{\otimes}\;\mathbb{D}^{\scriptstyle{opp}}\cong\mbox{End}_{{\mathbb{R}}}(V_{{\mathbb{R}}}^{mp+nq|mq+np})\hat{\otimes}\mbox{End}_{{\mathbb{R}}}(\mathbb{D})\;,$
(61)

where $\mbox{End}_{{\mathbb{R}}}(\mathbb{D})$ is the endomorphism superalgebra
of $\mathbb{D}$ considered as a vector superspace.

The (sueper)-Wedderburn theorem provides us with the general structure of
simple (super)-algebras over a field $\kappa$. It is suggestive to introduce
an equivalence relation on the superalgebras, such that $A$ and $B$ are
similar iff they are isomorphic to matrix superalgebras with coefficients in
the same division $\kappa$-superalgebra $\mathbb{D}$:

$A\sim B\iff
A\cong\mbox{End}_{\kappa}(V_{\kappa}^{m|n})\hat{\otimes}_{\kappa}\mathbb{D}\quad\&\quad
B\cong\mbox{End}_{\kappa}(V_{\kappa}^{p|q})\hat{\otimes}_{\kappa}\mathbb{D}\;.$
(62)

We denote the image of $A$ in the set of similarity classes by $[A]$.

######  Remark 5.

Throughout the paper, we interchangeably use the terms similarity classes and
Morita-classes. There is a general notion of Morita equivalence of algebras,
such that two (graded) algebras are Morita-equivalents iff their categories of
(graded) modules are equivalent. It turns out that the equivalence relation we
introduced is identical to the Morita-equivalence relation.

It is clear from the definition that the set of similarity classes of simple
(super)-algebras is in one-to-one correspondence with the set of division
(super)-algebras over $\kappa$. So, there are 3 classes of simple real
algebras and 10 classes of simple real superalgebras. As we will show later,
there is a unique class of simple complex algebras and two classes of simple
complex superalgebras. It is worth noting that the set of similarity classes
of simple algebras over $\kappa$ contain similarity classes of algebras over
all algebraic extensions of $\kappa$ up to its algebraic closure.

A natural operation on simple algebras is the tensor product. Over the real
numbers, we can write down the following tensor products

${\mathbb{R}}\otimes_{{\mathbb{R}}}{\mathbb{R}}\cong{\mathbb{R}},\quad\mathbb{H}\otimes_{{\mathbb{R}}}{\mathbb{R}}\cong\mathbb{H}\;,\quad\mathbb{H}\otimes_{{\mathbb{R}}}\mathbb{H}\cong\mbox{End}_{{\mathbb{R}}}({\mathbb{R}}^{4})\;,$
(63)

while
$\mathbb{C}\otimes_{{\mathbb{R}}}\mathbb{C}\cong\mathbb{C}\oplus\mathbb{C}$.
The subtlety here is that $\mathbb{C}$ is not a central simple algebra over
${\mathbb{R}}$. It is an important fact, which we do not prove here, that a
(graded) tensor product of two simple (super)-central (super)-algebras is a
simple (super)-central (super)-algebras. Moreover, Morita-classes of
(super)-central simple (super)-algebras (CSA, SCSSA) form a group. As we
mentioned, CSA (SCSSA) are closed under (graded) tensor product, and there
always exists an opposite due to the relation (61).

Finally, we can list the four relevant (graded) Brauer groups:

$Br({\mathbb{R}})\cong{\mathbb{Z}}_{2}\;,\quad
Br(\mathbb{C})\cong{\mathbb{Z}}_{1}\;,\quad
sBr({\mathbb{R}})\cong{\mathbb{Z}}_{8}\;,\quad
sBr(\mathbb{C})\cong{\mathbb{Z}}_{2}\;.$ (64)

Further, we notice that a tensor product over ${\mathbb{R}}$ of a real CSA
with a complex CSA gives us a complex CSA. Then, a pair of the Brauer groups
above form a monoid, see the blog by John Baez Baez2014 . We call it a Brauer
monoid $BR({\mathbb{R}})$: as a set, it is a disjoint union
$BR({\mathbb{R}})\cong Br({\mathbb{R}})\coprod Br(\mathbb{C})$. The Brauer
monoid of reals is isomorphic to the multiplicative monoid of three elements:

$BR({\mathbb{R}})\cong\\{-1,0,1\\}\;,$ (65)

where we identify $[\mathbb{H}]\to-1$, $[\mathbb{C}]\to 0$,
$[{\mathbb{R}}]\to+1$. Analogously, we can form a monoid $sBR({\mathbb{R}})$
of real and complex graded Brauer groups. $sBR({\mathbb{R}})$ is isomorphic to
the additive monoid on ten elements
$\\{0,1,2,3,4,5,6,7,8,\textbf{0},\textbf{1}\\}$, where we identify

$[C\ell_{r,-s}]\to r-|s|\;\mbox{mod}\;8\;,$ (66)
$[\mathbb{C}\ell_{n}]\to\textbf{n}\;\mbox{mod}\;2\;.$ (67)

Addition of lightface numbers is the same as in $\mathbb{Z}_{8}$, and addition
of boldface numbers is the same as in $\mathbb{Z}_{2}$. In order to add a
lightface number $x$ and a boldface number y, we should firstly project $x$
into $\mathbb{Z}_{2}$. For example, in this monoid $2+\textbf{1}=\textbf{1}$.

###  C.1 Wall’s theorem

In this section, we prove Walls’ theorem Wall1964 , following a more up-to-
date treatment by Deligne Deligne_noteson .

Let us first notice that the only division algebra $A_{\bar{\kappa}}$ over an
algebraically closed field $\bar{\kappa}$ is $\bar{\kappa}$ itself. Any
division algebra should be finite-dimensional, so let us consider a minimal
polynomial containing powers of $x\in A_{\bar{\kappa}}$.

$f(x)=x^{n}+a_{1}x^{n-1}+...a_{n}=0\;.$

Since $\bar{\kappa}$ is algebraically closed, $f(x)$ has at least one root $b$
and can be factorized $f(x)=(x-b)g(x)=0$. Since $f(x)$ was minimal, $g(x)\neq
0$ and $x=b$, what implies $x\in\bar{\kappa}$. So, the only division
$\bar{\kappa}$-algebra is $\bar{\kappa}$ itself.

Now, let us classify all division superalgebras over $\bar{\kappa}$. We recall
that the definition of a division superalgebra requires homogeneous non-zero
elements to be invertible. We can apply the same logic as in the previous
paragraph to even and odd components of a division $\bar{\kappa}$-superalgebra
and obtain the unique algebra $\bar{\kappa}[\epsilon]$, where $\epsilon$ is
odd and squares to 1. In particular, we obtain all complex division
superalgebras: $\mathbb{C}$ and $\mathbb{C}\ell_{1}$.

######  Theorem C.4.

(Wall, Deligne) The graded Brauer group $sBr({\mathbb{R}})$ is an iterated
extension of $\mathbb{Z}_{2}$ by ${\mathbb{R}}^{*}/{\mathbb{R}}^{*2}$ by the
ordinary group $Br({\mathbb{R}})$.

More explicitly, the theorem says that $sBr({\mathbb{R}})$ fits into the exact
sequence:

${1}$${sBr({\mathbb{R}})^{\prime}}$${sBr({\mathbb{R}})}$${\mathbb{Z}_{2}}$${1}$$\scriptstyle{\phi_{1}}$$\scriptstyle{\phi_{2}}$
(68)

while $sBr({\mathbb{R}})^{\prime}$ itself fits into the following exact
sequence:

${1}$${Br({\mathbb{R}})}$${sBr({\mathbb{R}})^{\prime}}$${{\mathbb{Z}}_{2}}$${1}$$\scriptstyle{\phi_{3}}$$\scriptstyle{\phi_{4}}$
(69)

###### Proof.

We first show the existence of the group homomorphism $\phi_{2}$. This map is
just an extension of scalars, it maps all CS superalgebras belonging to a
given similarity class to their complexifications. It induces the group
homomorphism: $sBr({\mathbb{R}})\to sBr(\mathbb{C})\cong\mathbb{Z}_{2}$, which
is surjective as $\mathbb{C}[\epsilon]$ is the image of
${\mathbb{R}}[\epsilon]$.

To make the sequence exact, the first term $sBr(k)^{\prime}$ has to be
isomorphic to the kernel of $\phi_{2}$. These are similarity classes of SCSSA
such that, after extension of scalars,

$A\to A\otimes{\mathbb{C}}\equiv A_{\mathbb{C}}$ (70)

become isomorphic to $\mbox{End}(\mathbb{C}^{m|n})$ .

Now, let us analyze $sBr(k)^{\prime}$. A matrix superalgebra
$\mbox{End}(\mathbb{C}^{m|n})$ has precisely two simple graded modules:
$\mathbb{C}^{m|n}$ and $\Pi\mathbb{C}^{m|n}\cong\mathbb{C}^{n|m}$. Let us
denote the set of simple modules over $A_{\mathbb{C}}$ by $I(A)$. We can
notice that simple modules over $A_{\mathbb{C}}$ can be seen as simple
$A$-modules. Also, $I(A)$ admits an action of the Galois group
$Gal(\mathbb{C}/{\mathbb{R}})\cong{\mathbb{Z}}_{2}$. Reversing the complex
structure does not necessarily lead to the same module, so that the Galois
action might exchange the elements of $I(A)$. Thus, we have a map

$\alpha_{A}:Gal(\mathbb{C}/{\mathbb{R}})\to\mathbb{Z}_{2}\;,$

where ${\mathbb{Z}}_{2}$ is the group of permutations of the two elements of
$I(A)$.

If $[A_{1}]\in sBr({\mathbb{R}})^{\prime}$ and $[A_{2}]\in
sBr({\mathbb{R}})^{\prime}$, then $[A_{1}\hat{\otimes}A_{2}]\in
sBr({\mathbb{R}})^{\prime}$. Also, one can check that
$\alpha_{A^{\prime}\otimes
A^{\prime\prime}}=\alpha_{A^{\prime}}+\alpha_{A^{\prime\prime}}$, so that the
map

$A\to\alpha_{A}$ (71)

descends to

$sBr({\mathbb{R}})^{\prime}\to\mbox{Hom}(Gal(\mathbb{C}/{\mathbb{R}}),\mathbb{Z}_{2})\cong{\mathbb{Z}}_{2}\;.$
(72)

In order to show that this is a surjective homomorphism, we provide an example
of a non-trivial element of
$\mbox{Hom}(Gal(\mathbb{C}/{\mathbb{R}}),\mathbb{Z}_{2})$. Assume
$A=C\ell_{2}$, its complexification
$\mathbb{C}\ell_{2}\cong\mbox{End}(\mathbb{C}^{1|1})$ has the following graded
representations (we pick a homogeneous basis):

$\rho_{1}^{\mathbb{C}}(e_{1})=\begin{pmatrix}0&1\\\ 1&0\\\
\end{pmatrix}\;,\quad\rho_{1}^{\mathbb{C}}(e_{2})=\begin{pmatrix}0&-i\\\
i&0\\\ \end{pmatrix}$ (73) $\rho_{2}^{\mathbb{C}}(e_{1})=\begin{pmatrix}0&1\\\
1&0\\\
\end{pmatrix}\;,\quad\rho_{2}^{\mathbb{C}}(e_{2})=-\begin{pmatrix}0&-i\\\
i&0\\\ \end{pmatrix}$ (74)

One can check that those two super-reps are non-equivalent. There is a
standard argument: there exists an element $e_{1}e_{2}\in{\cal
Z}(\mathbb{C}\ell_{2}^{0})$, which acts as a multiplication by $\pm i$ on the
even subrepresentations. Reversing the complex structure reverses the sign of
$\rho_{i}(e_{1})\rho_{i}(e_{2})$, what means that the Galois action permutes
the irreps inside of the super-irrep. However, permutation of even subspaces
is not an isomorphism of super-representations, so that complex conjugated
representations of $\mathbb{C}\ell_{2}$ are not equivalent.

The argument with a center holds for a general $A$ such that $[A]\in
sBr({\mathbb{R}})^{\prime}$. It is always true that ${\cal Z}(A^{0})$ acts on
even subspaces of elements from $I(A)$ by multiplication by characters
$(\chi_{1},\chi_{2})$ (where by a character of the module $M$ we mean a module
homomorphism $\mbox{Hom}(M,\mathbb{C})$). The Galois action is trivial iff one
of the characters is zero or they are both real numbers. The first case
corresponds to a purely even $A$, while the second case correspond to real CS
superalgebras having center isomorphic to ${\mathbb{R}}\oplus{\mathbb{R}}$.
Indeed, complexification here is a simple multiplication by the complex unit
and complex conjugation does not permute the characters.

Let us take any $[A]\in sBr({\mathbb{R}})^{\prime}$. The class of $A$ in
$sBr({\mathbb{R}})^{\prime}$ is that of some real division superalgebra
$\mathbb{D}$. If ${\cal Z}(A^{0})\cong{\mathbb{R}}\oplus{\mathbb{R}}$, then
${\cal Z}(\mathbb{D}^{0})\cong{\mathbb{R}}\oplus{\mathbb{R}}$, which is
impossible as $\mathbb{D}^{0}$ is a real division algebra. So, the kernel of
$\phi_{4}$ contains algebras similar to some real division algebra, which is
nothing but the ordinary Brauer group of reals. ∎

Using theorem C.4, we derive $|sBr({\mathbb{R}})|=8$. The simplest
representative is $C\ell_{1}$, which is clearly a real division superalgebra.
In fact, tensor powers of $C\ell_{1}$ exhaust all similarity classes of
central simple superalgebras. The proof of this statement is very clearly
presented in varadarajansupersymmetry . We also recommend a paper by Todd
Trimble Trimble2005 , where the theorem of Wall is proven by construction.

## References

  * (1) A. Atland and M. Zirnbauer, “Nonstandard Symmetry Classes In Mesoscopic Normal- Superconducting Hybrid Systems”, Phys. Rev. B 55, 1142-1161 (1997).
  * (2) V. Bargmann, “Note on Wigner’s Theorem on Symmetry Operations”, Journal of Mathematical Physics 5, 862 (1964).
  * (3) J. Baez “The Ten-Fold Way”, Blog Pages: 1, 2, 3, 4; arXiv:mat.RA/2011.14234.
  * (4) R. Gow, “Real-valued and 2-rational group characters”, J. Algebra 61 (1979) 388–413.
  * (5) P. Deligne, “Notes on Spinors” in: Quantum Fields and Strings: a Course for Mathematicians, AMS, (1999).
  * (6) F. Dyson, “The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics”, Journal of Mathematical Physics (1962).
  * (7) L. Fidkowski and A. Kitaev, “The effects of interactions on the topological classification of free fermion systems”, Phys. Rev. B 81, 134509 (2010) arXiv:cond-mat.str-el/0904.2197.
  * (8) D. S. Freed, “On Wigner’s Theorem”, arXiv:mat-ph/1112.2133.
  * (9) D. S. Freed and M. J. Hopkins, “Reflection positivity and invertible topological phases”, arXiv:hep-th/1604.06527.
  * (10) D. S. Freed and G. W. Moore, “Twisted equivariant matter”, Annales Henri Poincare 14, 1927-2023 (2013) arXiv:hep-th/1208.5055.
  * (11) P. Heinzner, A. Huckleberry and M. R. Zirnbauer, “Symmetry classes of disordered fermions”, Commun. Math. Phys. 257, 725-771 (2005) arXiv:math-ph/0411040.
  * (12) T. Ichikawa and Y. Tachikawa, “The super Frobenius-Schur indicator and finite group gauge theories on pin- surfaces”, arXiv:math.RT/2002.10642.
  * (13) G. W. Moore, ‘‘Quantum Symmetries and Compatible Hamiltonians’’, http://www.physics.rutgers.edu/ gmoore/695Fall2013/CHAPTER1-QUANTUMSYMMETRY-OCT5.pdf.
  * (14) G. W. Moore, ‘‘Quantum Symmetries and K-Theory’’, https://www.physics.rutgers.edu/ gmoore/PiTP-LecturesA.pdf.
  * (15) K. H. Neeb and G. Olafsson, ‘‘Antiunitary representations and modular theory’’, arXiv:math.RT/1704.01336.
  * (16) D. Rumynin and J. Taylor ‘‘Real Representations of C2-Graded Groups: The Antilinear Theory’’, arXiv:math.RT/2006.09765.
  * (17) T. Trimble ‘‘The Super Brauer Group and Super Division Algebras’’, web page.
  * (18) V. S. Varadarajan, ‘‘Supersymmetry for Mathematicians: An Introduction’’, AMS, (2004).
  * (19) C. T. C. Wall, ‘‘Graded Brauer Groups’’, Journal für die reine und angewandte Mathematik 213, (1964).
  * (20) S. Weinberg, ‘‘The Theory of Quantum Fields’’, Vol. I, CUP, (1995).
  * (21) E. P. Wigner, ‘‘Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren’’, Spinger, (1931).
  * (22) M. R. Zirnbauer, ‘‘On The Symmetry Classification Of Topological Insulators And Superconductors’’, Colloquium Talk.
  * (23) M. R. Zirnbauer, ‘‘Particle-Hole Symmetries in Condensed Matter’’, arXiv:math-ph/2004.07107.


# Comment on ”Universal Bound on Energy Cost of Bit Reset in Finite Time”

Jie Gu  Chengdu Research Institute of Education Science, Chengdu, Sichuan
610000, China  X.G. Zhang  Department of Physics, Center for Molecular
Magnetic Quantum Materials and Quantum Theory Project, University of Florida,
Gainesville, Florida 32611, USA

In Ref. [1], as a crucial step to their main result Eq. (5), the authors
proved an important identity (Eq. (6) therein),

$W_{\text{pn}}=T\left[\Delta S_{1}(p\|\gamma)+\Sigma\right],$ (1)

where $k_{B}=1$, $\Delta S_{1}=\sum_{i}p_{i}\ln(p_{i}/\gamma_{i})$ is the
relative entropy, and $\gamma$ the thermal state. This formula points out
that, the work penalty (also called dissipated work in, e.g., [2]), which is
the irreversible work on top of the quasi-static work due to finite-time
process, consists of two contributions: the change in the relative entropy
with respect to the thermal states and the entropy production. As detailed in
its Supplemental Material, Eq. (1) was proved by decomposing
$\dot{S}_{1}(p(t)\|\gamma(t))$ into two terms and identifying each term as
$\dot{\Sigma}$ and $\beta\dot{W}_{\text{pn}}$ respectively. Here we present a
quantum-mechanical generalization of Eq. (1) with an alternative derivation.

Consider an open quantum system interacting with a heat bath. The two end
points, i.e., the density matrices $\rho(0)$ and $\rho(\tau)$, are fixed. The
average work performed on the system associated with the quasi-static protocol
is

$W_{\text{qs}}=\Delta F,$ (2)

where $F$ is the equilibrium free energy, given by $F:=-T\ln Z$, and $Z$ the
partition function $Z:=\text{Tr}(-e^{\beta\hat{H}})$. The work associated to a
generic driving protocol is

$W=\Delta E-Q\\\ =\Delta E-T(\Delta S_{1}-\Sigma)\\\ =\Delta F_{1}+T\Sigma,$
(3)

where we have used the first law of thermodynamics, the definition of entropy
production $\Sigma=\Delta S_{1}-Q/T$, and the definition of nonequilibrium
1-free energy $F_{1}=E-TS_{1}$. The work penalty is then

$W_{\text{pn}}:=W-W_{\text{qs}}=T\left[\Delta
S_{1}(\rho||\rho^{\text{G}})+\Sigma\right],$ (4)

where we have plugged in Eq. (2) and rewritten the nonequilibrium free energy
as $F_{1}=TS_{1}(\rho\|\rho^{\text{G}})+F$ (see, e.g., [3]). Here
$\rho^{\text{G}}$ denotes the Gibbs state. This equation is universal and
applicable in quantum processes. By taking only diagonal elements in the
energy eigenbasis of the Hamiltonian, the equation above reduces to its
classical counterpart, Eq. (1).

## References

  * [1] Yi-Zheng Zhen, Dario Egloff, Kavan Modi, and Oscar Dahlsten.  Universal Bound on Energy Cost of Bit Reset in Finite Time.  Phys. Rev. Lett., 127(19):190602, nov 2021.
  * [2] Harry JD Miller, Matteo Scandi, Janet Anders, and Martí Perarnau-Llobet.  Work fluctuations in slow processes: quantum signatures and optimal control.  Phys. Rev. Lett., 123(23):230603, 2019.
  * [3] Takahiro Sagawa.  Entropy, Divergence, and Majorization in Classical and Quantum Thermodynamics.  jul 2020.