Topological Effects on Non-Relativistic Eigenvalue Solutions Under AB-Flux
Field with Pseudoharmonic- and Mie-type Potentials

Faizuddin<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of Physics, University of Science & Technology Meghalaya, Ri-Bhoi,
Meghalaya-793101, India

###### Abstract

: In this paper, we investigate the quantum dynamics of a Schrödinger particle
confined by the Aharonov-Bohm flux field with pseudoharmonic-type potential
under topological defects produced by a point-like global monopole. We solve
the three-dimensional radial Schrödinger wave equation analytically and
determines the exact eigenvalue solution of the quantum system. Afterwards, we
consider Mie-type potential in the quantum system and solve the radial
Schrödinger equation analytically. We then analyze the effects of topological
defects of the geometry and the magnetic flux field with these potentials on
the eigenvalue solutions. We show that the eigenvalue solutions shift more due
to the magnetic flux in addition to the topological defects which gives an
analogue of the Aharonov-Bohm effect for the bound state. Finally, we utilize
the eigenvalue solutions to some diatomic molecular potentials of physical
interest and analyze the topological effects and the magnetic flux field on
them.

Keywords: Topological defect, Non-relativistic wave equation, physical
potential, solutions of wave-equation: bound-state, geometric quantum phase,
special function

PACS Number(s): 03.65.Pm, 03.65.Ge, 14.80.Hv

##  1 Introduction

The searching for exact or approximate eigenvalue solutions of the non-
relativistic and relativistic wave equations have turned to become a principal
part from the beginning of quantum mechanics bb1 . These exact solutions are
important in different fields, such as atomic physics, nuclear and high energy
physics, quantum electrodynamics, and theory of molecular vibrations bb2 ; bb5
; bb8 ; bb9 ; bb11 ; bb12 ; bb14 ; bb15 ; bb16 . The exact or approximate
eigenvalue solutions of the stationary Schrödinger wave equation plays
important role to examine the correctness of models and approximations in
computational physics as well as in chemistry. In addition, the particle
interacts with potential of various kinds are used to describe importance of
several physical systems since they contain all the necessary information of a
quantum state under investigation. However, analytic solutions are possible
only for a few simple quantum systems such as the hydrogen atom and harmonic
oscillator problem that are given in many textbooks bb1 ; bb17 ; bb18 ; bb19 ;
bb20 ; bb21 . To obtain the exact and approximate solutions of the wave
equations, various methods or techniques have employed, such as the
Nikiforov–Uvarov method bb22 ; bb23 ; bb24 ; bb25 ; bb26 ; bb27  and its
functional analysis, supersymmetry quantum mechanics bb28 ; bb29 , asymptotic
iteration method bb30 ; bb31  and many more.

Furthermore, the study of quantum motions of charged particles in the presence
of an external uniform magnetic and Aharonov-Bohm (AB) flux fields, which are
perpendicular to the plane where the particles are confined have been carried
out since past few years. In fact, the study of non-relativistic and
relativistic charged particles that are confined to the magnetic field has
been growing research interested because of possible applications in different
fields, such as in graphene bb15 ; bb16 ; bb33 , semiconductor structures bb36
, environmental sciences bb37 , chemical physics bb38 , biology bb39 ,
molecular vibrational and rotational spectroscopy of molecular physics bb40 ,
and in cosmic string bb41 . Noted that the presence of the Aharonov-Bohm flux
field in quantum mechanical systems shows an analogue to the Aharonov-Bohm
(AB) effect YA ; MP . This AB-effect is a quantum mechanical phenomenon in
which an electrically charged particle is affected by flux field despite being
confined to a region in which both the magnetic and electric fields are zero.
The experimental confirmation of its existence was presented in Ref. NO .

Another kind of system that has been studied in quantum mechanical problems
are the topological defects which are produced in the early universe through
symmetry breaking TWBK . Topological defects are classified into cosmic
strings SZ2 , domain walls, and global monopoles MBAV ; ERBM . The quantum
mechanical systems in the background space-time generated by a cosmic string
space-time have been studied by several authors (see, Ref. bb41  and related
references there in). The spherically symmetric static point-like global
monopole space-time was first presented in Ref. MBAV  and have been studied by
researchers both in the relativistic limit (see, Refs. ALCO2 ; ALCO ; SR ),
and only a handful works in the non-relativistic limit CF ; CF2 ; VBB ; PN ;
FA ; FA2  in the quantum system.

In this work, we will study the non-relativistic particles under the influence
of the Aharonov-Bohm flux field in a point-like global monopole with
pseudoharmonic-type and then Mie-type potentials. In fact, we determine the
exact eigenvalue solutions with these potentials analytically that would be
useful for some physical systems. We apply these eigenvalue solutions for some
known interaction potential models and show that topological defects of the
point-like global monopole and the magnetic flux modified the results in
comparison to the flat space results with these potentials in the literature.

This paper is organised as follows: in section 2, we discuss the three-
dimensional radial Schrodinger wave equation in a point-like global monopole
geometry in the presence of AB-flux field with a potential. Then we solve this
radial equation with two types of potential, namely a pseudoharmonic-type
potential (sub-section 2.1) and the Mie-type potential (sub-section 2.2). In
section 3, we discuss few applications of the quantum system; and in section
4, we present our results. Throughout the analysis, we have used the natural
units $c=1=\hbar$.

##  2 Topological Effects on Radial Solution Under AB-flux Field with
Potential

We study the quantum motions of non-relativistic particle with potential in
the presence of AB-flux field in point-like global monopole. Afterwards, we
consider Mie-type potential and obtain the eigenvalue solution analytically
and discuss the effects of topological defects as well as the magnetic flux on
the eigenvalue solutions.

A static and spherically symmetric metric of point-like global monopole in the
spherical coordinates $(t,r,\phi,\theta)$ is given by CF ; CF2 ; ALCO ; SR ;
FA ; FA2 ; PN

$ds^{2}=-dt^{2}+\frac{dr^{2}}{\alpha^{2}}+r^{2}\,(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}),$
(1)

where $\alpha<1$ represents topological defect parameter of point-like global
monopole (PGM) space-time. This point-like global monopole space-time have
some interesting features including a naked curvature singularity on the axis
$R=R^{\,\mu}_{\mu}=\frac{2\,(1-\alpha^{2})}{r^{2}}$. Other properties of this
geometry are given in detail in Refs. ALCO ; SR .

The time-dependent Schrödinger wave equation in the presence of an
electromagnetic potential $\vec{A}$ is described by the following wave
equation CF ; CF2 ; PN ; FA ; FA2

$\displaystyle\Big{[}-\frac{1}{2\,M}\,\Big{(}\frac{1}{\sqrt{g}}\,D_{i}\,(\sqrt{g}\,g^{ij}\,D_{j})\Big{)}+V(r)\Big{]}\,\Psi=i\,\partial_{t}\,\Psi,$
(2)

where $E$ is the energy of the particle, $g$ is the determinant of the metric
tensor $g_{ij}$ with $g^{ij}$ its inverse, $D_{i}=(\partial_{i}-i\,e\,A_{i})$
with $e$ is the electric charges.

By the method of separation of variables, one can always write the total wave
function $\Psi(t,r,\theta,\phi)$ in terms of different variables. Suppose, we
choose a possible wave function in terms of a radial wave function $\psi(r)$
as :

$\Psi(t,r,\theta,\phi)=e^{-i\,E\,t}\,Y_{l,m}(\theta,\phi)\,\psi(r),$ (3)

where $Y_{l,m}(\theta,\phi)$ is the spherical harmonic function, and $l,m$ are
respectively the orbital and magnetic moment quantum numbers.

In this analysis, we have chosen the three-vector electromagnetic potential
$\vec{A}$ given by Refs. ALCO ; SR ; FA ; FA2

$A_{r}=0=A_{\theta},\quad
A_{\phi}=\frac{\Phi_{B}}{2\,\pi\,r\,\sin\theta},\quad\Phi_{B}=\Phi\,\Phi_{0},\quad\Phi_{0}=2\,\pi\,e^{-1},$
(4)

where $\Phi_{B}=const$ is the Aharonov-Bohm magnetic flux, $\Phi_{0}$ is the
quantum of magnetic flux, and $\Phi$ is the amount of magnetic flux which is a
positive integer. Note that the Aharonov–Bohm effect YA ; MP  is a quantum
mechanical phenomena and has been investigated in different branches of
physics including quantum systems (see, Ref. SR  and related references there
in).

Thereby, in the space-time background (1) and using Eqs. (3)–(4) into the Eq.
(2), we obtain the following differential equations for $\psi$ and $Y_{l,m}$,
respectively as

$\displaystyle\frac{1}{r^{2}}\,\frac{\partial}{\partial
r}\,\left(r^{2}\,\frac{\partial\psi}{\partial\,r}\right)+\left[\frac{2\,M}{\alpha^{2}}\,\Big{(}E-V(r)\Big{)}-\frac{l^{\prime}\,(l^{\prime}+1)}{\alpha^{2}\,r^{2}}\right]\,\psi=0,$
$\displaystyle\left[\frac{1}{\sin\theta}\,\frac{\partial}{\partial\theta}\,\left(\sin\theta\,\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\,\left(\frac{\partial}{\partial\phi}-i\,\Phi\right)^{2}+l^{\prime}\,(l^{\prime}+1)\right]\,Y_{l,m}=0,$
(5)

where $l^{\prime}=(l-\Phi)$ is the effective orbital quantum number.

The radial Schrödinger equation can be written as

$\displaystyle\psi^{\prime\prime}(r)+\frac{2}{r}\,\psi^{\prime}(r)+\frac{1}{\alpha^{2}}\,\Bigg{[}-\frac{l^{\prime}\,(l^{\prime}+1)}{r^{2}}+2\,M\,\Big{(}E-V(r)\Big{)}\Bigg{]}\,\psi(r)=0.$
(6)

The effective potential of the system (by changing the function
$\psi(r)=\frac{R(r)}{r}$ into the Eq. (8), one can find one dimensional
Schrodinger equation) is given by

$V_{eff}(r)=\frac{1}{\alpha^{2}}\,\Bigg{[}V(r)+\frac{(l-\Phi)\,(l-\Phi+1)}{2\,M\,r^{2}}\Bigg{]}.$
(7)

We can see that the effective potential of the quantum system depends on the
topological defects characterise by the parameter $\alpha$, and the magnetic
flux field $\Phi_{AB}$.

In this analysis, we have considered the following two different potentials of
physical interest in this quantum system and determines exact eigenvalue
solutions of the non-relativistic wave equation analytically in point-like
global monopole under AB-flux field.

###  2.1 Effects of Pseudoharmonic-type Potential

(a) $l=1=M=\beta=\gamma=V_{0}$, $\Phi=3/4$.

(b) $l=1=M=\beta=\gamma=V_{0}$, $\alpha=3/4$.

(c) $l=1=M=\beta=\gamma=V_{0}$.

(d) $l=1=M=\beta=\gamma=V_{0}$, $\alpha=3/4$, $\Phi=1$.

Figure 1: Effective potential of the quantum system using Pseudo-harmonic
potential.

In this section, we are interested on the following potential $V(r)$
superposition of harmonic oscillator and inverse quadratic potential given by
SMI ; SHD ; KJO ; SMI3 ; VK

$V(r)=V_{0}+\beta\,r^{2}+\frac{\gamma}{r^{2}},$ (8)

where $\beta,\gamma$ characterise the potential parameters and $V_{0}>0$ is a
constant potential term.

Thereby, substituting the potential (8) into the Eq. (7), we have obtained the
following radial equation:

$\psi^{\prime\prime}(r)+\frac{2}{r}\,\psi^{\prime}(r)+\Big{(}\Lambda-\frac{j^{2}}{r^{2}}-\omega^{2}\,r^{2}\Big{)}\,\psi(r)=0,$
(9)

where we set the parameters

$\displaystyle\Lambda=\frac{2\,M\,(E-V_{0})}{\alpha^{2}}\quad,\quad
j=\sqrt{\frac{(l-\Phi)\,(l-\Phi+1)+2\,M\,\gamma}{\alpha^{2}}}\quad,\quad\omega=\sqrt{\frac{2\,M\,\beta}{\alpha^{2}}}.$
(10)

The effective potential of the system is given by

$V_{eff}(r)=\frac{1}{\alpha^{2}}\,\Bigg{[}V_{0}+\beta\,r^{2}+\frac{\gamma}{r^{2}}+\frac{(l-\Phi)\,(l-\Phi+1)}{2\,M\,r^{2}}\Bigg{]}.$
(11)

We see that the effective potential under consideration depends on the
topological defect, the background curvature as well as the magnetic flux. We
have plotted few graphs (fig. 1) showing the influences of various factors on
the effective potential.

Transforming the above equation (9) via $\psi(r)=\frac{U(r)}{r^{3/2}}$, we
have

$U^{\prime\prime}(r)-\frac{1}{r}\,U^{\prime}(r)+\Bigg{[}\Lambda-\frac{(j^{2}-\frac{3}{4})}{r^{2}}-\omega^{2}\,r^{2}\Bigg{]}\,U(r)=0.$
(12)

Introducing a new variables via $s=\omega\,r^{2}$ into the above Eq. (12), we
have obtained the following second order differential equation:

$U^{\prime\prime}(s)+\left(\frac{1-4\,\mu^{2}}{4\,s^{2}}\right)\,U(s)+\frac{\nu}{s}\,U(s)-\frac{1}{4}\,U(s)=0,$
(13)

where different parameters are defined as

$2\,\mu=\sqrt{j^{2}+\frac{1}{4}}\quad,\quad\nu=\frac{\Lambda}{4\,\omega}.$
(14)

Equation (14) is the Whittaker differential equation KDM  and $U(s)$ is the
Whittaker function which can be written in terms of the confluent
hypergeometric function ${}_{1}F_{1}(s)$ as

$U(s)=s^{\frac{1}{2}+\mu}\,e^{-\frac{s}{2}}\,{}_{1}F_{1}\left(\mu-\nu+\frac{1}{2},2\,\mu+1;s\right).$
(15)

Since we are searching the bound-state solutions, thus the confluent
hypergeometric function
${}_{1}F_{1}\left(\mu-\nu+\frac{1}{2},2\,\mu+1;s\right)$ is a power series
solution of degree $n$, and the quantity $\Big{(}\mu-\nu+\frac{1}{2}\Big{)}$
must be a negative integer, that is, $\Big{(}\mu-\nu+\frac{1}{2}\Big{)}=-n$,
where $n=0,1,2,..$. Simplification of this condition will results the
following eigenvalue expression

$\displaystyle
E_{n,l}=V_{0}+\alpha\,\sqrt{\frac{8\,\beta}{M}}\,\Bigg{(}n+\frac{1}{2\,\alpha}\,\sqrt{(l-\Phi)\,(l-\Phi+1)+2\,M\,\gamma+\frac{\alpha^{2}}{4}}+\frac{1}{2}\Bigg{)}.$
(16)

The normalized radial wave functions are given by

$\psi_{n,l}(s)=D_{n,l}\,\Bigg{(}\frac{2\,M\,\beta}{\alpha^{2}}\Bigg{)}^{3/8}\,s^{\mu-\frac{1}{4}}\,e^{-\frac{s}{2}}\,{}_{1}F_{1}\left(\mu-\nu+\frac{1}{2},2\,\mu+1;s\right),$
(17)

where $D_{n,l}$ is a constant which can be determined by the normalization
condition for the radial wave function

$\frac{1}{\alpha}\,\int^{\infty}_{0}\,r^{2}\,dr\,|\psi(r)|^{2}=1.$ (18)

To solve the integrals of the radial wave function, we can write the confluent
hypergeometric function in terms of the associated Laguerre polynomials by the
relation GBA

${}_{1}F_{1}\left(-n,2\,\mu+1;x\right)=\frac{n!\,(2\,\mu)!}{(n+2\,\mu)!}\,L^{(2\,\mu)}_{n}(x).$
(19)

Then, taking into account $s=\omega\,r^{2}$, and with the help of AP  to solve
the integrals, the normalization constant is given by

$D_{n,l}=\frac{1}{(2\,\mu)!}\,\sqrt{\frac{2\,\alpha\,(n+2\,\mu)!}{n!}}=\frac{1}{\left(\sqrt{j^{2}+\frac{1}{4}}\right)!}\,\sqrt{\frac{2\,\alpha\,\left(n+\sqrt{j^{2}+\frac{1}{4}}\right)!}{n!}}.$
(20)

(a) $l=1=M=\beta=\gamma=V_{0}$, $\Phi=3/4$.

(b) $l=1=M=\beta=\gamma=V_{0}$.

(c) $M=1=\beta=\gamma=V_{0}=\Phi$.

(d) $l=1=M=\beta=\gamma=V_{0}$, $\Phi=0$.

Figure 2: The energy eigenvalue $E_{n,l}$ with topological defect parameter
$\alpha$ for different values of other parameters.

Equation (16) is the energy levels and Eqs. (17)–(20) are the normalized
radial wave functions of a non-relativistic particle confined by the AB-flux
field with pseudo-harmonic-type potential under topological defects produced
by point-like global monopole geometry. One can see that the energy levels and
the radial wave function are influenced by the topological defects
characterise by the parameter $\alpha$, and the magnetic flux field
$\Phi_{AB}$ and gets modified in comparison to flat space results obtained in
Refs. SMI ; SHD ; KJO ; SMI3 ; VK . We have plotted a few graphs of the energy
levels $E_{n,l}$ (fig. 2) and the normalized radial wave function
$\psi_{n,l}(s)$ (fig. 3) for different values of various parameters.

(a) $l=1=M=\beta=\gamma$, $\Phi=3/4$.

(b) $l=1=M=\beta=\gamma$, $\alpha=3/4$.

(c) $l=1=M=\beta=\gamma$.

(d) $M=1=\beta=\gamma$, $\alpha=3/4$, $\Phi=1$.

Figure 3: The normalized radial wave function $\psi_{n,l}$ for different
values of the parameters.

Below, we will discuss few special cases of the above quantum system. For
example,

Case I: Without Topological defect and Potential
$V(r)=\Big{(}\frac{A}{r^{2}}+B\,r^{2}\Big{)}$.

In this case, we want to study the above quantum system without topological
defects. In that case, for $\alpha\to 1$, the space-time geometry (1) under
consideration becomes Minkowski flat space. Furthermore, we set the constant
potential term $V_{0}=0$ in the potential expression (8).

Therefore, the energy eigenvalue in that case from (16) becomes

$\displaystyle
E_{n,l}=\sqrt{\frac{2\,B}{M}}\,\Bigg{(}2\,n+1+\sqrt{\Big{(}l-\Phi+\frac{1}{2}\Big{)}^{2}+2\,M\,A}\Bigg{)}.$
(21)

which is similar to the energy eigenvalue obtained in Ref. FC  provided zero
magnetic flux $\Phi\to 0$ here. Thus, the magnetic flux considered in the
quantum system shifts the eigenvalue solution of a non-relativistic particle
in comparison to the known result obtained in Ref. FC  with potential of the
form $V(r)=\Big{(}\frac{A}{r^{2}}+B\,r^{2}\Big{)}$. Overall we can see that
the energy eigenvalue (16) gets more modified in comparison to the result in
Ref. FC  due to the presence of the topological defects characterise by the
parameter $\alpha$, and the constant potential term $V_{0}$ in addition to the
magnetic flux field $\Phi_{AB}$ in the quantum system.

The normalized radial wave functions in that case becomes

$\psi_{n,l}(s)=N_{n,l}\,\Bigg{(}\frac{2\,M\,B}{\alpha^{2}}\Bigg{)}^{3/8}\,s^{\bar{\mu}-\frac{1}{4}}\,e^{-\frac{s}{2}}\,{}_{1}F_{1}\left(-n,1+2\,\bar{\mu};s\right),$
(22)

where

$\displaystyle N_{n,l}$ $\displaystyle=$
$\displaystyle\frac{1}{\left(\sqrt{\vartheta^{2}+\frac{1}{4}}\right)!}\,\sqrt{\frac{2\,\left(n+\sqrt{\vartheta^{2}+\frac{1}{4}}\right)!}{n!}},\quad\vartheta=\sqrt{(l-\Phi)\,(l-\Phi+1)+2\,M\,A},$
$\displaystyle 2\,\bar{\mu}$ $\displaystyle=$
$\displaystyle\sqrt{\vartheta^{2}+\frac{1}{4}},\quad\bar{\nu}=\frac{\bar{\Lambda}}{4\,\bar{\omega}},\quad\bar{\Lambda}=2\,M\,E_{n,l},\quad\bar{\omega}=\sqrt{2\,M\,B}.$
(23)

Case II: Without Topological Defects but Potential
$V(r)=\Big{(}\frac{A}{r^{2}}+B\,r^{2}+C\Big{)}$

We discuss the above quantum mechanical system without topological defects,
that is, $\alpha\to 1$. In that case, the space-time geometry (1) under
consideration reduces to Minkowski flat space. In addition, the chosen
potential here is of the form $V(r)=\Big{(}\frac{A}{r^{2}}+B\,r^{2}+C\Big{)}$.

Therefore, for $\alpha\to 1$ and with the chosen potential form, one will have
the following energy eigenvalue expression

$\displaystyle
E_{n,l}=C+\sqrt{\frac{2\,B}{M}}\,\Bigg{(}2\,n+1+\sqrt{\Big{(}l-\Phi+\frac{1}{2}\Big{)}^{2}+2\,M\,A}\Bigg{)}$
(24)

which is similar to the result obtained in flat space background in Ref. AA
provided zero magnetic flux $\Phi\to 0$ here. Thus, the magnetic flux
considered in the quantum system shifts the eigenvalue solution of a non-
relativistic particle in comparison to the known result obtained in AA  with
this chosen potential. Hence, overall we can see that the topological defects
of point-like global monopole characterise by the parameter $\alpha$, and the
magnetic flux field modified the eigenvalue solution in comparison to the
result obtained in Ref. AA  with this potential form. The normalized radial
wave function will be the same obtained in Eqs. (22)–(23) with now
$\bar{\Lambda}=2\,M\,(E_{n,l}-C)$.

###  2.2 Effects of Mie-Type Potential

In this section, we are interested on another kind of potential equal to
repulsive Coulomb plus inverse square potential given by SMI ; KJO ; SMI3 ;
SMI2 ; SMI4

$V(r)=V_{0}+\frac{\delta}{r}+\frac{\gamma}{r^{2}},$ (25)

where $\delta,\gamma$ characterises the potential parameter.

For this type of potential, the effective potential of the system becomes

$V_{eff}(r)=\frac{1}{\alpha^{2}}\,\Bigg{[}V_{0}+\frac{\delta}{r}+\frac{\gamma}{r^{2}}+\frac{(l-\Phi)\,(l-\Phi+1)}{2\,M\,r^{2}}\Bigg{]}.$
(26)

We can see that the effective potential of the system depends on the
topological defect ans well as the magnetic flux. We have plotted a few graph
(fig. 4) showing the influences of various parameters on the effective
potential.

(a) $l=1=M=\delta=\gamma=V_{0}$, $\Phi=3/4$.

(b) $l=1=M=\delta=\gamma=V_{0}$, $\alpha=3/4$.

(c) $l=1=M=\delta=\gamma=V_{0}$.

(d) $l=1=M=\delta=\gamma=V_{0}$, $\alpha=4/5$, $\Phi=1$.

Figure 4: Effective potential of the quantum system using Mie-type potential.

Thereby, substituting the potential (25) into the Eq. (6), we have obtained
the following radial equation:

$\psi^{\prime\prime}(r)+\frac{2}{r}\,\psi^{\prime}(r)+\Big{(}-\Delta^{2}-\frac{j^{2}}{r^{2}}-\frac{2\,\kappa}{r}\Big{)}\,\psi(r)=0,$
(27)

where $j$ is defined in Eq. (11) and

$\displaystyle\Delta^{2}=\frac{2\,M\,(V_{0}-E)}{\alpha^{2}}\quad,\quad\kappa=\frac{M\,\delta}{\alpha^{2}}.$
(28)

We perform a change of the function $\psi(r)=\frac{R(r)}{\sqrt{r}}$ into the
Eq. (28), we have

$R^{\prime\prime}(r)+\frac{1}{r}\,R^{\prime}(r)+\Big{(}-\Delta^{2}-\frac{\tau^{2}}{r^{2}}-\frac{2\,\kappa}{r}\Big{)}\,R(r)=0,$
(29)

where
$\tau=\sqrt{j^{2}+\frac{1}{4}}=\sqrt{\frac{(l-\Phi)\,(l-\Phi+1)+2\,M\,\gamma}{\alpha^{2}}+\frac{1}{4}}$.

Finally performing a change of variables via $\xi=2\,\Delta\,r$ into the Eq.
(36), we have

$R^{\prime\prime}(\xi)+\frac{1}{\xi}\,R^{\prime}(\xi)+\Bigg{(}-\frac{\tau^{2}}{\xi^{2}}-\frac{\kappa}{\Delta}\,\frac{1}{\xi}-\frac{1}{4}\Bigg{)}\,R(\xi)=0.$
(30)

It is known that the wave function $R(\xi)$ is well-behaved at the origin
$\xi\to 0$, since it is a singular point of the Eq. (30). Suppose, a possible
solution to the Eq. (30) is given by

$R(\xi)=\xi^{\tau}\,e^{-\frac{\xi}{2}}\,F(\xi),$ (31)

where $F(\xi)$ is an unknown function.

Substituting this solution (31) into the Eq. (30), we have obtained the
following equation:

$\xi\,F^{\prime\prime}(\xi)+\Big{(}1+2\,\tau-\xi\Big{)}\,F^{\prime}(\xi)+\Bigg{(}-\tau-\frac{\kappa}{\Delta}-\frac{1}{2}\Bigg{)}\,F(\xi)=0.$
(32)

Equation (32) is the confluent hypergeometric differential equation form MA ;
GBA . The solution to this equation (32) can be expressed in terms of a
confluent hypergeometric function
$F(\xi)={}_{1}F_{1}\Big{(}\tau+\frac{\kappa}{\Delta}+\frac{1}{2},2\,\tau+1;\xi\Big{)}$
which is well-behaved for $\xi\to\infty$. As stated earlier for bound-state
solutions, the function ${}_{1}F_{1}$ should be a finite degree polynomial of
degree $n$, and the quantity
$\Big{(}\tau+\frac{\kappa}{\Delta}+\frac{1}{2}\Big{)}=-n$, where $n=0,1,2,..$.

After simplification of the condition

$\Big{(}\tau+\frac{\kappa}{\Delta}+\frac{1}{2}\Big{)}=-n$

, one can obtain the following expression of the energy eigenvalues:

$E_{n,l}=V_{0}-\frac{M\,\delta^{2}}{2\,\Bigg{(}\Big{(}n+\frac{1}{2}\Big{)}\,\alpha+\sqrt{(l-\Phi)\,(l-\Phi+1)+2\,M\,\gamma+\frac{\alpha^{2}}{4}}\Bigg{)}^{2}}.$
(33)

Equation (33) is the energy levels of a Schrödinger particle confined by the
Aharonov-Bohm flux field with Mie-type potential in a point-like global
monopole space-time background. Thus, we can see that this energy eigenvalue
(33) is influenced by the topological defects of point-like global monopole
characterise by the parameter $\alpha$, and the magnetic flux field
$\Phi_{AB}$ and gets modified in comparison to those results obtained in Refs.
SMI ; KJO ; SMI3 ; SMI2 ; SMI4  in flat space background with this Mie-type
potential. We have plotted graphs of the energy levels $E_{n,l}$ (fig. 5) and
the radial wave function $\psi_{n,l}(\xi)$ (fig. 6) for different values of
various parameters.

(a) $l=1=M=\delta=\gamma=V_{0}$, $\Phi=3/4$.

(b) $M=1=\delta=\gamma=V_{0}$.

(c) $l=1=M=\delta=\gamma=V_{0}$, $\Phi=1$.

(d) $l=1=M=\delta=\gamma=V_{0}$, $\Phi=0$.

Figure 5: The energy Eigenvalue $E_{n,l}$ with topological defect parameter
$\alpha$ for various values of other parameters.

The radial wave function is therefore given by

$R_{n,l}(\xi)=\xi^{\tau}\,e^{-\frac{\xi}{2}}\,{}_{1}F_{1}\Big{(}-n,1+2\,\tau;\xi\Big{)}.$
(34)

That can be written as

$\displaystyle\psi_{n,l}(\xi)$ $\displaystyle=$
$\displaystyle\sqrt{2\,\Delta}\,\xi^{\tau-1/2}\,e^{-\frac{\xi}{2}}\,{}_{1}F_{1}\Big{(}-n,1+2\,\tau;\xi\Big{)}$
(35) $\displaystyle=$
$\displaystyle\sqrt{\frac{2\,M\,\delta}{\alpha^{2}\,\Big{(}n+\tau+\frac{1}{2}\Big{)}}}\,\xi^{\tau-1/2}\,e^{-\frac{\xi}{2}}\,{}_{1}F_{1}\Big{(}-n,1+2\,\tau;\xi\Big{)}.$

If one would analyze the quantum system without topological defects, that is,
$\alpha\to 1$. In that case, the space-time geometry (1) under consideration
becomes Minkowski flat space.

(a) $l=1=M=\delta=\gamma$, $\alpha=3/4=\Phi$.

(b) $l=1=M=\delta=\gamma$, $\Phi=3/4$.

(c) $l=1=M=\delta=\gamma$, $\alpha=4/5$.

(d) $l=1=M=\delta=\gamma$.

Figure 6: The radial wave function $\psi_{n,l}$ with $\xi$ for various values
of different parameters.

Therefore, for $\alpha\to 1$, the energy eigenvalue (33) becomes

$E_{n,l}=V_{0}-\frac{M\,\delta^{2}}{2\,\Bigg{(}n+\sqrt{\Big{(}l-\Phi+\frac{1}{2}\Big{)}^{2}+2\,M\,\gamma}+\frac{1}{2}\Bigg{)}^{2}}$
(36)

which is the similar to the result obtained in Refs. AA ; RS  provided zero
magnetic flux $\Phi\to 0$ here. Thus, we can see that the presence of the
magnetic flux field modified the energy spectrum of a non-relativistic
particle in comparison to those results obtained Refs. AA ; RS . Furthermore,
the presence of the topological defect characterise by the parameter $\alpha$
modified the eigenvalue (33) more in addition to the magnetic flux in
comparison to those results obtained in Refs. AA ; RS .

##  3 Applications to Some Diatomic Molecular potentials

The above results of the studied quantum system previous are now being
utilized to develop solutions to some specific types of physical potential
models which have application in practical problems. The pseudo-harmonic and
Kratzer potentials are successfully employed GL  to study the eigenvalues
spectra of a class of diatomic molecules in flat space background. Here, we
will study the quantum system in curved space-time background under
topological defects with these known potentials.

###  3.1 Harmonic Oscillator Potential

The harmonic oscillator potential can be recovered from the potential (9) by
setting the potential parameters $\gamma=0$, $V_{0}=0$, and
$\beta=\frac{1}{2}\,M\,\omega^{2}_{0}$, we have,
$V(r)=\frac{1}{2}\,M\,\omega^{2}_{0}\,r^{2}$, a harmonic oscillator potential.
Here $\omega_{0}$ is the oscillator frequency.

Therefore, using the harmonic oscillator potential in the quantum system, one
will find the following energy eigenvalue expression:

$\displaystyle
E_{n,l}=2\,\alpha\,\omega_{0}\,\Bigg{(}n+\frac{1}{2\,\alpha}\,\sqrt{(l-\Phi)\,(l-\Phi+1)+\frac{\alpha^{2}}{4}}+\frac{1}{2}\Bigg{)}.$
(37)

Equation (37) is the energy levels of harmonic oscillator in the presence of
the Aharonov-Bohm flux fields in point-like global monopole.

In absence of magnetic flux field, $\Phi_{AB}\to 0$, the energy eigenvalues
from Eq. (37) becomes

$\displaystyle
E_{n,l}=\alpha\,\omega_{0}\,\Bigg{(}2\,n+\sqrt{\frac{l\,(l+1)}{\alpha^{2}}+\frac{1}{4}}+1\Bigg{)}.$
(38)

which is similar to those results obtained in Refs. CF ; CF2 . Thus, we can
see that the magnetic flux field shifts the eigenvalue solution of harmonic
oscillator in comparison to the results in Refs. CF ; CF2 . The energy
eigenvalue $E_{n,l}$ depends on the geometric quantum phase $\Phi_{B}$ which
shows an analogous of the Aharonov-Bohm effect YA ; MP .

###  3.2 Pseudoharmonic Potential

The pseudoharmonic potential can be recovered from the potential (9) by
setting the parameters $\beta=\frac{D_{e}}{r^{2}_{0}}$,
$\gamma=D_{e}\,r^{2}_{0}$, and $V_{0}=-2\,D_{e}$. Thus, we have of the
following potential form SMI ; RSS ; FC2

$V(r)=\frac{D_{e}}{r^{2}_{0}}\,r^{2}+\frac{D_{e}\,r^{2}_{0}}{r^{2}}-2\,D_{e}=D_{e}\,\Bigg{(}\frac{r}{r_{0}}-\frac{r_{0}}{r}\Bigg{)}^{2}.$
(39)

Here $D_{e}$ is the dissociation energy between atoms in a solid and $r_{0}$
is the equilibrium inter-nuclear separation. This potential is of great
importance not only in physics but also in chemistry and is useful to describe
the interactions of some diatomic molecules AC ; SMI5 .

Therefore, using the above potential, one can find the energy eigenvalues of
the quantum system as

$\displaystyle
E_{n,l}=-2\,D_{e}+\alpha\,\sqrt{\frac{2\,D_{e}}{M\,r^{2}_{0}}}\,\Bigg{(}2\,n+\sqrt{\frac{(l-\Phi)\,(l-\Phi+1)+2\,M\,D_{e}\,r^{2}_{0}}{\alpha^{2}}+\frac{1}{4}}+1\Bigg{)}.$
(40)

The normalized radial wave functions are given by

$\psi_{n,l}(s)=B_{n,l}\,\Bigg{(}\frac{2\,M\,D_{e}}{\alpha^{2}\,r^{2}_{0}}\Bigg{)}^{3/8}\,s^{\mu-\frac{1}{4}}\,e^{-\frac{s}{2}}\,{}_{1}F_{1}(-n,1+2\,\mu;s),\quad
B_{n,l}=\frac{1}{\left(\sqrt{j^{2}+\frac{1}{4}}\right)!}\,\sqrt{\frac{2\,\alpha\,\left(n+\sqrt{j^{2}+\frac{1}{4}}\right)!}{n!}},$
(41)

where $\tau=2\,\mu=\sqrt{j^{2}+1/4}$,
$j=\sqrt{\frac{(l-\Phi)\,(l-\Phi+1)+2\,M\,D_{e}\,r^{2}_{0}}{\alpha^{2}}}$.

Equation (40) is the energy levels and Eq. (41) is the normalized radial wave
function of a non-relativistic particle confined by the Aharonov-Bohm flux
field with pseudoharmonic potential in point-like global monopole. One can see
that the eigenvalue solution Eqs. (40)–(41) get modified by the topological
defects of the geometry characterise by the parameter $\alpha$, and the AB-
flux field in comparison to the those results obtained in Refs. SMI ; RSS ;
FC2 .

For zero magnetic flux field, $\Phi_{AB}\to 0$, the eigenvalue solution from
Eqs. (40)–(41) becomes

$\displaystyle E_{n,l}$ $\displaystyle=$
$\displaystyle-2\,D_{e}+\alpha\,\sqrt{\frac{2\,D_{e}}{M\,r^{2}_{0}}}\,\Bigg{(}2\,n+\sqrt{\frac{l\,(l+1)+2\,M\,D_{e}\,r^{2}_{0}}{\alpha^{2}}+\frac{1}{4}}+1\Bigg{)}$
$\displaystyle\psi_{n,l}(s)$ $\displaystyle=$ $\displaystyle
B_{n,l}\,\Bigg{(}\frac{2\,M\,D_{e}}{\alpha^{2}\,r^{2}_{0}}\Bigg{)}^{3/8}\,s^{\Bigg{(}\frac{1}{2}\,\sqrt{\frac{l\,(l+1)+2\,M\,D_{e}\,r^{2}_{0}}{\alpha^{2}}+\frac{1}{4}}-\frac{1}{4}\Bigg{)}}\,e^{-\frac{s}{2}}\,{}_{1}F_{1},$
$\displaystyle B_{n,l}$ $\displaystyle=$
$\displaystyle\frac{1}{\left(\sqrt{\frac{l\,(l+1)+2\,M\,D_{e}\,r^{2}_{0}}{\alpha^{2}}+\frac{1}{4}}\right)!}\,\sqrt{\frac{2\,\alpha\,\left(n+\sqrt{\frac{l\,(l+1)+2\,M\,D_{e}\,r^{2}_{0}}{\alpha^{2}}+\frac{1}{4}}\right)!}{n!}},$
(42)

We can see that the global effects of the geometry characterise by the
parameter $\alpha$ of point-like global monopole and the magnetic flux are
present explicitly in the eigenvalue solution and modified the result in
comparison to flat space case with this pseudo-harmonic potential obtained in
Refs. SMI ; RSS ; FC2 .

###  3.3 The Kratzer-Fues or Modified Kratzer Potential

The modified Kratzer potential can easily be recovered from the potential (25)
by setting the potential parameters $\delta=-2\,D_{e}\,r_{0}$,
$\gamma=D_{e}\,r^{2}_{0}$, and $V_{0}=D_{e}$, we have SMI6

$V(r)=D_{e}+\frac{D_{e}\,r^{2}_{0}}{r^{2}}-\frac{2\,D_{e}\,r_{0}}{r}=D_{e}\,\Bigg{(}\frac{r-r_{0}}{r}\Bigg{)}^{2}.$
(43)

This potential also known as Kratzer-Fues potential and has been used by
several authors KJO ; SMI2 ; SMI4 ; RSS ; FC2 ; VK  to describe molecular
structures and interactions.

Thus, using this potential, the energy eigenvalue of the quantum system

$E_{n,l}=D_{e}-\frac{2\,M\,D^{2}_{e}\,r^{2}_{0}}{\Bigg{(}n\,\alpha+\frac{\alpha}{2}+\sqrt{(l-\Phi)\,(l-\Phi+1)+2\,M\,D_{e}\,r^{2}_{0}+\frac{\alpha^{2}}{4}}\Bigg{)}^{2}}.$
(44)

The radial wave function is therefore given by

$R_{n,l}(\xi)=\xi^{\tau}\,e^{-\frac{\xi}{2}}\,{}_{1}F_{1}\Big{(}-n,1+2\,\tau;\xi\Big{)},$
(45)

where
$\tau=\sqrt{\frac{(l-\Phi)\,(l-\Phi+1)+2\,M\,D_{e}\,r^{2}_{0}}{\alpha^{2}}+\frac{1}{4}}$.

Equation (43) is the energy levels and Eq. (44)–(45) is the radial wave
function of a non-relativistic particle confined by the Aharonov-Bohm flux
field with modified Kratzer potential in point-like defect. One can see that
the eigenvalue solution gets modified by the topological defects of the
geometry characterise by the parameter $\alpha$, and the magnetic flux field
with this potential in comparison to those results obtained in Refs. SMI ; KJO
; SMI3 ; SMI2 ; RSS ; FC2 ; SMI4 ; SMI6 ; VK .

In absence of topological defects, that is, $\alpha\to 1$, the energy
eigenvalue from Eq. (44) becomes

$E_{n,l}=D_{e}-\frac{2\,M\,D^{2}_{e}\,r^{2}_{0}}{\Bigg{(}n+\frac{1}{2}+\sqrt{(l-\Phi)\,(l-\Phi+1)+2\,M\,D_{e}\,r^{2}_{0}+\frac{1}{4}}\Bigg{)}^{2}}.$
(46)

The radial wave functions are given by

$R_{n,l}(\xi)=\xi^{\tau_{0}}\,e^{-\frac{\xi}{2}}\,{}_{1}F_{1}\Big{(}-n,1+2\,\tau_{0};\xi\Big{)},$
(47)

where
$\tau_{0}=\sqrt{(l-\Phi)\,(l-\Phi+1)+2\,M\,D_{e}\,r^{2}_{0}+\frac{1}{4}}$.

Equation (46) is the energy spectra and Eq. (47) is the radial wave function
of a non-relativistic particle confined by the AB-flux field with modified
Kratzer potential in flat space background.

###  3.4 The Kratzer Potential

The Kratzer potential can be recovered from potential (25) by setting the
potential parameters $\delta=-2\,D_{e}\,r_{0}$, $\gamma=D_{e}\,r^{2}_{0}$, and
$V_{0}=0$, we have AK ; EF ; HA ; FC2 ; gg1 ; gg2 ; gg3 ; gg4

$V(r)=\frac{D_{e}\,r^{2}_{0}}{r^{2}}-\frac{2\,D_{e}\,r_{0}}{r}=2\,D_{e}\,\Bigg{[}\frac{1}{2}\,\Big{(}\frac{r_{0}}{r}\Big{)}^{2}-\Big{(}\frac{r_{0}}{r}\Big{)}\Bigg{]}.$
(48)

This potential has extensively been used to describe molecular structures and
interactions in chemistry.

Thereby, using Kratzer potential in the quantum, one can obtain the bound-
states energy eigenvalue

$E_{n,l}=-\frac{2\,M\,D^{2}_{e}\,r^{2}_{0}}{\Bigg{(}\Big{(}n+\frac{1}{2}\Big{)}\,\alpha+\sqrt{(l-\Phi)\,(l-\Phi+1)+2\,M\,D_{e}\,r^{2}_{0}+\frac{\alpha^{2}}{4}}\Bigg{)}^{2}}.$
(49)

The radial wave function is given by

$R_{n,l}(\xi)=\xi^{\tau}\,e^{-\frac{\xi}{2}}\,{}_{1}F_{1}\Big{(}-n,1+2\,\tau;\xi\Big{)},$
(50)

where
$\tau=\sqrt{\frac{(l-\Phi)\,(l-\Phi+1)+2\,M\,D_{e}\,r^{2}_{0}}{\alpha^{2}}+\frac{1}{4}}$.

Equation (49) is the energy levels and Eq. (50) is the radial wave function of
a non-relativistic particle confined by the Aharonov-Bohm flux field with
Kratzer potential in point-like defect. One can see that the eigenvalue
solution is influenced by the topological defects characterise by the
parameter $\alpha$, and the magnetic flux field. This eigenvalue solution gets
modified in comparison to those results obtained in Refs. HA ; FC2 ; gg1 ; gg2
; gg3 ; gg4  due to the topological defects, and the magnetic flux field.

##  4 Conclusions

In this analysis, we studied a non-relativistic particle under the influence
of the Aharonov-Bohm flux field with potential in a curved space-time via
point-like global monopole and derived the radial wave equation of the
Schrödinger equation. In sub-section 2.1, we have considered pseudo-harmonic
potential and solved the radial equation analytically. The exact eigenvalue
solution of the schrödinger equation is obtained where the energy levels are
given by Eq. (16) and the normalized radial wave function by Eq. (17)–(20). We
have seen that the presence of topological defect characterises by the
parameter $\alpha$ which causes the space-time geometry to be a curved one,
and the magnetic flux field influences the eigenvalue solution and gets
modified in comparison to those results obtained in Refs. SMI ; SHD ; KJO ;
SMI3 ; VK  in Minkowski flat space-time background with this pseudo-harmonic
potential. We have discussed the obtained eigenvalue solution without
topological effects, and shown that only the magnetic flux field shifted the
energy levels and the wave function in comparison to those results obtained in
Ref. FC  and in Ref. AA  with this potential (constant potential term
$V_{0}=0$ in the previous one). In sub-section 2.2, we have considered Mie-
type potential and solved the same quantum analytically. The energy
eigenvalues are given by Eq. (33) and the radial wave function by Eq. (35).
Here also, we have seen that the presence of topological defect characterises
by the parameter $\alpha$, and the magnetic flux field influences the
eigenvalue solution. This solution gets modified in comparison to those
results obtained in Ref. SMI ; KJO ; SMI3 ; SMI2 ; SMI4  with this Mie-type
potential in Minkowski flat space-time. Also, the obtained eigenvalue solution
is analyzed without topological effects and shown that only the magnetic flux
field shifted the results in comparison to those obtained in Ref. AA ; RS  in
flat space background with this potential.

In section 3, we have utilized the eigenvalue solutions of the quantum system
obtained with pseudoharmonic- and Mie-type potentials in point-like defect to
some interaction potential models which are used for diatomic molecular
structure. These potential models include harmonic oscillator potential (sub-
section 3.1), pseudoharmonic potential (sub-section 3.2), modified Kratzer or
Kratzer-Fues potential (sub-section 3.3), and Kratzer potential (sub-section
3.4). With these potentials, we have obtained the energy eigenvalues and the
radial wave functions of non-relativistic particles confined by the AB-flux
field. One can see in sub-section 3.1, that only the magnetic flux field
shifted the known eigenvalue solution; whereas in sub-section 3.2--3.4, we
have seen that both the topological defects of the geometry characterises by
the parameter $\alpha$, and the magnetic flux field $\Phi_{AB}$ modified the
known eigenvalue solutions obtained in flat space background in the
literature.

Throughout the analysis, one can see a shifting in the orbital quantum number
$l$, that is, $l\to l^{\prime}=\left(l-\frac{e\,\Phi_{B}}{2\pi}\right)$ called
effective orbital quantum number. This is because the non-relativistic
particles are confined by the the AB-flux field but there is no direct
interaction of the particles with external fields. This energy eigenvalue thus
depends on the geometric quantum phase $\Phi_{B}$, and is a periodic function
of it, that is,
$E_{n,l}(\Phi_{B}\pm\,\Phi_{0}\,\tau)=E_{n,l\mp\tau}(\Phi_{B})$, where
$\tau=0,1,2,...$ with a periodicity $\Phi_{0}$. This dependence of the
eigenvalue solution on the geometric quantum phase shows an analogous of the
Aharonov-Bohm (AB) effect YA ; MP  for the bound-state. It is well-known in
condensed matter physics that the dependence of the energy eigenvalue on the
geometric quantum phase gives rise to a persistent currents which we will
discuss in the future work.

## Conflict of Interest

There is no conflict of interests regarding publication of this manuscript.

## Funding Statement

No fund has received for this manuscript.

## Data Availability Statement

No new data are generated in this manuscript.

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#  Rota–Baxter and averaging operators on racks and rack algebras

V. G. Bardakov, V. A. Bovdi  Sobolev Institute of Mathematics, 4 Acad. Koptyug
avenue, 630090, Novosibirsk, Russia.  Novosibirsk State Agrarian University,
Dobrolyubova street, 160, Novosibirsk, 630039, Russia.  Regional Scientific
and Educational Mathematical Center of Tomsk State University, 36 Lenin Ave.,
Tomsk, Russia<EMAIL_ADDRESS>United Arab Emirates University, P.O. Box
15551, Al Ain, Abu Dhabi, United Arab Emirates<EMAIL_ADDRESS>
###### Abstract.

In the present article we define and investigate relative Rota–Baxter
operators and relative averaging operators on racks and rack algebras. Also,
if $B$ is a Rota–Baxter or averaging operator on a rack $X$, then we can
extend $B$ by linearity to the rack algebra $\Bbbk[X]$. On the other side, we
have definitions of Rota–Baxter and averaging operators on arbitrary algebra.
We find connections between these operators. In particular, we prove that if
$B:X\to X$ is an averaging operator on a rack, then its linear extension on a
rack algebra $\Bbbk[X]$ gives an averaging operator.

Keywords: Rack, quandle, rack extension, rack algebra, quandle algebra,
relative averaging operator, relative Rota–Baxter operator.

Mathematics Subject Classification 2010: 17A30, 20E22.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Relative Rota–Baxter operators on racks
  4. 4 Rack extensions and averaging operators on racks
  5. 5 Properties of averaging operators on racks
  6. 6 Rota–Baxter and averaging operators on rack algebras
  7. 7 Some generalization of conjugation quandle and core quandle
  8. 8 Open problems
    1. 8.1 Other operators
    2. 8.2 From Lie racks to Leibniz algebra

##  1\. Introduction

The study of operators on different algebraic systems is a classic direction
in algebra. The most studied operators over these objects are their
endomorphisms and automorphisms. To define an endomorphism of an algebraic
system, it is sufficient to define it on the generating set, and then we
extend it to all elements. However, there are other types of operators, the
construction of which is not so simple. Examples of such operators include the
Reynolds operator and the averaging operator, which arose independently in
different areas of mathematics. Let $X$ be a locally compact Hausdorff space
and let $C_{0}(X)$ denotes the Banach algebra of all continuous functions on
$X$ vanishing at $\infty$. A Reynolds operator $T:C_{0}(X)\to C_{0}(X)$ (see
[38]) is a linear operator, satisfying the Reynolds identity:

$T(a)T(b)=T\left(T(a)b+aT(b)-T(a)T(b)\right),\qquad(a,b\in C_{0}(X)).$

Obviously, a projection $T$ is a Reynolds operator if $T$ is averaging, i.e.

$T(a)T(b)=T\left(aT(b)\right),\qquad(a,b\in C_{0}(X)).$

Reynolds introduced this operator in connection with the theory of turbulence.
Averaging operators are studied in the works of Birkoff [15, 16] and further
by Kelley [31], Rota [39, 40], Seever [43] and many others (see [26, 21, 30,
19, 36, 20, 17], as well as references therein).

One type of environment of many interesting operators is the Rota-Baxter
operator. The first appearance of Rota–Baxter operators (shortly, RB-
operators) for commutative algebras can be traced back to Baxter’s seminal
paper [13] in 1960. Since then, the theory of Rota-Baxter operators has
undergone extensive development by various authors in different fields of
mathematics. It should be noted that Rota-Baxter operators play an important
role due to their connection with a number of mathematical concepts, including
the Yang-Baxter equation [14, 42], Loday algebras [3], and double Poisson
algebras [41]. A historical overview and connections of these operators with
other mathematical concepts can be found in the monograph [24].

A Rota–Baxter operator on groups were introduced in [25], a group $G$ with a
Rota–Baxter operator $B$ is called a Rota–Baxter group (shortly, RB-group). In
[25] it was shown that if $(G,B)$ is a Rota–Baxter Lie group, then the tangent
map $B$ at identity is a Rota–Baxter operator of weight $1$ on the Lie algebra
of the Lie group $G$. The properties of $RB$-groups are actively studied in
[11]. In [12], a connection between $RB$-groups, the Yang–Baxter equation and
skew braces was found. In [22] it was proven that the Rota–Baxter operators on
the group ring $\Bbbk[G]$ are in one-to-one correspondence with the
Rota–Baxter operators of weight 1 on $G$. In [28] relative Rota–Baxter groups
were defined as groups with a relative Rota–Baxter operator. In this case, the
relative Rota–Baxter operator depends not only on the group, but also on the
group of its automorphisms. In the case when this group of automorphisms is a
group of inner automorphisms, we obtain the Rota–Baxter operator. Some
properties of relative Rota–Baxter operators are studied in [37]. Relative
Rota–Baxter operators on an arbitrary Hopf algebra were defined in [8].

Following Kurosh [33] in the sequel of this article we called a non-empty set
with a binary algebraic operation a groupoid. Our attention concentrated on
the investigation of Rota–Baxter and averaging operators on such groupoids as
groups, racks and quandles. Obviously, for any groupoid one can define
averaging operator. Recall that a quandle is a groupoid with a binary
operation that satisfies axioms encoding the three Reidemeister moves of
planar diagrams of links in the $3$-space. If we forget on the first
Reidemeister move we get a groupoid which is called by rack. Racks and
quandles were introduced by Joyce [29] and Matveev [35], who showed that link
quandles are complete invariants of non-split links up to orientation of the
ambient $3$-space. It is worth emphasizing that racks and quandles are also
useful in certain areas of mathematics: in group theory, set-theoretic
solutions of the Yang-Baxter equations and Hopf algebras, and discrete
integrable systems (see [1]). Automorphisms of quandels, which reveal much
about their internal structure, are studied in detail in a series of articles
[4, 5].

In the present article we define and study relative Rota–Baxter and averaging
operators on racks and racks algebras. In particular, we are studying the
following question. Let $(X,*)$ be a rack (quandle) and $B\colon X\to X$ is a
relative Rota–Baxter or averaging operator on $X$. Is it possible to define
new algebraic binary operation $\circ$ on $X$ such that $(X,\circ)$ is a rack
(quandle), $B$ is a relative Rota–Baxter or averaging operator on $(X,\circ)$
and $B$ induces a rack homomorphism $B:(X,\circ)\to(X,*)$?

Also, if $B$ is a Rota–Baxter or averaging operator on $X$, then we can extend
$B$ by linearity to the rack algebra $\Bbbk[X]$. On the other side, we have
definitions of Rota–Baxter and averaging operators on arbitrary algebra. The
question on connection between these two definitions arises. We prove that if
$B:X\to X$ is an averaging operator on a rack, then its linear extension on a
rack algebra $\Bbbk[X]$ gives an averaging operator. The situation with
Rota–Baxter operators is more complicated. We give conditions under which the
linear extension of a Rota–Baxter operator $B:X\to X$ on $\Bbbk[X]$ (we shall
call this extension by rack Rota–Baxter operator) gives an algebraic
Rota–Baxter operator on the algebra $\Bbbk[X]$.

The paper is structured as follows. The next section is devoted to review some
facts about Rot–Baxter operators, relative Rot–Baxter operators, racks and
quandles. We give definition and prove some properties of relative Rota–Baxter
operators on racks in Section 3.

In Section 4 we define semi-direct product of racks, that gives an answer on a
question [6, Problem 4.15], further we introduce a notion of (relative)
averaging operators on racks. We proved that if $(X,*)$ is a rack and $B:X\to
X$ is an averaging operator, then it defines new algebraic operation $\circ$
on $X$ such that $(X,\circ)$ is a rack, $B$ is an averaging operator on
$(X,\circ)$ and $B$ induces a rack homomorphism $B:(X,\circ)\to(X,*)$.

In Section 5 we formulate some properties of averaging operators. In
particular, for Cartesian product $X:=X_{1}\times X_{2}$ of quandles $X_{1}$
and $X_{2}$ we define operators $P_{i}:X\to X_{i}$ and prove that they are
averaging operators on $X$. Also we define averaging operator on union of
averaging racks.

In Section 6 we consider two approach to definition of Rota–Baxter and
averaging operators on rack algebras. We find conditions under which linear
extension of RB-operator on $X$ gives an RB-operator on algebra $\Bbbk[X]$. We
prove that any averaging operator on $X$ can be extended to an averaging
operator on $\Bbbk[X]$.

In Section 7 we suggested some generalizations of conjugation and core
quandles on Rota–Baxter groups.

At the end of the article we formulate some questions for further research.

##  2\. Preliminaries

A groupoid $(G,\cdot)$ is a non-empty set $G$ with a binary algebraic
operation ‘$\cdot$’ (see [33]). A groupoid $G$ is called idempotent if $a\cdot
a=a$ for any $a\in G$.

Let $\Bbbk$ be a field and let $A$ be an algebra over $\Bbbk$. A linear
operator $R:A\to A$ is called a Rota–Baxter operator of weight
$\lambda\in\Bbbk$ if

(1) $R(x)R(y)=R(R(x)y+xR(y)+\lambda xy),\qquad\qquad(x,y\in A).$

An algebra endowed with a Rota–Baxter operator is called a Rota–Baxter
algebra.

In [25] were defined Rota–Baxter operators of weights $\pm 1$ on a group.

######  Definition 2.1 ([25]).

Let $G$ be a group and let $B:G\to G$ be a map.

  * (i)

$B$ is called a Rota–Baxter operator (RB-operator) of weight 1  if

$B(g)B(h)=B(gB(g)hB(g)^{-1}),\qquad\qquad(g,h\in G);$

  * (ii)

$B$ is called a Rota–Baxter operator (RB-operator) of weight $-1$ if

$B(g)B(h)=B(B(g)hB(g)^{-1}g),\qquad\qquad(g,h\in G).$

A group endowed with an RB-operator is called a Rota–Baxter group (RB-group).

######  Example 2.2.

Let $G$ be a group. The following maps:

$B_{[0]}(g):=e;\qquad B_{[-1]}(g):=g^{-1},\qquad\qquad(g\in G)$

are $RB$-operators which are called elementary Rota–Baxter operators.

A group $G$ is called Rota–Baxter-elementary, if any Rota-Baxter operator on
$G$ is elementary. It is evident that every cyclic group of prime order is
elementary.

######  Proposition 2.3 ([25]).

Let $(G,B)$ be a Rota–Baxter group. Define on $G$ the operation

$g\circ h:=gB(g)hB(g)^{-1},\qquad\qquad(g,h\in G)$

The following conditions hold:

  * (i)

$(G,\circ)$ is a group;

  * (ii)

the map $B$ is an RB-operator on the group $(G,\circ)$;

  * (iii)

$B:(G,\circ)\to G$ is a group homomorphism of RB-groups $(G,\circ,B)$ and
$(G,B)$.

######  Definition 2.4 ([28]).

Let $H$ and $G$ be groups. Let $\phi:G\to\operatorname{Aut}(H)$ be a group
homomorphism and let $B:H\to G$ be a map. A relative Rota–Baxter group is a
quadruple $(H,G,\phi,B)$, where

(2)
$B(h_{1})B(h_{2})=B(h_{1}\phi_{B(h_{1})}(h_{2})),\qquad\quad(h_{1},h_{2}\in
H)$

in which $\phi_{g}:=\phi(g)\in\operatorname{Aut}(H)$ and $g\in G$. The map $B$
is called a relative RB-operator on $H$ with respect to $(G,\phi)$.

In particular, if $H=G$ and $\phi_{g}:G\to\operatorname{Inn}(G)$, in which
$\phi_{g}(h):=ghg^{-1}$ for some fixed $g\in G$, we get a classical
Rota–Baxter group $(H,B)$.

The following definition can be found in the papers Joyce [29] and Matveev
[35].

######  Definition 2.5.

A groupoid $(Q,*)$ is called quandle if the following axioms hold:

  * (Q1)

$x*x=x$ for all $x\in Q$;

  * (Q2)

for each $x,y\in Q$ there exists a unique $z\in Q$ such that $x=z*y$, in this
case we write $z:=x\bar{*}y$;

  * (Q3)

$(x*y)*z=(x*z)*(y*z)$ for all $x,y,z\in Q$.

If the groupoid $(Q,*)$ satisfying only conditions (Q2) and (Q3), then it is
called a rack.

######  Remark 2.6.

The rack (quandle) defined in Definition 2.5 is called a right rack (quandle).
If we take instead (Q2) and (Q3) the following two axioms:

  * (LQ2)

for each $x,y\in Q$ there exists a unique $z\in Q$ such that $x=y*z$, in this
case we write $z:=y\bar{*}x$;

  * (LQ3)

$z*(x*y)=(z*x)*(z*y)$ for all $x,y,z\in Q$;

then we get the definition of left rack (quandle).

Used quandles and racks in of our article are presented in the following.

######  Example 2.7.

Let $(G,\cdot)$ be a group and $\phi\in\operatorname{Aut}(G)$. Let $C_{n}$ be
a cyclic group of order $n\geq 1$,

  * (i)

if $a*_{m}b:=b^{-m}ab^{m}$ for any $a,b\in G$ and $m\in\mathbb{Z}$, then
$\operatorname{Conj\,}_{m}(G):=(G,*_{m})$ is called the $m$-conjugation
quandle of $G$. If $m=1$ then
$\operatorname{Conj\,}(G):=\operatorname{Conj\,}_{1}(G)$ is called conjugation
quandle of $G$;

  * (ii)

if $a*b:=ba^{-1}b$ for any $a,b\in G$, then $\operatorname{Core}(G):=(G,*)$ is
called the core quandle of $G$. In particular, the core quandle
$\operatorname{R}_{n}:=(C_{n},*)$ is called the dihedral quandle;

  * (iii)

if $a*b:=\phi(ab^{-1})b$ for any $a,b\in G$, then
$\operatorname{Alex}(G,\phi):=(G,*)$ is called the generalized Alexander
quandle of $G$ with respect to $\phi$.

A quandle $(Q,*)$ is called trivial if $x*y=x$ for all $x,y\in Q$. Unlike
groups, a trivial quandle can have arbitrary number of elements. We denote the
$n$-element trivial quandle by $\mathrm{T}_{n}$ and an arbitrary trivial
quandle by $\mathrm{T}$.

If $(Q,*)$ is a quandle, then from axiom (Q3) we obtain that

$(x*y)*x=x*(y*x),\qquad\qquad(x,y\in Q).$

Furthermore, the axioms (Q2) and (Q3) are equivalent to existing the map

(3) $S_{x}:Q\to Q\qquad(y\mapsto y*x,\quad y\in Q)$

being an automorphism of $Q$ for each $x\in Q$. These automorphisms are called
inner automorphisms, and the group generated by all such automorphisms is
denoted by $\operatorname{Inn}(X)$. A quandle $Q$ is called connected if it
admits a transitive action by its group of inner automorphisms. For example,
dihedral quandles of odd order are connected, whereas that of even order are
disconnected. A quandle $X$ is called involutary if
$S_{x}^{2}=\mathrm{id}_{Q}$ for each $x\in Q$. For example, all core quandles
are involutary. A quandle (resp. rack) $(Q,*)$ is called commutative if
$x*y=y*x$ for all $x,y\in Q$. The dihedral quandle $\operatorname{R}_{3}$ is
commutative and no trivial quandle $\mathrm{T}_{n}$ with $n>1$ (i.e. more than
one element) is commutative.

##  3\. Relative Rota–Baxter operators on racks

To define a relative RB-operators on racks we recall construction of relative
RB-operators on groups (see [28]).

Let $H$ and $G$ be groups and let $\Phi:G\to\operatorname{Aut}(H)$ be a
homomorphism. It is easy to check that $(G\times H,*)$ is a group in which the
multiplication $*$ is defined by

(4)
$(g_{1},h_{1})*(g_{2},h_{2}):=\left(g_{1}g_{2},(h_{1}\Phi_{g_{2}})h_{2}\right),\qquad(g_{1},g_{2}\in
G,h_{1},h_{2}\in H)$

where $h_{1}\Phi_{g_{2}}$ is the image of the element $h_{1}$ under the action
of the automorphism $\Phi_{g_{2}}:=\Phi(g_{2})$. The group $G\ltimes_{\Phi}H$
is called a semi-direct product of $G$ and $H$ with respect to $\Phi$. In
particular, if $G=\operatorname{Aut}(H)$ then the group
$\operatorname{Hol}(H):=\operatorname{Aut}(H)\ltimes_{\Phi}H$ is called the
holomorph of the group $H$.

Now let $B:H\to G$ be a map and let

$\Gamma_{B}:=\\{(B(h),h)\mid h\in H\\}\subseteq G\times H$

be its graphic. The groupoid $(\Gamma_{B},*)$ in which

$(B(h),h)*(B(k),k):=\left(B(h)B(k),(h\Phi_{B(k)})k\right),\qquad(h,k\in H)$

is a group if and only if $B$ satisfies the following analogous of the
relative RB-operator

(5) $B(h)B(k)=B\left((h\Phi_{B(k)})k\right),\qquad\quad(h,k\in H).$

######  Remark 3.1.

The original definition of relative RB-operator on groups (see (2)) does not
coincide with (5) because we have a different multiplication in a semi-direct
product. If we define the multiplication on $G\ltimes_{\Phi}H$ by the formula

$(g_{1},h_{1})*(g_{2},h_{2}):=\left(g_{1}g_{2},h_{1}\cdot\Phi_{g_{1}}(h_{2})\right),\qquad(g_{1},g_{2}\in
G,\;h_{1},h_{2}\in H),$

we get the relative RB-operator (2). Our approach is motivated by the using
right racks and quandles.

Using this construction we define relative Rota–Baxter operators on racks.

Let $(X,\cdot)$ and $(A,\circ)$ be racks. Let
$\Phi:A\to\operatorname{Conj\,}(\operatorname{Aut}(X))$ be a rack homomorphism
such that $a\mapsto\Phi_{a}\in\operatorname{Aut}(X)$. Define on the set
$A\times X$ the following multiplication:

$(a,x)*(b,y):=(a\circ b,(x\Phi_{b})\cdot y),\qquad(x,y\in X,a,b\in A).$

Further, let $B:X\to A$ be a map and let

$\Gamma_{B}:=\\{(B(x),x)\mid x\in X\\}\subset A\times X$

be its graphic. The multiplication on $A\times X$ induces multiplication on
$\Gamma_{B}$:

$(B(x),x)*(B(y),y):=\left(B(x)B(y),(x\Phi_{B(y)})y\right).$

If $B(x)B(y)=B\left((x\Phi_{B(y)})y\right)$, then the set $\Gamma_{B}$ is
closed under multiplication $*$ and we get a groupoid $(\Gamma_{B},*)$.

Now we are ready to introduce the following.

######  Definition 3.2.

Let $(X,\cdot)$ and $(A,*)$ be racks and let
$\Phi:A\to\operatorname{Conj\,}(\operatorname{Aut}(X))$ be a rack
homomorphism. A relative Rota–Baxter operator (RRB-operator) on $(X,\cdot)$
with respect to $(A,\Phi)$ is a map $B:X\to A$ such that

(6) $B(x)*B(y)=B\left((x\Phi_{B(y)})\cdot y\right),\qquad\qquad(x,y\in X)$

where $\Phi_{B(y)}:=\Phi(B(y))$ is the image of
$B(y)\in\operatorname{Aut}(X)$. The quadruple $(X,A,\Phi,B)$ is called a
relative Rota-Baxter rack (RRB-rack). If $X$ and $A$ are quandles, then the
quadruple $(X,A,\Phi,B)$ is called a relative Rota-Baxter quandle (RRB-
quandle).

If $\Phi={\bf 1}$ is the identity map (i.e.
$\Phi(a)=\Phi_{a}=\mathrm{id}\in\operatorname{Aut}(X)$ for any $a\in A$), then
from (6) we have $B(x)*B(y)=B\bigl{(}x\cdot y\bigr{)}$ for any $x,y\in X$.

This yields the following.

######  Lemma 3.3.

Let $(X,\cdot)$ and $(A,*)$ be two racks. If $f:X\to A$ is a rack
homomorphism, then the quadruple $(X,A,{\bf 1},f)$ is a relative Rota–Baxter
rack.

If $A=X$ and $\Phi:X\to\operatorname{Conj\,}(\operatorname{Inn}(X))$, then we
introduce the following.

######  Definition 3.4.

Let $(X,*)$ be a rack. A map $B:X\to X$ such that

$B(x)*B(y)=B\left((x*B(y))*y\right),\qquad\qquad(x,y\in X)$

is called a Rota–Baxter operator (RB-operator) on $X$ and $(X,B)$ is called an
RB-rack.

######  Remark 3.5.

If $(X,*)$ is a left rack, then one can define an RB-operator on $X$ as a map
$B:X\to X$ such that

$B(x)*B(y)=B\left(x*(B(x)*y)\right),\qquad\qquad(x,y\in X).$

Using these properties of RRB-racks we obtain the following.

######  Theorem 3.6.

Let $(X,\cdot)$ and $(A,*)$ be two racks and let
$\Phi:A\to\operatorname{Conj\,}(\operatorname{Aut}(X))$ be a rack
homomorphism, $a\mapsto\Phi_{a}\in\operatorname{Aut}(X)$. Let $(X,A,\Phi,B)$
be an RRB-rack. Set

$x\circ y:=(x\Phi_{B(y)})\cdot y\in X,\qquad\qquad(x,y\in X).$

  * (i)

If $\operatorname{Im}\Phi=\\{\mathrm{id}\\}$, then

$B(x)*B(y)=B(x\cdot y),\qquad\qquad(x,y\in X).$

Hence, $B:X\to A$ is a homomorphism and $x\circ y=x\cdot y$.

  * (ii)

If $\operatorname{Im}\Phi\not=\\{\mathrm{id}\\}$ and for any $x,y,z\in X$
holds

$\bigl{(}(x\Phi_{B(y)})\Phi_{B(z)}\bigr{)}\cdot
z=\Big{(}\bigl{(}x\Phi_{B(z)}\bigr{)}\Phi_{B((y\Phi_{B(z)})\cdot z)}\Big{)}\\\
\cdot\bigl{(}z\Phi_{B((y\Phi_{B(z)})\cdot z)}),$

then $(X,\circ)$ is a rack.

Moreover, if $(X,A,\Phi,B)$ is a RRB-quandle and $x\Phi_{B(x)}=x$ for any
$x\in X$, then $(X,\circ)$ is a quandle.

###### Proof.

(i) Is evident.

(ii) Let us check the quandle axioms for the groupoid $(X,\circ)$. To get
(Q1), we have to show that

$x\circ x=(x\Phi_{B(x)})\cdot x=x,\qquad\qquad(x\in X)$

which is possible only if $x\Phi_{B(x)}=x$, i.e. the automorphism
$\Phi_{B(x)}$ fixes each $x\in X$.

Since $\Phi_{B(a)}$ is an automorphism and $(X,\cdot)$ is a rack, for any
$a,b\in X$ there exists a unique $x:=\Phi^{-1}_{B(a)}(b\bar{\cdot}a)\in X$
such that $x\circ a=x\Phi_{B(a)})\cdot a=b$. Hence (Q2) holds.

Now, by direct calculation we have

$\begin{split}(x\circ y)\circ z&=\bigl{(}(x\Phi_{B(y)})\cdot y\bigr{)}\circ
z\\\ &=\bigl{(}((x\Phi_{B(y)})\cdot y\bigr{)}\Phi_{B(z)})\cdot
z\qquad\qquad\mbox{($\Phi_{B(z)}$ is an automorphism)}\\\
&=\bigl{(}(x\Phi_{B(y)})\Phi_{B(z)}\cdot(y\Phi_{B(z)})\bigr{)}\cdot z\\\
&=\bigl{(}((x\Phi_{B(y)})\Phi_{B(z)})\cdot
z\bigr{)}\cdot\bigl{(}(y\Phi_{B(z)})\cdot z\bigr{)};\\\ (x\circ z)\circ(y\circ
z)&=\bigl{(}(x\Phi_{B(z)})\cdot z\bigr{)}\circ\bigl{(}(y\Phi_{B(z)})\cdot
z\bigr{)}\\\ &=\bigl{(}((x\Phi_{B(z)})\cdot
z\bigr{)}\Phi_{B((y\Phi_{B(z)})\cdot z)})\cdot\bigl{(}(y\Phi_{B(z)})\cdot
z\bigr{)}.\end{split}$

Using the right cancellation (axiom (Q2)), it follows that

$\begin{split}\bigl{(}(x\Phi_{B(y)})\Phi_{B(z)}\bigr{)}\cdot
z&=\bigl{(}(x\Phi_{B(z)})\cdot z\bigr{)}\Phi_{B((y\Phi_{B(z)})\cdot z)}\\\
&=\bigl{(}x\Phi_{B(z)}\bigr{)}\Phi_{B((y\Phi_{B(z)})\cdot
z)})\cdot\bigl{(}z\Phi_{B((y\Phi_{B(z)})\cdot z)})\end{split}$

because $\Phi_{B((y\Phi_{B(z)})\cdot z)}$ is an automorphism. Consequently,
(Q2) holds. ∎

######  Remark 3.7.

From the proof follows that the equality in (ii) is equivalent to the equality

$\bigl{(}(x\Phi_{B(y)})\Phi_{B(z)}\bigr{)}\cdot z=\bigl{(}(x\Phi_{B(z)})\cdot
z\bigr{)}\Phi_{B((y\Phi_{B(z)})\cdot z)}$

From Theorem 3.6 we have the following.

######  Corollary 3.8.

Let $(X,\cdot,B)$ be an RB-rack. Define on $X$ an operation:

$x\circ y:=(x\cdot B(y))\cdot y,\qquad\qquad(x,y\in X).$

If for any $x,y,z\in X$ holds

$\begin{split}\bigl{(}(x\cdot B(y))&\cdot B(z)\bigr{)}\cdot z\\\
&=\bigl{(}(x\cdot B(z))\cdot z\bigr{)}\cdot B((y\cdot B(z))\cdot
z),\end{split}$

then $(X,\circ)$ is a rack.

Moreover, if $(X,\cdot,B)$ is an RB-quandle and for any $x\in X$ holds the
equation $x\cdot B(x)=x$, then the groupoid $(X,\circ)$ is a quandle.

Some RB-operators are presented in the following.

######  Example 3.9.

  * (i)

Let $X$ be a quandle, $p\in X$. Then the map $B_{p}(x):=p$ ($x\in X$) is an
RB-operator.

  * (ii)

For a trivial quandle $\operatorname{T}$ any map $B:T\to T$ is an RB-operator.

The next properties of RB-operators are evident.

######  Proposition 3.10.

Let $(X,\cdot,B)$ be a Rota–Baxter rack (quandle). Then

  * (i)

the image $\operatorname{Im}B$ of $B$ is a rack (quandle);

  * (ii)

if $(X_{1},B_{1})$ and $(X_{1},B_{1})$ are two Rota–Baxter racks, then the map

$B((x_{1},x_{2})):=(B_{1}(x_{1}),B_{2}(x_{2})),\qquad(x_{i}\in X_{i})$

is a Rota–Baxter operator on $X:=X_{1}\times X_{2}$.

##  4\. Rack extensions and averaging operators on racks

In the first part of the present section we suggest some approach to solution
of the problem of a description semi-direct product of quandles (see [6,
Problem 4.15]). A particular case of this construction was introduced in [32].
Note that, rack extensions were investigated in [1, 2, 9, 18, 27].

######  Theorem 4.1.

Let $(X,\cdot)$ and $(A,*)$ be two racks and let
$\Phi:A\to\operatorname{Conj\,}(\operatorname{Aut}(X))$ be a rack
homomorphism. On the set $Y:=A\times X$ we define the following operation:

$(a,x)\circ(b,y):=(a*b,x\Phi_{b}),\qquad\qquad(a,b\in A,x,y\in X)$

in which $x\Phi_{b}$ means the action of the automorphism $\Phi_{b}=\Phi(b)$
on the element $x$.

The gropoid $(Y,\circ)$ is a rack. Additionally, if $X$ and $A$ are quandles
and $\Phi_{a}$ lies in the kernel of $\Phi$ for any $a\in A$, then $(Y,\circ)$
is a quandle.

###### Proof.

Let us check the rack axioms. The equation $u\circ v=w$ with unknown
$u=(a,x)\in Y$ and given $v=(b,y),w=(c,z)\in Y$ has a unique solution. Indeed,
it has the form

$(a*b,x\Phi_{b})=(c,z).$

The element $a=c\bar{*}b$ is a unique solution for the equation $a*b=c$,
because $A$ is a rack. Furthermore, $\Phi_{b}$ is an automorphism, so the
equation $x\Phi_{b}=c$ has a unique solution $x=c\Phi^{-1}_{b}$. Hence the
axiom (Q2) holds.

Now, it is easy to check that

$\begin{split}(u\circ v)\circ
w&=(a*b,x\Phi_{b})\circ(c,z)=\left((a*b)*c,(x\Phi_{b})\Phi_{c}\right),\\\
(u\circ w)\circ(v\circ
w)&=(a*c,x\Phi_{c})\circ(b*c,y\Phi_{c})=\left((a*c)*(b*c),(x\Phi_{c})\Phi_{b*c}\right),\end{split}$

so we have to prove that $(u\circ v)\circ w=(u\circ w)\circ(v\circ w)$ for any
$u=(a,x)$, $v=(b,y)$, $w=(c,z)\in Y$. Since $A$ is a rack, holds the equality
$(a*b)*c=(a*c)*(b*c)$. Using the fact that $\Phi$ is a rack homomorphism to a
conjugacy quandle, we obtain that

$\Phi_{b*c}=\Phi_{c}^{-1}\Phi_{b}\Phi_{c}\qquad\text{and}\qquad(x\Phi_{b})\Phi_{c}=(x\Phi_{c})\Phi_{b*c},$

so (Q3) holds.

If $X$ and $A$ are quandles and $\Phi_{a}$ lies in the kernel of $\Phi$, then
$u\circ u=(a*a,x\Phi_{a})=(a,x\;\mathrm{id})=u$, so the axiom (Q1) holds. ∎

######  Definition 4.2.

Let the conditions of Theorem 4.1 hold. The rack $A\ltimes_{\Phi}X:=(A\times
X,\circ)$ is called a semi-direct product of $X$ and $A$ with respect to
$\Phi$.

In particular, if $A=\operatorname{Conj\,}(\operatorname{Aut}(X))$ and
$\Phi=\mathrm{id}$, then
$\operatorname{Hol}(X):=\operatorname{Conj\,}(\operatorname{Aut}(X))\ltimes_{\Phi}X$
is called the holomorph of $X$.

Let us define a relative averaging operator on racks.

######  Definition 4.3.

Let $(X,\cdot)$ and $(A,*)$ be racks and let
$\Phi:A\to\operatorname{Conj\,}(\operatorname{Aut}(X))$ be a rack
homomorphism. A map $B:X\to A$ is called a relative averaging operator on $X$
with respect to $(A,\Phi)$, if

$B(x)*B(y)=B(x\Phi_{B(y)}),\qquad(x,y\in X)$

in which $x\Phi_{B(y)}$ is the image of the element $x$ under the action of
the automorphism $\Phi_{B(y)}$. The quadruple $(X,A,\Phi,B)$ is called a
relative averaging rack (RA-rack).

An analogue of the corresponding theorems for Rota–Baxter operators and
relative Rota–Baxter operators for groups (see [25, 28]) is the following.

######  Theorem 4.4.

Let $(X,\cdot)$ and $(A,*)$ be two racks and let
$\Phi:A\to\operatorname{Conj\,}(\operatorname{Aut}(X))$ be a rack
homomorphism. Let $B:X\to A$ be a map and let

$\Gamma_{B}:=\\{(B(x),x)\mid x\in X\\}\subseteq A\ltimes_{\Phi}X.$

The map $B:X\to A$ is a relative averaging operator on $X$ with respect to
$(A,\Phi)$ if and only if $\Gamma_{B}$ is a subrack of $A\ltimes_{\Phi}X$.

###### Proof.

Evidently, $\Gamma_{B}$ is a subset of $A\ltimes_{\Phi}X$, so we have

$(B(x),x)\circ(B(y),y):=(B(x)*B(y),x\Phi_{B(y)}),\qquad(x,y\in X).$

But $(B(x),x)\circ(B(y),y)\in(\Gamma_{B},\circ)$ if and only if
$B(x)*B(y)=B(x\Phi_{B(y)})$ for any $x,y\in X$. ∎

If $X=A$ and $\Phi:X\to\operatorname{Inn}(X)$ such that $\Phi_{B(y)}=S_{B(y)}$
is the inner automorphism of $X$, then we define the following.

######  Definition 4.5.

Let $(X,\cdot)$ be a rack and let $B:X\to X$ be a map. If

$B(x)\cdot B(y)=B(x\cdot B(y)),\qquad(x,y\in X)$

then $B$ is called a right averaging operator on $X$. If

$B(x)\cdot B(y)=B(B(x)\cdot y),\qquad(x,y\in X)$

then $B$ is called a left averaging operator on $X$.

In the sequel of our article we mainly consider right averaging operators
which is called simply averaging operators. A pair $(X,B)$ is called an right
(left) averaging rack, if $X$ is a rack and $B$ is a right (left) averaging
operator. We call a right averaging rack simply an averaging rack.

######  Remark 4.6.

In racks the multiplication on the right is an automorphism, but
multiplication on the left, in general, is not. Consequently, the theory of
right averaging operators is different from the theory of left averaging
operators. It is interesting to construct a theory of left averaging
operators.

Some properties of averaging operators on racks are similar to the properties
of RB-operators on groups (see Proposition 2.3), which shows the following.

######  Theorem 4.7.

Let $(X,B)$ be an averaging rack and let $(X,\circ_{B})$ be a groupoid, in
which

(7) $x\circ_{B}y:=x\cdot B(y),\qquad(x,y\in X).$

The following conditions hold:

  * (i)

$(X,\circ_{B})$ is a rack. Moreover, if $x\cdot B(x)=x$ for all $x\in X$, then
$(X,\circ_{B})$ is a quandle;

  * (ii)

$B$ is an averaging operator on the rack $(X,\circ_{B})$;

  * (iii)

the map $B:(X,\circ_{B})\to(X,\cdot)$ is a homomorphism.

###### Proof.

(i) Let us check the rack axioms. The equation $x\circ_{B}B(a)=b$ has a unique
solution $x=b\,\bar{\cdot}\,B(a)$, i.e. (Q2) holds.

From the equations

$\begin{split}(x\circ_{B}y)\circ_{B}z&=(x\cdot B(y))\circ_{B}z=(x\cdot
B(y))\cdot B(z)=(x\cdot B(z))\cdot(B(y)\cdot B(z))\\\ &=(x\cdot
B(z))\cdot(B(x\cdot B(y));\qquad(\text{by definition of $B$})\\\
(x\circ_{B}z)\circ_{B}(y\circ_{B}z)&=(x\cdot B(z))\circ_{B}(y\cdot
B(z))=(x\cdot B(z))\cdot B(y\cdot B(z))\end{split}$

follows that $(x\circ_{B}y)\circ_{B}z=(x\circ_{B}z)\circ_{B}(y\circ_{B}z)$ for
any $x,y,z\in X$, so (Q3) holds.

Further, by the assumption $x\circ_{B}x=x\cdot B(x)=x$, so (Q1) holds.

(ii) An averaging operator $B$ on $(X,\circ_{B})$ must satisfies the identity

$B(x)\circ_{B}B(y)=B(x\circ_{B}B(y)),\qquad(x,y\in X)$

which is by (7) has the form

$B(x)\cdot B\left(B(y)\right)=B\left(x\cdot
B\left(B(y)\right)\right),\qquad(x,y\in X).$

If we put $z=B(y)$, we get $B(x)\cdot B\left(z\right)=B\left(x\cdot
B(z)\right)$. Since $B$ is an averaging operator on $(X,\cdot)$, this identity
holds.

(iii) By the definition of $B$ and operation $\circ_{B}$, we have

$B(x)\cdot B\left(y\right)=B\left(x\circ_{B}y\right),\qquad(x,y\in X).$

It is means that $B$ is a homomorphism. ∎

######  Problem 4.8.

Let $(X,\cdot,B)$ be an averaging rack. Then $(X,\circ_{B})$ is a rack, where
$\circ_{B}$ as in (7). The set $(X,\cdot,\circ_{B})$ with two operations
$\cdot$ and $\circ_{B}$, is an analogue of a skew brace (see [23]). Which
axioms connect the rack operations $\cdot$ and $\circ_{B}$?

##  5\. Properties of averaging operators on racks

The next properties of averaging operators are evident.

######  Lemma 5.1.

Let $(X,\cdot,B)$ be an averaging rack (quandle).

  * (i)

the image $\operatorname{Im}B$ of $B$ is a rack (quandle);

  * (ii)

if $(X_{1},B_{1})$ and $(X_{1},B_{1})$ are two averaging racks, then the map

$B((x_{1},x_{2})):=(B_{1}(x_{1}),B_{2}(x_{2})),\qquad(x_{i}\in X_{i})$

is an averaging operator on $X:=X_{1}\times X_{2}$.

Let $X_{1}$ and $X_{2}$ be quandles and let $X_{1}\times
X_{2}:=\\{(x^{\prime},x^{\prime\prime})\mid x^{\prime}\in
X_{1},x^{\prime\prime}\in X_{2}\\}$ be their Cartesian product with a
component-wise multiplication. To define operators projectivizations, we fix
elements $y_{0}\in X_{1}$, $z_{0}\in X_{2}$, and define the following maps:

(8) $\begin{split}P_{1}&:X_{1}\times X_{2}\to X_{1}\times X_{2},\quad
P_{1}\big{(}(x^{\prime},x^{\prime\prime})\big{)}=(x^{\prime},z_{0});\\\
P_{2}&:X_{1}\times X_{2}\to X_{1}\times X_{2},\quad
P_{2}\big{(}(x^{\prime},x^{\prime\prime})\big{)}=(y_{0},x^{\prime\prime}).\end{split}$

Clearly, each $P_{i}(X)\cong X_{i}$ and does not depend on the choice of
$y_{0}$ and $z_{0}$, respectively.

######  Remark 5.2.

In the case of racks, the image $P_{i}(X)$ can be non-isomorphic to $X_{i}$.

If $G=G_{1}\times G_{2}$ is a direct product of groups $G_{1}$ and $G_{2}$,
then the projections $P_{i}:G\to G_{i}$ ($i=1,2$) are Rota–Baxter operators. A
similar result for quandles is the following.

######  Proposition 5.3.

Let $X_{1}$ and $X_{2}$ be quandles and let $X:=X_{1}\times X_{2}$ be their
Cartesian product with a component-wise multiplication. Each operator
$P_{i}:X\to X_{i}$ (see (8)) is an averaging operator on $X$.

###### Proof.

Consider the case of the operator $P_{1}$. If
$x=(x_{1},x_{2}),y=(y_{1},y_{2})\in X$, then

$P_{1}(x)*P_{1}(y)=P_{1}(x*P_{1}(y)).$

Indeed, if $z_{0}\in X_{2}$ is our fixed element, then

$\begin{split}P_{1}(x)*P_{1}(y)&=(x_{1},z_{0})*(y_{1},z_{0})=(x_{1}*y_{1},z_{0});\\\
P_{1}(x*P_{1}(y))&=P_{1}((x_{1},x_{2})*(y_{1},z_{0}))=P_{1}((x_{1}*y_{1},x_{2}*z_{0}))=(x_{1}*y_{1},z_{0}).\end{split}$

The case of the operator $P_{2}$ is similar to the first one. ∎

Recall a construction of disjoint union of racks (see, for example, [5,
Proposition 9.2]).

######  Definition 5.4.

Let $(X_{1},*)$, $(X_{2},\circ)$ be racks and let ${\rm Conj}_{-1}\left({\rm
Aut}(X_{1})\right)$, ${\rm Conj}_{-1}\left({\rm Aut}(X_{2})\right)$ be
$(-1)$-conjugation quandles of $X_{1}$ and $X_{2}$, respectively (see Example
2.7(i)). Let

$\sigma:X_{1}\to{\rm Conj}_{-1}\left({\rm
Aut}(X_{2})\right),~{}~{}\tau:X_{2}\to{\rm Conj}_{-1}\left({\rm
Aut}(X_{1})\right)$

be rack homomorphisms. The set $X:=X_{1}\sqcup X_{2}$ with the operation:

$x\star y:=\begin{cases}x*y,&x,y\in X_{1};\\\ x\circ y,&x,y\in X_{2};\\\
{\tau(y)}(x),&x\in X_{1},y\in X_{2};\\\ {\sigma(y)}(x),&x\in X_{2},y\in
X_{1},\end{cases}$

is a rack if and only if the following conditions hold:

  * (i)

$\tau(z)(x)*y=\tau\left(\sigma(y)(z)\right)(x*y)$; ($x,y\in X_{1}$ and $z\in
X_{2}$)

  * (ii)

$\sigma(z)(x)\circ y=\sigma\left(\tau(y)(z)\right)(x\circ y)$, ($x,y\in X_{2}$
and $z\in X_{1}$).

The rack $X_{1}\underset{\sigma,\tau}{\sqcup}X_{2}:=(X_{1}\sqcup X_{2},\star)$
is called a union of racks $X_{1}$ and $X_{2}$ (with respect to
$\sigma,\tau$).

For simplicity we write $\tau_{y}(x)$ instead $\tau(y)(x)$ and $\sigma_{y}(x)$
instead $\sigma(y)(x)$, respectively.

A subrack $Y$ of a rack $(X,*)$ is called normal if $y*x\in Y$ for all $y\in
Y$ and $x\in X$. Evidently, if $Y$ is a normal subrack of $X$, then $Y$ is a
union of orbits in $X$. Racks $X_{1},X_{2}$ are obviously normal in
$X=X_{1}\underset{\sigma,\tau}{\sqcup}X_{2}$, therefore if $X$ is a connected
rack, then it cannot be presented as a union of two subracks. If $X$ is not
connected, $X_{1}$ is an orbit of some element from $X$, $X_{2}=X\setminus
X_{1}$, then $X=X_{1}\underset{\sigma,\tau}{\sqcup}X_{2}$ for appropriate
$\sigma,\tau$. So, a rack $X$ can be presented as a union of two racks if and
only if $X$ is not connected. The dihedral quandle ${\rm R}_{2n}$ of an even
order can be presented as a union ${\rm
R}_{n}\underset{\sigma,\tau}{\sqcup}{\rm R}_{n}$ of two subquandles (see [5])
for appropriates $\sigma,\tau$.

######  Theorem 5.5.

Let $(X_{1},B_{1})$ and $(X_{2},B_{2})$ be racks with the averaging operators
$B_{1}$ and $B_{2}$, respectively. Let
$X=X_{1}\underset{\sigma,\tau}{\sqcup}X_{2}$ for some $\sigma,\tau$. Let
$B:X\to X$ be a map in which

$B(z):=\begin{cases}B_{1}(z),&\qquad\text{for}\quad z\in X_{1};\\\
B_{2}(z),&\qquad\text{for}\quad z\in X_{2}.\end{cases}$

The map $B$ is an averaging operator if and only if for any $z_{1}\in X_{1}$
and $z_{2}\in X_{2}$ hold the following:

$\begin{split}\tau_{B_{2}(z_{2})}(B_{1}(z_{1}))&=B_{1}(\tau_{B_{2}(z_{2})}(z_{1}));\\\
\sigma_{B_{2}(z_{1})}(B_{1}(z_{2}))&=B_{1}(\sigma_{B_{2}(z_{1})}(z_{2})).\end{split}$

###### Proof.

Let us check the identity

(9) $B(z_{1})\star B(z_{2})=B(z_{1}\star B(z_{2})),\qquad(z_{1},z_{2}\in X).$

If $z_{1},z_{2}\in X_{1}$, or $z_{1},z_{2}\in X_{2}$, then (9) follows from
the definition of $B$. If $z_{1}\in X_{1},z_{2}\in X_{2}$, then

$B_{1}(z_{1})\star B_{2}(z_{2})=B_{1}(z_{1}\star B_{2}(z_{2})),$

which is equivalent to
$\tau_{B_{2}(z_{2})}(B_{1}(z_{1}))=B_{1}(\tau_{B_{2}(z_{2})}(z_{1}))$.

Let $z_{1}\in X_{2}$ and $z_{2}\in X_{1}$. It is easy to check that
$B_{2}(z_{1})\star B_{1}(z_{2})=B_{2}(z_{1}\star B_{1}(z_{2}))$, which is
equivalent to
$\sigma_{B_{2}(z_{2})}(B_{1}(z_{1}))=B_{1}(\sigma_{B_{2}(z_{2})}(z_{1}))$.∎

##  6\. Rota–Baxter and averaging operators on rack algebras

Let $A$ be an algebra over a field $\Bbbk$. As we seen in Section 2, the RB-
operator of weight $\lambda\in\Bbbk$ on $A$ is a linear map $R:A\to A$ such
that

(10) $R(a)R(b)=R(R(a)b+aR(b)+\lambda ab),\qquad\qquad(a,b\in A).$

Further, we will call this operator $R$ by algebraic Rota–Baxter operator
(shortly ARB-operator) of weight $\lambda$ on $A$.

######  Example 6.1.

For any algebra $A$, the zero operator: $R(a)=0$ ($a\in A$); and the operator:
$R(a)=-\lambda a$ ($a\in A$), are algebraic Rota–Baxter operators. These
operators are called trivial Rota–Baxter operators.

Let $B$ be an RB-operator on a rack $X$ and $A=\Bbbk[X]$ be its rack algebra
over a field $\Bbbk$. We can extend $B$ by linearity to $A$:

(11) $B\left(\sum_{x\in X}\alpha_{x}\,x\right):=\sum_{x\in
X}\alpha_{x}\,B(x),\qquad\qquad(\alpha_{x}\in\Bbbk,x\in X).$

We will call such operators by rack Rota–Baxter operators on $A$. If $X$ is a
quandle, we will call such operators by quandle Rota–Baxter operators.

######  Remark 6.2.

Further we denote a RB-operator on a rack and its extension on rack algebra by
the same symbol. It is clear from the context where this operator acts.

Also, on rack algebra there exist so called monomial operators, who occupy the
middle position between algebraic and rack RB-operators.

######  Definition 6.3.

An algebraic RB-operator $R:\Bbbk[X]\to\Bbbk[X]$ which acts by the formula

$R\left(\sum_{x\in X}\alpha_{x}\,x\right)=\sum_{x\in
X}\alpha_{x}\,R(x)=\sum_{x\in
X}(\alpha_{x}\beta_{x})y_{x},\qquad\qquad(\alpha_{x},\beta_{x}\in\Bbbk,y_{x}\in
X)$

is said to be a monomial RB-operator on $\Bbbk[X]$.

Now we are considering the following question. Let $(X,*)$ be a rack and let
$B:X\to X$ be an RB-operator on $X$. Under which conditions $B$ induces an
algebraic RB-operator on $\Bbbk[X]$?

An answer gives.

######  Proposition 6.4.

Let $(X,*)$ be a rack and $B:X\to X$ be an $RB$-operator on $X$. The linear
extension of $B$ to $\Bbbk[X]$ is an algebraic RB-operator of weight $\lambda$
(i.e. satisfy (10)) if and only if $\lambda=-1$ and for any $x,y\in X$ we have
either

$\begin{cases}B(x)*B(y)&=B(B(x)*y);\\\ B(x*y)&=B(x*B(y)),\\\
\end{cases}\qquad\text{or}\qquad\begin{cases}B(x)*B(y)&=B(x*B(y));\\\
B(x*y)&=B(B(x)*y).\\\ \end{cases}$

###### Proof.

Let us show that the equality (10) holds if it holds for elements of $X$.
Indeed, if $u=\sum_{x\in X}\alpha_{x}\,x$ and $v=\sum_{y\in
X}\beta_{y}\,y\in\Bbbk[X]$ in which $\alpha_{x},\beta_{y}\in\Bbbk$, then using
(11) we have

$B(u)*B(v)=\sum_{x,y\in X}\alpha_{x}\beta_{y}\,\left(B(x)*B(y)\right),$
$B(B(u)*v)+B(u*B(v))+\lambda B(u*v)=B\left(\sum_{x,y\in
X}\alpha_{x}\beta_{y}\,(B(x)*y)\right)+B\left(\sum_{x,y\in
X}\alpha_{x}\beta_{y}\,(x*B(y))\right)+$ $+\lambda\left(\sum_{x,y\in
X}\alpha_{x}\beta_{y}\,\left(B(x)*B(y)\right)\right).$

Hence, if for any $x,y\in X$ holds (10), then it holds for $u,v$.

Let us find conditions under which the equality (10) holds. Since all elements
$B(x)$, $B(y)$ lies in $X$, the left hand side $B(x)*B(y)$ of (10) must be
equal to

$B(B(x)*y)~{}\mbox{or}~{}B(x*B(y))~{}\mbox{or}~{}\lambda B(x*y).$

But the last case is not possible since $B(B(x)*y)+B(x*B(y))\not=0$. Hence,

$\begin{cases}B(x)*B(y)&=B(B(x)*y);\\\ -\lambda B(x*y)&=B(x*B(y)),\\\
\end{cases}\qquad\text{or}\qquad\begin{cases}B(x)*B(y)&=B(x*B(y));\\\ -\lambda
B(x*y)&=B(B(x)*y),\\\ \end{cases}$

which is possible only if $\lambda=-1$. ∎

######  Proposition 6.5.

Let $\Bbbk[X]$ be the quandle algebra of a quandle $X$ over a field $\Bbbk$.
Let $B:\Bbbk[X]\to\Bbbk[X]$ be a quandle and algebraic RB-operator on the
quandle algebra $\Bbbk[X]$. Then for any $x\in X$ holds

$B(B(x)*x)=B(x*B(x)).$

###### Proof.

Since $B$ is a quandle and algebraic RB-operator, one of the systems from
Proposition 6.4 holds. Suppose that the first system holds. Then for $x=y$ we
have

$\begin{cases}B(x)*B(x)&=B(B(x)*x),\\\ B(x*x)&=B(x*B(x)),\\\
\end{cases}\Leftrightarrow\begin{cases}B(x)&=B(B(x)*x),\\\ B(x)&=B(x*B(x)).\\\
\end{cases}$

Hence, $B(B(x)*x)=B(x*B(x))$. If the second system holds, we get the same
equality. ∎

Taking in attention Example 6.1 we get than

######  Corollary 6.6.

For any rack algebra $\Bbbk[X]$ the identity map is a rack and algebraic
Rota–Baxter operator of weight $\lambda=-1$.

Now we are considering averaging operators.

######  Proposition 6.7.

Any averaging operator on a rack is extended by linearity to an averaging
operator on the rack algebra.

###### Proof.

Consider only right averaging operators, proof for left averaging operators is
similar.

Let $B:X\to X$ be an averaging operator on the rack $(X,*)$, i.e.

$B(x)*B(y)=B(x*B(y)),\qquad\qquad(x,y\in X).$

For any $u=\sum_{x\in X}\alpha_{x}\,x$ and $v=\sum_{y\in
X}\beta_{y}\,y\in\Bbbk[X]$ where $\alpha_{x},\beta_{y}\in\Bbbk$, we have that

$\begin{split}B(u)*B(v)&=\sum_{x,y\in
X}\alpha_{x}\beta_{y}\,\left(B(x)*B(y)\right);\\\
B(u*B(v))&=B\left(\sum_{x,y\in
X}\alpha_{x}\beta_{y}\,\left(x*B(y)\right)\right)=\sum_{x,y\in
X}\alpha_{x}\beta_{y}\,B\left(x*B(y)\right).\end{split}$

Since $B$ is an averaging operator on $X$, we conclude that
$B(u)*B(v)=B(u*B(v))$, so $B$ induced an averaging operator on $\Bbbk[X]$. ∎

######  Question 6.8.

Let $\Bbbk[\operatorname{R}_{n}]$ be a quandle algebra of a dihedral quandle
$\operatorname{R}_{n}$ over a field $\Bbbk$. Find:

– Averaging operators on $\operatorname{R}_{n}$ and on
$\Bbbk[\operatorname{R}_{n}]$;

– Rota–Baxter operators on $\operatorname{R}_{n}$;

– (monomial) Rota–Baxter operators on $\Bbbk[\operatorname{R}_{n}]$.

##  7\. Some generalization of conjugation quandle and core quandle

In Example 2.7, for each group $G$ we constructed the conjugation quandle
$\operatorname{Conj\,}(G)$ and the core quandle $\operatorname{Core}(G)$. The
following definition generalizes these constructions.

######  Definition 7.1.

Let $(G,B)$ be a Rota–Baxter group. Define the following two groupoids:

  * (i)

$B$-conjugation groupoid: $(\operatorname{Conj\,}_{B}(G),*_{B})$ in which
$x*_{B}y:=B(y)xB(y)^{-1}$, for $x,y\in G$;

  * (ii)

$B$-core groupoid: $(\operatorname{Core}_{B}(G),*_{B})$ in which
$x*_{B}y:=B(y)x^{-1}B(y)$ for $x,y\in G$.

Evidently, $(a*_{B}b)*_{B}b=a$ for all $a,b\in G$ in
$\operatorname{Core}_{B}(G)$.

######  Example 7.2.

Let $(G,B)$ be an RB-group. Let $(\operatorname{Conj\,}_{B}(G),*_{B})$ and
$(\operatorname{Core}_{B}(G),*_{B})$ be groupoids defined in Definition 7.1.
Let $B_{[-1]}$ and $B_{[0]}$ be maps from Example 2.2.

  * (i)

If $B:=B_{[-1]}$, then the operation $*_{B}$ in the groupoid
$\operatorname{Conj\,}_{B}(G)$ has the form $x*_{B}y:=y^{-1}xy$ and
$\operatorname{Conj\,}_{B}(G)=\operatorname{Conj\,}(G)$ is the conjugation
quandle.

  * (ii)

If $B:=B_{[-1]}$, then the operation $*_{B}$ in $\operatorname{Core}_{B}(G)$
has the form $x*_{B}y:=y^{-1}x^{-1}y^{-1}$. Clearly, in general,
$(\operatorname{Core}_{B}(G),*_{B})$ is not a rack.

  * (iii)

If $B:=B_{[0]}$ then the operation in $\operatorname{Conj\,}_{B}(G)$ has the
form $x*_{B}y:=x$ that means that $\operatorname{Conj\,}_{B}(G)$ is a trivial
quandle.

  * (iv)

If $B:=B_{[0]}$ then the operation in $\operatorname{Core}_{B}(G)$ has the
form $x*_{B}y:=x^{-1}$ and $(\operatorname{Core}_{B}(G),*_{B})$ is a rack. If
$G$ is an elementary abelian $2$-group, then $\operatorname{Core}_{B}(G)$ is a
quandle.

Let $\mathcal{RB}(G)$ be a set of all $RB$-operators on a group $G$. If
$\mathcal{G}$ is the set of all groupoids on the set $G$, then there exist the
following two maps:

$\Psi_{1},\Psi_{2}:G\times\mathcal{RB}(G)\to\mathcal{G},$

which are defined (see Definition 7.1) by the rules:

(12) $\begin{split}\Psi_{1}((G,B))&\mapsto(G,x*_{B}y:=B(y)xB(y)^{-1}),\\\
\Psi_{2}((G,B))&\mapsto(G,x*_{B}y:=B(y)x^{-1}B(y)),\qquad\quad(x,y\in
G).\end{split}$

In Proposition 2.3 for each RB-group $(G,B)$ was defined a new operation
$\circ$ on $G$, such that $(G,B,\circ)$ becomes a group. Conditions on an RB-
group $(G,B)$ under which groupoids $(G,B,*_{B})$ from Definition 7.1 become
racks or/and quandles, provide the following.

######  Theorem 7.3.

Let $(G,B)$ be a Rota–Baxter group. Let $\zeta(G)$ be the center of $G$. Let
$(\operatorname{Conj\,}_{B}(G),*_{B})$ and
$(\operatorname{Core}_{B}(G),*_{B})$ be groupoids from Definition 7.1. The
following conditions hold:

  * (i)

if $B(b)^{-1}b\in\zeta(G)$ for any $b\in G$, then the $B$-conjugation groupoid
$(\operatorname{Conj\,}_{B}(G),*_{B})$ is a rack. Moreover, if $B(a)\in
C_{G}(a)$ for any $a\in G$, then the $B$-conjugation groupoid
$(\operatorname{Conj\,}_{B}(G),*_{B})$ is a quandle;

  * (ii)

if

$B(c)B(b)^{-1}B(c)=B\left(B(c)^{-1}B(b)^{-1}B(c)^{-1}\right),\qquad\qquad(b,c\in
G)$

then the $B$-core groupoid $(\operatorname{Core}_{B}(G),*_{B})$ is a rack.
Moreover, if $\left(aB(a)\right)^{2}=1$ for any $a\in G$, then
$(\operatorname{Core}_{B}(G),*_{B})$ is a quandle.

###### Proof.

(i) For each $a,b\in G$ there is a unique $x=B(a)^{-1}bB(a)\in G$ such that
$x*_{B}a=b$. Since

$\begin{split}(a*_{B}b)*_{B}c&=B(c)B(b)aB(b)^{-1}B(c)^{-1};\\\
(a*_{B}c)*_{B}(b*_{B}c)&=B(c)bab^{-1}B(c)^{-1},\end{split}$

we obtain that $B(b)aB(b)^{-1}=bab^{-1}$ by (Q3). This yields that
$B(b)^{-1}b\in\zeta(G)$. According to (Q2), we have that $B(a)aB(a)^{-1}=a$,
so $B(a)\in C_{G}(a)$.

(ii) The proof is similar. ∎

Let $I$ be a set. Let $(G,B_{i})$, $i\in I$ be RB-groups, such that the set of
their images

(13) $\\{\Psi_{k}(G,B_{i}),\mid i\in
I,k=1,2;\quad\Psi_{1},\Psi_{2}\quad\text{from \eqref{ZZAQAZ}}\\}$

consists of quandles which were constructed in Theorem 7.3. Let $(G,*_{i},i\in
I)$ be the algebraic system with the set of quandle operations
$*_{i}:=*_{B_{i}}$. This algebraic system is called an $I$-quandle system.

Define a product of quandle operations $*_{i}$ (see [7]) by the rule:

(14) $g*_{s}*_{t}h:=(g*_{s}h)*_{t}h,\qquad\qquad(g,h\in G).$

According to [7], a pair $(G,*_{s}*_{t})$ in which $s,t\in I$ is a quandle if
and only if

(15) $(g*_{s}h)*_{t}q=(g*_{t}q)*_{s}(h*_{t}q),\qquad\qquad(g,h\in G,s,t\in
I).$

An $I$-quandle system $(G,*_{i},i\in I)$ in which (15) holds for any $s,t\in
I$ is called a multi-quandle (see Turaev [44]). Turaev used multi-quandles for
construction of some knot invariants. A similar conception of multi-groups
were introduced and studied in [10, 34].

######  Proposition 7.4.

Let $I$ be a set. Let $(G,B_{i})$, $i\in I$ be RB-groups such that
$(G,*_{i},i\in I)$ is an $I$-quandle system, in which

(16) $x*_{i}y:(=x*_{B_{i}}y)=B_{i}(y)xB_{i}(y)^{-1},\qquad(x,y\in G,i\in I).$

This quandle system is a multi-quandle if and only if

$B_{t}(q)^{-1}B_{s}\left(B_{t}(q)hB_{t}(q)^{-1}\right)^{-1}B_{t}(q)B_{s}(h)$

lies in the center $\zeta(G)$.

###### Proof.

Let $*_{i}$ be a quandle operation on $G$ for each $i$ defined by (16).
Clearly,

$\begin{split}(g*_{s}h)*_{t}q&=g^{B_{s}(h)^{-1}B_{t}(q)^{-1}};\\\
(g*_{t}q)*_{s}(h*_{t}q)&=g^{B_{t}(q)^{-1}B_{s}\left(B_{t}(q)hB_{t}(q)^{-1}\right)^{-1}}.\end{split}$

According to (15), we obtain that
$g^{B_{s}(h)^{-1}B_{t}(q)^{-1}}=g^{B_{t}(q)^{-1}B_{s}\left(B_{t}(q)hB_{t}(q)^{-1}\right)^{-1}}$
and

$B_{t}(q)^{-1}B_{s}\left(B_{t}(q)hB_{t}(q)^{-1}\right)^{-1}B_{t}(q)B_{s}(h)\in\zeta(G).$

∎

######  Proposition 7.5.

Let $I$ be a set. Let $(G,B_{i})$, $i\in I$ be RB-groups such that
$(G,*_{i},i\in I)$ is an $I$-quandle system, in which

(17) $x*_{i}y:(=x*_{B_{i}}y)=B_{i}(y)x^{-1}B_{i}(y),\qquad(x,y\in G,i\in I).$

This quandle system is a multi-quandle if and only if for any $h,q\in G$ and
$s,t\in I$ holds

$\begin{split}B_{t}(q)B_{s}(h)^{-1}gB_{s}(h)^{-1}B_{t}(q)=&B_{s}\left(B_{t}(q)h^{-1}B_{t}(q)\right)\times\\\
&\times
B_{t}(q)^{-1}gB_{t}(q)^{-1}B_{s}\left(B_{t}(q)h^{-1}B_{t}(q)\right).\end{split}$

###### Proof.

Let $*_{i}$ be a quandle operation on $G$ for each $i$ defined by (17).
Clearly,

$\begin{split}(g*_{s}h)*_{t}q&=B_{t}(q)B_{s}(h)^{-1}gB_{s}(h)^{-1}B_{t}(q);\\\
(g*_{t}q)*_{s}(h*_{t}q)&=B_{s}\left(B_{t}(q)h^{-1}B_{t}(q)\right)B_{t}(q)^{-1}gB_{t}(q)^{-1}B_{s}\left(B_{t}(q)h^{-1}B_{t}(q)\right).\end{split}$

According to (15), we obtain that

$\begin{split}B_{t}(q)B_{s}(h)^{-1}gB_{s}(h)^{-1}B_{t}(q)&=B_{s}\left(B_{t}(q)h^{-1}B_{t}(q)\right)\times\\\
&\times
B_{t}(q)^{-1}gB_{t}(q)^{-1}B_{s}\left(B_{t}(q)h^{-1}B_{t}(q)\right).\end{split}$

∎

##  8\. Open problems

######  Question 8.1.

We know definition of Rota–Baxter operator on Lie algebra. Is it possible to
define Rota–Baxter operators on a Lie super-algebra?

###  8.1. Other operators

On arbitrary algebra $A$ it is possible defined the following operators:

  * (i)

Reynolds operator: $R(x)R(y)=R(xR(y)+R(x)y-R(x)R(y))$;

  * (ii)

Nijenhuis operator: $R(x)R(y)=R(xR(y)+R(x)y-R(xy))$;

  * (iii-a)

Right averaging operator: $R(x)R(y)=R(xR(y))$;

  * (iii-b)

Left averaging operator: $R(x)R(y)=R(R(x)y)$;

  * (vi)

Differential operator of weight $\lambda$: $d(xy)=d(x)y+xd(y)+\lambda
d(x)d(y)$.

######  Question 8.2.

Is it possible to define these operators on groups (quandles, racks)?

######  Question 8.3.

Let $(G,B)$ be a Rota–Baxter group. Under which conditions $B$ defines a
$RB$-operator on the conjugacy quandle $\operatorname{Conj\,}(G)$, on the core
quandle $\operatorname{Core}(G)$, on the generalized Alexander quandle
$Alex(G,\varphi)$, respectively?

###  8.2. From Lie racks to Leibniz algebra

In [25] it has been proven that if $(G,B)$ is a Rota–Baxter Lie group, then
the tangent map $B$ at identity is a Rota–Baxter operator of weight $1$ on the
Lie algebra of the Lie group $G$.

A pointed rack $(X,*,1)$ is a rack $(X,*)$, with a distinguished element $1\in
X$ such that

$x*1=x,\quad\text{and}\quad 1*x=1\qquad(x\in X).$

Imitating the notion of a Lie group, the smooth version of a pointed rack is
called Lie rack. Let $(X,*,1)$ be a smooth pointed rack and let $T_{1}X$ be
the tangent space of it in $1$. In [32] was proved that there exists a
bilinear mapping $[\cdot,\cdot]:T_{1}X\times T_{1}X\to T_{1}X$ such that
$(T_{1}X,[\cdot,\cdot])$ is a Leibniz algebra.

######  Question 8.4.

Let $(X,*,1)$ be a smooth pointed rack with a relative Rota–Baxter or
averaging operator $B$, and $(T_{1}X,[\cdot,\cdot])$ be the corresponding
Leibniz algebra. What operator induces $B$ on this algebra?

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