# A Solvable Model of a Nonlinear extension of Quantum Mechanics

Alan Chodos<EMAIL_ADDRESS>Dept. of Physics, University of Texas at
Arlington, 502 Yates Street, Box 19059, Arlington, TX 76019  Fred Cooper
<EMAIL_ADDRESS>The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM
87501, USA  Theoretical Division and Center for Nonlinear Studies, Los Alamos
National Laboratory, Los Alamos, NM 87545

###### Abstract

We introduce a particular nonlinear generalization of quantum mechanics which
has the property that it is exactly solvable in terms of the eigenvalues and
eigenfunctions of the Hamiltonian of the usual linear quantum mechanics
problem. We hope that this simple example will elucidate some of the issues of
interpreting nonlinear generalization of quantum mechanics that have been put
forth to resolve questions about quantum measurement theory.

###### pacs:

3.65Ud, 3.65Ta, 05.45.-a, 11.10.Lm

††preprint: nonlinearQM.tex

##  I Introduction

There are a number of reasons to entertain modifications of quantum mechanics.
One is simply to push the envelope Weinberg89 ; Kaplan22 : is it possible that
the Schrödinger equation is an approximation, albeit a very good one, to
something else? What could that something else be?

Another is to try to cure some of the perceived difficulties with quantum
mechanics as we know it: can we introduce collapse of the wave function as
part of the dynamics without appeal to a separate postulate Ghiradi86 ;
Pearle89 ; Bassi13 ? Can we avoid probabilities by finding hidden variables
Hance22  that secretly determine the results of measurement, without running
afoul of Bell’s theorem and related restrictions? (Bell said Bell  “if [a
hidden-variable theory] is local it will not agree with quantum mechanics, and
if it agrees with quantum mechanics it will not be local.”) Yet another
motivation is the problem of uniting quantum mechanics with a theory of
gravity. Most such attempts assume that quantum ideas remain intact and that
gravity should be modified. But a minority view Diosi97 ; Penrose96 ;
Penrose14  holds that quantum mechanics should also be subject to change. In
particular it has been suggested that, to incorporate gravity, quantum
mechanics might have to become non-linear. To explore some of these issues in
a manageable context, in this paper we introduce a particularly simple non-
linearity into the Schrödinger equation, which has the advantage that we can
completely solve for its effects, whatever the Hamiltonian. However, the non-
linearity raises questions of interpretation, which we discuss in the
conclusions. Because the interpretation is not settled, we cannot definitively
yet say whether our modification suffers from some of the problems of previous
attempts to introduce non-linearities, notably, for example, Weinberg’s scheme
Weinberg89  , which was shown, by Gisin Gisin90  and Polchinski Polchinski 91
, to involve superluminal communication and other alleged pathologies.

To proceed, we replace the single Schrödinger equation

${\rm i}\frac{\partial}{\partial t}\ket{\psi}=H\ket{\psi}.$ (1)

with a pair of nonlinear Schrödinger equations

$\displaystyle{\rm i}\frac{\partial}{\partial t}\ket{\psi}$
$\displaystyle=H\ket{\psi}+g\ket{\phi}\bra{\phi}\ket{\psi}$ $\displaystyle{\rm
i}\frac{\partial}{\partial t}\ket{\phi}$
$\displaystyle=H\ket{\phi}+g^{\ast}\ket{\psi}\bra{\psi}\ket{\phi}$ (2)

Here $g$ is a coupling constant, which we take, in general, to be complex:
$g=a+{\rm i}b$. Most of our analysis considers the case $b\neq 0$. The $b=0$
case is discussed briefly in the Appendix.

We stress that our equations are valid for any dynamical system to which the
Hamiltonian applies, be it non-relativistic quantum mechanics, or relativistic
field theory, or something else. As we shall see, assuming the eigenvalue
problem for $H$ is solved, we are able to provide complete solutions for
$\ket{\psi}$and $\ket{\phi}$. . But we have assigned a pair of state vectors
to our system, not just one. We defer for now the question of how to make use
of this embarrassment of riches.

##  II Solution of the Nonlinear equations

Eqs. (I) have one immediate set of solutions: choose two orthogonal vectors,
$\ket{A(t)}$ and $\ket{B(t)}$, each of which solves the original Schrödinger
equation. Then $\ket{\psi}=\ket{A}$ and $\ket{\phi}=\ket{B}$ clearly solve the
new equations. This is an exceptional case; we are interested in the general
case in which $\Braket{\phi}{\psi}\neq 0$.

To solve these equations when $\Braket{\phi}{\psi}\neq 0$. we consider the
inner products:

$N=\Braket{\psi}{\psi}+\Braket{\phi}{\phi};~{}\tau=\Braket{\psi}{\psi}-\Braket{\phi}{\phi};~{}\gamma=\Braket{\phi}{\psi};\delta=\gamma^{\ast}\gamma.$
(3)

As a consequence of Eqs (I) and their complex conjugates, these quantities
obey equations that are independent of $H$:

$\frac{dN}{dt}=0;~{}\dot{\tau}=4b\delta;~{}\dot{\gamma}={\rm
i}g\gamma\tau;~{}\dot{\delta}=-2b\tau\delta.$ (4)

From the differential equations alone, there are two solutions for $\tau$ and
$\delta$:

$\displaystyle\tau$
$\displaystyle=2\omega_{0}\tanh(2b\omega_{0}t);~{}~{}\delta=\omega_{0}^{2}~{}\text{sech}^{2}(2b\omega_{0}t)$
$\displaystyle\tau$
$\displaystyle=2\omega_{0}\coth(2b\omega_{0}t);~{}~{}\delta=-\omega_{0}^{2}~{}\text{cosech}^{2}(2b\omega_{0}t)$
(5)

Here $\omega_{0}$ is constant and in the future for simplicity we will let
$\xi=2b\omega_{0}t$.

But $\delta$ must be positive, and we want $\tau$ to be non-singular; hence we
are limited to the first solution.

Note that we have the relationship

$\omega_{0}^{2}=\frac{\tau^{2}}{4}+\delta,$ (6)

From the equation for $\gamma$ we obtain:

$\gamma=\omega_{0}e^{i\theta}(\cosh\xi)^{{\rm i}\frac{g}{b}}$ (7)

where $\theta$ is an arbitrary phase. Writing $g=a+{\rm i}b$ we can separate
the real and imaginary parts of $\gamma$

$\displaystyle\gamma$
$\displaystyle=\omega_{0}\text{sech}(\xi)\left(\cos\left(\theta+\frac{a\log\left(\cosh^{2}(\xi)\right)}{2b}\right)+i\sin\left(\theta+\frac{a\log\left(\cosh^{2}(\xi)\right)}{2b}\right)\right))$
(8) $\displaystyle=\omega_{0}\sech\xi e^{{\rm
i}\varphi}=\gamma_{1}+i\gamma_{2},$

where
$\varphi=\left(\theta+\frac{a\log\left(\cosh^{2}(\xi)\right)}{2b}\right)$.

It is convenient to parameterize $\ket{\psi}$ and $\ket{\phi}$ as follows:

$\displaystyle\ket{\psi}$
$\displaystyle=\gamma^{1/2}\sum_{n}(\hat{\psi}_{n}(t)e^{-{\rm
i}E_{n}t})\ket{n}$ (9) $\displaystyle\ket{\phi}$
$\displaystyle=(\gamma^{\ast})^{1/2}\sum_{n}(\hat{\phi}_{n}(t)e^{-{\rm
i}E_{n}t})\ket{n}$ (10)

Here $\ket{n}$ is a complete set of eigenvectors of $H,$ with eigenvalues
$E_{n}$. For notational convenience, we assume the spectrum of $H$is discrete,
but this is not necessary.

We find that $\hat{\psi}$ and $\hat{\phi}$ obey the coupled equations

$\displaystyle{\rm i}\dot{\hat{\psi}}_{n}-\frac{g}{2}\tau\hat{\psi}_{n}$
$\displaystyle=g\hat{\phi}_{n}|\gamma|=g\hat{\phi}_{n}\delta^{\frac{1}{2}},$
$\displaystyle{\rm
i}\dot{\hat{\phi}}_{n}+\frac{g^{\ast}}{2}\tau\hat{\phi}_{n}$
$\displaystyle=g^{\ast}\hat{\psi}_{n}|\gamma|=g^{\ast}\hat{\psi}_{n}\delta^{\frac{1}{2}},$
(11)

which after some algebra, lead to the uncoupled second-order equations:

$\displaystyle\ddot{\hat{\psi}}_{n}+g^{2}\omega_{0}^{2}\hat{\psi}_{n}=0$
$\displaystyle\ddot{\hat{\phi}}_{n}+g^{\star
2}\omega_{0}^{2}\hat{\phi}_{n}=0.$ (12)

The general solution to these equations is

$\displaystyle\hat{\psi}_{n}$ $\displaystyle=A_{n}e^{-\omega_{0}bt}e^{{\rm
i}a\omega_{0}t}+B_{n}e^{+\omega_{0}bt}e^{-{\rm i}a\omega_{0}t}$
$\displaystyle\hat{\phi}_{n}$ $\displaystyle=C_{n}e^{+\omega_{0}bt}e^{{\rm
i}a\omega_{0}t}+D_{n}e^{-\omega_{0}bt}e^{-{\rm i}a\omega_{0}t}.$ (13)

We must satisfy the first order equations Eq. (II) for $\hat{\phi}_{n}$ and
$\hat{\psi}_{n}$ and make sure that $\Braket{\phi}{\psi}=\gamma$ This
determines $C_{n},D_{n}$.

$C_{n}=-A_{n};~{}~{}D_{n}=B_{n},$ (14)

and also imposes the normalization condition:

$\sum_{n}(B_{n}^{\ast}B_{n}-A_{n}^{\ast}A_{n})=1.$ (15)

and the orthogonality condition

$\sum_{n}B_{n}^{\ast}A_{n}=0.$ (16)

It is useful to rescale $A_{n},~{}B_{n}$ by $\sinh\vartheta,\cosh\vartheta$
respectively and then impose the normalizations.

$\sum_{n}({A}_{n})^{\star}A_{n}=\sum_{n}({B}_{n})^{\star}B_{n}=1.$ (17)

One has that $\vartheta$ is related to the norm $N$:

$N=2\omega_{0}\cosh[2\vartheta].$ (18)

We can summarize the above as follows. Let $\ket{A}$ and $\ket{B}$ be two
normalized orthogonal solutions of the original Schrödinger equation. i.e.

$\ket{A}=\sum_{n}{A}_{n}e^{-iE_{n}t}\ket{n};~{}~{}\ket{B}=\sum_{n}{B}_{n}e^{-iE_{n}t}\ket{n}$
(19)

Then

$\displaystyle\ket{\psi}$
$\displaystyle=\gamma^{1/2}\left[e^{ig\omega_{0}t}\sinh\vartheta\ket{A}+e^{-ig\omega_{0}t}\cosh\vartheta\ket{B}\right]$
$\displaystyle\ket{\phi}$
$\displaystyle={\gamma^{\ast}}^{1/2}\left[-e^{ig^{\ast}\omega_{0}t}\sinh\vartheta\ket{A}+e^{-ig^{\star}\omega_{0}t}\cosh\vartheta\ket{B}\right]$
(20)

Note that $\gamma=\Braket{\phi}{\psi}$ goes to zero as
$t\rightarrow\pm\infty$, so that in those regions we expect that $\ket{\psi}$
and $\ket{\phi}$ obey the usual Schrödinger equation. In fact we find that
indeed as $t\rightarrow-\infty$,

$\displaystyle\ket{\psi}$
$\displaystyle\rightarrow\gamma^{1/2}e^{ig\omega_{0}t}\sinh\vartheta\ket{A}\rightarrow\sqrt{2\omega_{0}}e^{i\hat{\theta}/2}\sinh\vartheta\ket{A}$
$\displaystyle\ket{\phi}$
$\displaystyle\rightarrow{\gamma^{\ast}}^{1/2}e^{-ig^{\star}\omega_{0}t}\cosh\vartheta\ket{B}\rightarrow\sqrt{2\omega_{0}}e^{-i\hat{\theta}/2}\cosh\vartheta\ket{B}.$
(21)

whereas as $t\rightarrow+\infty$,

$\displaystyle\ket{\psi}$
$\displaystyle\rightarrow\gamma^{1/2}e^{-ig\omega_{0}t}\cosh\vartheta\ket{B}\rightarrow\sqrt{2\omega_{0}}e^{i\hat{\theta}/2}\cosh\vartheta\ket{B}$
$\displaystyle\ket{\phi}$
$\displaystyle\rightarrow-{\gamma^{\ast}}^{1/2}e^{-ig^{\star}\omega_{0}t}\sinh\vartheta\ket{A}\rightarrow-\sqrt{2\omega_{0}}e^{-i\hat{\theta}/2}\sinh\vartheta\ket{A}.$
(22)

We see we can write this in terms of an $S$ matrix which is unitary and which
takes the state at $t\rightarrow-\infty$ to the state at
$t\rightarrow+\infty$. We find

$\left(\begin{array}[]{c}\ket{\psi}_{+}\\\ \ket{\phi}_{+}\\\
\end{array}\right)=S\left(\begin{array}[]{c}\ket{\psi}_{-}\\\
\ket{\phi}_{-}\\\ \end{array}\right)$ (23)

where

$S=\left(\begin{array}[]{cc}0&e^{i\hat{\theta}}\\\ -e^{-i\hat{\theta}}&0\\\
\end{array}\right)$ (24)

where $\hat{\theta}=\theta-\frac{a}{b}\ln 2$. We can now introduce the density
matrix $\rho$ via

$\rho=\frac{1}{N}\left({\ket{\psi}}{\bra{\psi}}+{\ket{\phi}}{\bra{\phi}}\right)$
(25)

which statisfies $Tr\rho=1$. Unlike the individual $\ket{\psi}$ and
$\ket{\phi}$, $\rho$ satisfies the linear evolution equation of the usual
Schrödinger equation.

${\rm i}\frac{\partial\rho}{\partial t}=[H,\rho].$ (26)

Using the explicit forms of $\ket{\psi}$ and $\ket{\phi}$ Eqs. (II) we obtain

$\rho=\frac{2\omega_{0}}{N}\left[\sinh^{2}\vartheta\ket{A}\bra{A}+\cosh^{2}\vartheta\ket{B}\bra{B}\right].$
(27)

Thus we see that the eigenstates of $\rho$ are the vectors
$\left(\ket{A},\ket{B}\right)$ with eigenvalues
$\frac{2\omega_{0}}{N}\left(\sinh^{2}\vartheta,\cosh^{2}\vartheta\right)$.

We find by straightforward calculation that

$Tr\rho^{2}=1-\frac{2\cal S}{N^{2}}$ (28)

where the positive quantity $\cal S$, called the Schwarz parameter is given
by:

$\displaystyle\cal S$
$\displaystyle=\bra{\psi}\ket{\psi}\bra{\phi}\ket{\phi}-\bra{\psi}\ket{\phi}\bra{\phi}\ket{\psi}$
(29)
$\displaystyle=4\omega^{2}_{0}\left(\sinh^{2}\vartheta\cosh^{2}\vartheta\right).$

which is clearly independent of time. We see that $\cal S$ can vanish if and
only if $\ket{\psi}$ and $\ket{\phi}$ are proportional which requires
$\vartheta=0$. In that case both $\ket{\psi}$ and $\ket{\phi}$ are
proportional to $\ket{B}$ and the system is in a “pure” state. Because $\cal
S$ is a constant, $\ket{\psi}$ and $\ket{\phi}$ are always proportional or
they never are.

##  III Lagrangian Formalism

We have not been able to find a Lagrangian that will yield eqs. (I) directly,
and we are not sure if one exists. However, once we have solved for
$\gamma(t)$ we can linearize eqs. (I) to obtain

${\rm i}\frac{\partial}{\partial t}\ket{\psi}=H\ket{\psi}+g\gamma\ket{\phi}$
(30) ${\rm i}\frac{\partial}{\partial
t}\ket{\phi}=H\ket{\phi}+g^{\ast}\gamma^{\ast}\ket{\psi}$ (31)

This version of our equations can be derived from the action:

$\displaystyle\Gamma$ $\displaystyle=\int dt~{}\cal{L}$ $\displaystyle=\int
dt~{}\left[\bra{\psi}\left({\rm i}\frac{\partial}{\partial
t}-H\right)\ket{\psi}\right.$ $\displaystyle\left.+\bra{\phi}\left({\rm
i}\frac{\partial}{\partial t}-H\right)\ket{\phi}\right.$ ,
$\displaystyle\left.-(r_{1}+ir_{2})\bra{\psi}\ket{\phi}-(r_{1}-ir_{2})\bra{\phi}\ket{\psi}\right],$
(32)

from the equations:

$\frac{\delta\Gamma}{\delta\bra{\psi}}=0;~{}~{}\frac{\delta\Gamma}{\delta\bra{\phi}}=0.$
(33)

Here $g\gamma=r_{1}+{\rm i}r_{2}=(a\gamma_{1}-b\gamma_{2})+{\rm
i}(b\gamma_{1}+a\gamma_{2})$, and $\gamma_{1},\gamma_{2}$ are given in Eq.
(8).

##  IV Dissipation Function Formulation

Although the linearized equations have a Lagrangian formalism, if we naively
generalize Eq. (III) we obtain a real action which only gives the real part of
the full equations. That is the naive generalization is

$\displaystyle\Gamma$ $\displaystyle=\int dt~{}\cal{L}$ (34)
$\displaystyle=\int dt~{}\left[\bra{\psi}\left({\rm i}\frac{\partial}{\partial
t}-H\right)\ket{\psi}\right.$ $\displaystyle\left.+\bra{\phi}\left({\rm
i}\frac{\partial}{\partial
t}-H\right)\ket{\phi}-(g+g^{\ast})/2\bra{\phi}\ket{\psi}\bra{\psi}\ket{\phi}\right],$

which only depends on the the real part of $g$.

Since the nonlinear terms in our equation are similar to a complex self
consistent potential, the equations we are studying here share some properties
with the usual nonlinear Schrödinger equation in an external complex
potential,. That problem has been studied using a Dissipation Functional
Formalism Mertens  . So at least formally we can derive our nonlinear complex
equations using this formalism. Again writing the complex coupling constant
here as $g=a+{\rm i}b$, we find now that the real part of Eqs. (I) can be
obtained from the simple action:

$\displaystyle\Gamma$ $\displaystyle=\int dt~{}\cal{L}$ (35)
$\displaystyle=\int dt~{}\left[\bra{\psi}\left({\rm i}\frac{\partial}{\partial
t}-H\right)\ket{\psi}\right.$ $\displaystyle\left.+\bra{\phi}\left({\rm
i}\frac{\partial}{\partial
t}-H\right)\ket{\phi}-a\bra{\phi}\ket{\psi}\bra{\psi}\ket{\phi}\right],$

by varying this action with respect to $\bra{\psi}$ and $\bra{\phi}$. In
analogy with Mertens , we find that there is a dissipation functional

$F=\int dt~{}\mathcal{F}$ (36)

such that the full Eqs. (I) can be obtained from:

$\displaystyle\frac{\delta\Gamma}{\delta\bra{\psi}}$
$\displaystyle=-\frac{\delta F}{\delta\bra{\psi_{t}}}$
$\displaystyle\frac{\delta\Gamma}{\delta\bra{\phi}}$
$\displaystyle=-\frac{\delta F}{\delta\bra{\phi_{t}}}$ (37)

Here we use the notation $\bra{\phi_{t}}=\partial_{t}\bra{\phi}$ etc. In the
$``x"$ representation this would translate to

$\frac{\delta\Gamma}{\delta\bra{\partial_{t}\psi}}\rightarrow\frac{\delta\Gamma}{\delta\partial_{t}\psi^{\star}(x,t)}.$
(38)

The appropriate $\mathcal{F}$ is given by

$\mathcal{F}=-{\rm
i}{b}\left[\bra{\phi}\ket{\psi}\bra{\psi_{t}}\ket{\phi}-\bra{\phi}\ket{\psi_{t}}\bra{\psi}\ket{\phi}-\bra{\psi}\ket{\phi}\bra{\phi_{t}}\ket{\psi}+\bra{\psi}\ket{\phi_{t}}\bra{\phi}\ket{\psi}\right]$
(39)

The complex conjugate of Eqs. (I) can be obtained from the complex conjugate
of Eqs. (IV). So at least formally we can derive Eqs. (I) using this
formalism.

##  V Discussion

In ordinary quantum mechanics, a dynamical system is described by a state
vector evolving according to a particular Hamiltonian. In our modified
version, in contrast, we need two orthogonal such state vectors, $\ket{A}$ and
$\ket{B}$ , and the evolution is given by a linear combinations of these,
$\ket{\psi}$ and $\ket{\phi}$ with time-dependent coefficients that depend on
2 additional parameters, $\omega_{0}$ and $\vartheta$. So a given dynamical
system is somewhat more intricate than ordinary quantum mechanics would lead
us to believe.

In order fully to determine either $\ket{\psi}$ or $\ket{\phi}$ we need to
specify their asymptotic values both as $t\rightarrow-\infty$ and as
$t\rightarrow+\infty$. Once we so specify one of $\ket{\psi}$ or $\ket{\phi}$
, the other is then determined. This circumstance has some similarities with
the work of Aharonov, Bergmann and Lebowitz (ABL) Aharonov64  , and also
Aharonov-Vaidman , who consider systems that are both pre- and post-selected,
in the context of ordinary quantum theory, and ask for the resulting
probabilities if an observable is measured at some intermediate time $t$.

From an operational point of view, though, it seems much more natural to
specify the system at some initial time, and then to give a recipe for the
outcome of a measurement at some later time, without having to know the state
of the system in the far future. But then, in our formulation, we have to
specify twice as much information as is normally given in ordinary quantum
mechanics. We need to know what the significance of this extra information is.
If an experimentalist prepares the system, say in its ground state, what does
this mean in terms of $\ket{\psi}$ and $\ket{\phi}$? If both $\ket{\psi}$ and
$\ket{\phi}$ need to be in the ground state, then we are demanding
$\vartheta=0$ . But maybe that is too restrictive. The proper rule for mapping
our formalism onto a physical system is a major unresolved issue.

One possible scenario is that the extra information encoded in $\ket{\psi}$
and $\ket{\phi}$ can be used to make the result of a measurement
deterministic. Then the probabilistic rules of quantum mechanics would emerge
when averaging over the extra information. It remains to be seen whether such
a scenario can be realized. If it can, that might open the way to coupling
this modified form of quantum mechanics to a purely classical form of general
relativity, since in a deterministic theory collapse of the wave function is
avoided.

On a more poetic note, one could imagine that $\ket{\psi}$ and $\ket{\phi}$
exist in separate universes, so that we live, for example, in the $\ket{\psi}$
universe, which communicates with the $\ket{\phi}$ universe via the non-linear
terms. While our universe evolves from $\ket{A}$ to $\ket{B}$ the $\ket{\phi}$
universe evolves the other way, from $\ket{B}$ to $\ket{A}$ (or evolves the
same way, but backwards in time). Related to this, we observe that if there is
an anti-unitary operator that commutes with $H$, then our equations (I) enjoy
a time-reversal symmetry, which involves both $t$ going to $-t$ and the
interchange of $\ket{\psi}$ and $\ket{\phi}$.

##  Appendix A The Case of a Single State Vector

Even simpler than the case we consider in the main text is when we have only
one state vector, coupled non-linearly to itself, that is

${\rm i}\frac{\partial}{\partial
t}\ket{\psi}=H\ket{\psi}+g\ket{\psi}\Braket{\psi}{\psi}.$ (40)

From Eq. (40) and its complex conjugate, we immediately derive

${\rm i}\frac{\partial}{\partial
t}\Braket{\psi}{\psi}=(g-g^{*})\Braket{\psi}{\psi}^{2}.$ (41)

If $g$ is real, then $\Braket{\psi}{\psi}=N$ where $N$ is a constant. In that
case, the general solution is just

$\ket{\psi}=\sum_{n}c_{n}e^{-i(E_{n}+gN)t}\ket{n}.$ (42)

where the $(\ket{n})$ are eigenstates of $H$ with eigenvalues $E_{n}$, and the
$c_{n}$ are constants obeying

$\sum_{n}|c_{n}|^{2}=N.$ (43)

If $g$ is complex, $g=a+ib$ then the solution to Eq. (41) is

$\Braket{\psi}{\psi}=\frac{-1}{2b(t-t_{0})}.$ (44)

This only makes sense if $2b(t-t_{0})<0$. If $b>0$ , the system ceases to
exist after $t=t_{0}$. Conversely, if $b<0$ , the system comes into existence
with infinite norm at $t=t_{0}$, and the norm decays inversely with $t$
thereafter. These pathologies are associated with the fact that, unlike for
the two state-vector case, the self-coupling is non-hermitian if $g$ is
complex. In the case $b<0$ this behavior is reminiscent of the evolution of
the universe from an initial big bang.

As in the two state-vector case, one can go on to construct the general
solution for $\ket{\psi}$ in the form of a function of time multiplying a
solution of the usual Schrödinger equation.

##  Appendix B  The case $g$ is real ($b=0$)

In this case as we have shown there is a Lagrangian description of the
problem. In the main text we have concentrated on the general case of complex
$g$. It is not straightforward to obtain the solution to the special case of
real $g$ by setting $b=0$ in the solution. Rather, we return to equations (4)
and note that when $b=0$, $\tau$ as well as $\delta$ are constants:

$\tau=\tau_{0}=const.$ (45)

which leads to an immediate solution for $\gamma=\Braket{\phi}{\psi}$

$\gamma=\gamma_{0}e^{{\rm i}g\tau_{0}t}$ (46)

where $\gamma_{0}$ is a constant. Note that this leads to
$\delta=|\gamma_{0}|^{2}$. As in the general case, it is convenient to
parameterize the state vectors as

$\displaystyle\ket{\psi}$
$\displaystyle=\gamma^{1/2}\sum_{n}(\hat{\psi}_{n}(t)e^{-{\rm
i}E_{n}t})\ket{n}$ (47) $\displaystyle\ket{\phi}$
$\displaystyle=(\gamma^{\ast})^{1/2}\sum_{n}(\hat{\phi}_{n}(t)e^{-{\rm
i}E_{n}t})\ket{n}$ (48)

which imply the equations

$\displaystyle{\rm i}\dot{\hat{\psi}}_{n}-\frac{g}{2}\tau_{0}\hat{\psi}_{n}$
$\displaystyle=g\hat{\phi}_{n}|\gamma_{0}|,$ (49) $\displaystyle{\rm
i}\dot{\hat{\phi}}_{n}+\frac{g}{2}\tau_{0}\hat{\phi}_{n}$
$\displaystyle=g\hat{\psi}_{n}|\gamma_{0}|$ (50)

which after some algebra, lead to the uncoupled second-order equations:

$\displaystyle\ddot{\hat{\psi}}_{n}+g^{2}\omega_{0}^{2}\hat{\psi}_{n}=0$
$\displaystyle\ddot{\hat{\phi}}_{n}+g^{2}\omega_{0}^{2}\hat{\phi}_{n}=0,$ (51)

where

$\omega_{0}^{2}=\frac{\tau_{0}^{2}}{4}+\delta,$ (52)

The general solution to these equations is

$\displaystyle\hat{\psi}_{n}$ $\displaystyle=A_{n}e^{{\rm
i}g\omega_{0}t}+B_{n}e^{-{\rm i}g\omega_{0}t}$ $\displaystyle\hat{\phi}_{n}$
$\displaystyle=C_{n}e^{{\rm i}g\omega_{0}t}+D_{n}e^{-{\rm i}g\omega_{0}t}$
(53)

where

$C_{n}=-\frac{1}{k}~{}~{}A_{n};~{}~{}D_{n}=kB_{n},$ (54)

and

$k=\frac{\sqrt{\delta}}{\omega_{0}+\frac{\tau_{0}}{2}}=\frac{(\omega_{0}-\tau_{0}/2)}{\sqrt{\delta}}.$
(55)

The two forms of $k$ are equivalent because of the definition of $\omega_{0}$.
The requirement that $\Braket{\phi}{\psi}=\gamma$ leads to the conditions

$\sum_{n}(k~{}B_{n}^{\ast}B_{n}-\frac{1}{k}A_{n}^{\ast}A_{n})=1.$ (56)

and the orthogonality condition

$\sum_{n}B_{n}^{\ast}A_{n}=\sum_{n}A_{n}^{\ast}B_{n}=0.$ (57)

###### Acknowledgements.

We are grateful to Sean Carroll and David Fairlie for conversations. AC would
like to thank the Santa Fe Institute for hospitality while part of this work
was done.

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The Racah Algebra as a Subalgebra of the Bannai–Ito Algebra

The Racah Algebra as a Subalgebra
of the Bannai–Ito Algebra

Hau-Wen HUANG

H.-W. Huang

Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
<EMAIL_ADDRESS>
Received May 22, 2020, in final form July 31, 2020; Published online August
10, 2020

Assume that ${\mathbb{F}}$ is a field with
$\operatorname{char}{\mathbb{F}}\not=2$. The Racah algebra $\Re$ is a unital
associative ${\mathbb{F}}$-algebra defined by generators and relations. The
generators are $A$, $B$, $C$, $D$ and the relations assert that
$[A,B]=[B,C]=[C,A]=2D$ and each of $[A,D]+AC-BA$, $[B,D]+BA-CB$, $[C,D]+CB-AC$
is central in $\Re$. The Bannai–Ito algebra $\mathfrak{BI}$ is a unital
associative ${\mathbb{F}}$-algebra generated by $X$, $Y$, $Z$ and the
relations assert that each of $\\{X,Y\\}-Z$, $\\{Y,Z\\}-X$, $\\{Z,X\\}-Y$ is
central in $\mathfrak{BI}$. It was discovered that there exists an
${\mathbb{F}}$-algebra homomorphism $\zeta\colon\Re\to\mathfrak{BI}$ that
sends $A\mapsto\frac{(2X-3)(2X+1)}{16}$, $B\mapsto\frac{(2Y-3)(2Y+1)}{16}$,
$C\mapsto\frac{(2Z-3)(2Z+1)}{16}$. We show that $\zeta$ is injective and
therefore $\Re$ can be considered as an ${\mathbb{F}}$-subalgebra of
$\mathfrak{BI}$. Moreover we show that any Casimir element of $\Re$ can be
uniquely expressed as a polynomial in $\\{X,Y\\}-Z$, $\\{Y,Z\\}-X$,
$\\{Z,X\\}-Y$ and $X+Y+Z$ with coefficients in ${\mathbb{F}}$.

Bannai–Ito algebra; Racah algebra; Casimir elements

81R10; 81R12

##  1 Introduction

Throughout this paper we adopt the following conventions: Assume that
${\mathbb{F}}$ is a field with $\operatorname{char}{\mathbb{F}}\not=2$. Let
${\mathbb{N}}$ denote the set of all nonnegative integers. The bracket $[\,,]$
stands for the commutator and the curly bracket $\\{\,,\\}$ stands for the
anticommutator. An algebra is meant to be an associative algebra with unit $1$
and a subalgebra is a subset of the parent algebra which is closed under the
operations and has the same unit.

The Racah algebra [19, 23] and the Bannai–Ito algebra [26] are the
${\mathbb{F}}$-algebras defined by generators and relations to give the
algebraic interpretations of the Racah polynomials and the Bannai–Ito
polynomials, respectively. At first, the description of those relations
involved several parameters. In recent papers [7, 9, 16, 18] the role of the
parameters is replaced by the central elements. The contemporary Racah and
Bannai–Ito algebras are defined as follows: The Racah algebra $\Re$ is an
${\mathbb{F}}$-algebra generated by $A$, $B$, $C$, $D$ and the relations
assert that

$\displaystyle[A,B]=[B,C]=[C,A]=2D$

and each of

$\displaystyle\alpha=[A,D]+AC-BA,\qquad\beta=[B,D]+BA-
CB,\qquad\gamma=[C,D]+CB-AC$

is central in $\Re$. Note that

$\displaystyle\delta=A+B+C$

is also central in $\Re$. The Bannai–Ito algebra $\mathfrak{BI}$ is an
${\mathbb{F}}$-algebra generated by $X$, $Y$, $Z$ and the relations assert
that each of

$\displaystyle\kappa=\\{X,Y\\}-Z,\qquad\lambda=\\{Y,Z\\}-X,\qquad\mu=\\{Z,X\\}-Y$

is central in $\mathfrak{BI}$. The applications to the Racah problems for
$\mathfrak{su}(2)$, $\mathfrak{su}(1,1)$, $\mathfrak{sl}_{-1}(2)$ and the
connections to the Laplace–Dunkl and Dirac–Dunkl equations on the $2$-sphere
have been explored in [3, 6, 8, 11, 12, 13, 14, 15, 17, 19, 20, 23]. For more
information and recent progress, see [2, 4, 5, 7, 9, 10, 22].

A result of [16] made the following link between the Racah algebra $\Re$ and
the Bannai–Ito algebra $\mathfrak{BI}$. The standard realization for
$\mathfrak{BI}$ is a representation $\pi\colon\mathfrak{BI}\to{\rm
End}({\mathbb{F}}[x])$ given in [26, Section 4]. Inspired by $\pi$, a
representation $\tau\colon\Re\to{\rm End}({\mathbb{F}}[x])$ was constructed in
[16, Section 2] as well as an ${\mathbb{F}}$-algebra homomorphism
$\zeta\colon\Re\to\mathfrak{BI}$ that sends

$\displaystyle A\mapsto\frac{(2X-3)(2X+1)}{16},\qquad
B\mapsto\frac{(2Y-3)(2Y+1)}{16},\qquad C\mapsto\frac{(2Z-3)(2Z+1)}{16}.$

Briefly $\tau$ is the composition of $\zeta$ followed by $\pi$. The main
result of this paper is to prove that $\zeta$ is injective. To see this we
derive the following results. We show that the monomials

$\displaystyle A^{i}B^{j}C^{k}D^{\ell}\alpha^{r}\beta^{s}\qquad\text{for all
$i,j,k,\ell,r,s\in{\mathbb{N}}$}$ (1.1)

are an ${\mathbb{F}}$-basis for $\Re$ and the monomials

$\displaystyle X^{i}Y^{j}Z^{k}\kappa^{r}\lambda^{s}\mu^{t}\qquad\text{for all
$i,j,k,r,s,t\in{\mathbb{N}}$}$ (1.2)

are an ${\mathbb{F}}$-basis for $\mathfrak{BI}$. We consider the following
${\mathbb{F}}$-subspaces of $\mathfrak{BI}$ induced from the basis (1.2) for
$\mathfrak{BI}$: Let
$w_{X},w_{Y},w_{Z},w_{\kappa},w_{\lambda},w_{\mu}\in{\mathbb{N}}$ be given.
For each $n\in{\mathbb{N}}$ let $\mathfrak{BI}_{n}$ denote the
${\mathbb{F}}$-subspace of $\mathfrak{BI}$ spanned by
$X^{i}Y^{j}Z^{k}\kappa^{r}\lambda^{s}\mu^{t}$ for all
$i,j,k,r,s,t\in{\mathbb{N}}$ with

$\displaystyle w_{X}i+w_{Y}j+w_{Z}k+w_{\kappa}r+w_{\lambda}s+w_{\mu}t\leq n.$

We show that the sequence $\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$ is an
${\mathbb{N}}$-filtration of $\mathfrak{BI}$ if and only if

$\displaystyle\max\\{w_{Z},w_{\kappa}\\}\leq
w_{X}+w_{Y},\qquad\max\\{w_{X},w_{\lambda}\\}\leq
w_{Y}+w_{Z},\qquad\max\\{w_{Y},w_{\mu}\\}\leq w_{Z}+w_{X}.$

We apply the basis (1.1) for $\Re$ and the ${\mathbb{N}}$-filtration
$\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$ of $\mathfrak{BI}$ associated
with

$\displaystyle(w_{X},w_{Y},w_{Z},w_{\kappa},w_{\lambda},w_{\mu})=(4,4,6,8,9,9)$

to conclude the injectivity of $\zeta$.

We regard the Racah algebra $\Re$ as an ${\mathbb{F}}$-subalgebra of
$\mathfrak{BI}$ via $\zeta$. Let $\mathfrak{C}$ denote the commutative
${\mathbb{F}}$-subalgebra of $\Re$ generated by $\alpha$, $\beta$, $\gamma$,
$\delta$. Extending the setting [15, Section 2], each element of

$\displaystyle
D^{2}+A^{2}+B^{2}+\frac{(\delta+2)\\{A,B\\}-\big{\\{}A^{2},B\big{\\}}-\big{\\{}A,B^{2}\big{\\}}}{2}+A(\beta-\delta)+B(\delta-\alpha)+\mathfrak{C}$

is called a Casimir element of $\Re$ [22]. Each Casimir element of $\Re$ is
central in $\Re$. We locate the expressions for the $D_{6}$-symmetric Casimir
elements [22, Section 5] of $\Re$ in terms of

$\displaystyle\iota=X+Y+Z$

and $\kappa$, $\lambda$, $\mu$. Note that $\iota$, $\kappa$, $\lambda$, $\mu$
are in the centralizer of $\Re$ in $\mathfrak{BI}$. Furthermore we apply the
${\mathbb{N}}$-filtration $\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$ of
$\mathfrak{BI}$ associated with

$\displaystyle(w_{X},w_{Y},w_{Z},w_{\kappa},w_{\lambda},w_{\mu})=(1,1,2,0,0,0)$

to prove that for any Casimir element $\Omega$ of $\Re$ there exists a unique
four-variable polynomial $P(x_{1},x_{2},x_{3},x_{4})$ over ${\mathbb{F}}$ such
that

$\displaystyle\Omega=P(\iota,\kappa,\lambda,\mu).$

The outline of this paper is as follows: In Sections 2 and 3 we present the
required backgrounds on $\Re$ and $\mathfrak{BI}$, especially the basis (1.1)
for $\Re$ and the criterion for $\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$
as an ${\mathbb{N}}$-filtration of $\mathfrak{BI}$. In Section 4 we review the
homomorphism $\zeta\colon\Re\to\mathfrak{BI}$ and evaluate the image of $D$
under $\zeta$. In Section 5 we give the proof for the injectivity of $\zeta$.
In Section 6 we show that each Casimir element of $\Re$ can be uniquely
expressed as a polynomial in $\iota$, $\kappa$, $\lambda$, $\mu$ over
${\mathbb{F}}$.

##  2 The Racah algebra $\boldsymbol{\Re}$

######  Definition 2.1 ([7, 16, 19, 23]).

The Racah algebra $\Re$ is an ${\mathbb{F}}$-algebra defined by generators and
relations in the following way. The generators are $A$, $B$, $C$, $D$. The
relations assert that

$\displaystyle[A,B]=[B,C]=[C,A]=2D$ (2.1)

and each of

$\displaystyle[A,D]+AC-BA,\qquad[B,D]+BA-CB,\qquad[C,D]+CB-AC$

is central in $\Re$.

We define $\alpha$, $\beta$, $\gamma$, $\delta$ as the following elements of
$\Re$:

$\displaystyle\alpha=[A,D]+AC-BA,$ (2.2) $\displaystyle\beta=[B,D]+BA-CB,$
(2.3) $\displaystyle\gamma=[C,D]+CB-AC,$ (2.4) $\displaystyle\delta=A+B+C.$
(2.5)

######  Lemma 2.2 ([22, Lemma 3.2]).

The following $(i)$–$(iii)$ hold:

  1. $(i)$

The ${\mathbb{F}}$-algebra $\Re$ is generated by $A$, $B$, $C$.

  2. $(ii)$

Each of $\alpha$, $\beta$, $\gamma$, $\delta$ is central in $\Re$.

  3. $(iii)$

The sum of $\alpha$, $\beta$, $\gamma$ is equal to zero.

######  Proposition 2.3.

The ${\mathbb{F}}$-algebra $\Re$ has a presentation with generators $A$, $B$,
$C$, $D$, $\alpha$, $\beta$ and relations

$\displaystyle BA=AB-2D,$ (2.6) $\displaystyle CB=BC-2D,$ (2.7) $\displaystyle
CA=AC+2D,$ (2.8) $\displaystyle DA=AD-AB+AC+2D-\alpha,$ (2.9) $\displaystyle
DB=BD-BC+AB-\beta,$ (2.10) $\displaystyle DC=CD-AC+BC-2D+\alpha+\beta,$ (2.11)
$\displaystyle\alpha A=A\alpha,\qquad\alpha B=B\alpha,\qquad\alpha
C=C\alpha,\qquad\alpha D=D\alpha,$ (2.12) $\displaystyle\beta
A=A\beta,\qquad\beta B=B\beta,\qquad\beta C=C\beta,\qquad\beta
D=D\beta,\qquad\beta\alpha=\alpha\beta.$ (2.13)

###### Proof.

Relations (2.6)–(2.8) are immediate from (2.1). Relation (2.9) follows from
(2.2), (2.6). Relation (2.10) follows from (2.3), (2.6) and (2.7). Relation
(2.11) follows from (2.4), (2.7) and Lemma 2.2(iii). Relations (2.12) and
(2.13) follow from Lemma 2.2(ii). ∎

######  Theorem 2.4.

The elements

$\displaystyle A^{i}B^{j}C^{k}D^{\ell}\alpha^{r}\beta^{s}\qquad\text{for all
$i,j,k,\ell,r,s\in{\mathbb{N}}$}$ (2.14)

are an ${\mathbb{F}}$-basis $\Re$.

###### Proof.

To prove the result we invoke the diamond lemma [1, Theorem 1.2]. The
relations (2.6)–(2.13) are regarded as a reduction system. The
${\mathbb{F}}$-linear combinations of (2.14) are exactly the irreducible
elements under the reduction system. There are no inclusion ambiguities in the
reduction system. The nontrivial overlap ambiguities involve the words $CBA$,
$DBA$, $DCB$, $DCA$. In any reduction ways, we eventually obtain that

$\displaystyle CBA=ABC+2AB-2BC-2AD+2BD-2CD-2\beta,$ $\displaystyle
DBA=ABD-2D^{2}+A^{2}B-AB^{2}-2AD+2BD+2AB-2BC-A\beta-B\alpha-2\beta,$
$\displaystyle
DCB=BCD-2D^{2}+B^{2}C-BC^{2}-2BD+2CD-2AC+2BC+B\alpha+B\beta-C\beta$
$\displaystyle\hphantom{DCB=}{}-4D+2\alpha+2\beta,$ $\displaystyle
DCA=ACD+2D^{2}-A^{2}C+AC^{2}-2AD-2AC+2BC+2CD+A\alpha+A\beta-C\alpha$
$\displaystyle\hphantom{DCA=}{}-4D+2\alpha+2\beta.$

Hence each of the overlap ambiguities is resolvable.

Let $M$ denote the free monoid with the alphabet set
$S=\\{A,B,C,D,\alpha,\beta\\}$. Let $\ell\colon M\to{\mathbb{N}}$ denote the
length function of $M$. Consider an element $w=s_{1}s_{2}\cdots s_{n}\in M$
where $s_{1},s_{2},\ldots,s_{n}\in S$. An operation on $w$ is called an
elementary operation if it is one of the following actions on $w$:

  1. $\bullet$

We interchange $s_{i}$ and $s_{j}$ where $1\leq i<j\leq n$ and the position of
$s_{j}$ is left to the position of $s_{i}$ in the list

$\displaystyle A,\quad B,\quad C,\quad D,\quad\alpha,\quad\beta.$

  2. $\bullet$

Choose $s_{i}\in\\{B,C,D\\}$ and replace $s_{i}$ by the left neighbor of
$s_{i}$ in the list

$\displaystyle A,\quad B,\quad C,\quad D.$

We define a binary relation $\preceq$ on $M$ as follows: For any $u,w\in M$ we
say that $u\rightarrow w$ whenever $\ell(u)<\ell(w)$ or $u$ is obtained from
$w$ by an elementary operation. For any $u,w\in M$ we define $u\preceq w$ if
there exist $u_{0},u_{1},\ldots,u_{k}\in M$ with $k\in{\mathbb{N}}$ such that

$\displaystyle u=u_{0}\rightarrow u_{1}\rightarrow\cdots\rightarrow
u_{k-1}\rightarrow u_{k}=w.$

By construction $\preceq$ is a partial order relation on $M$ satisfying the
descending chain condition. Moreover $\preceq$ is a monoid partial order on
$M$ compatible with the reduction system (2.6)–(2.13). Therefore, by diamond
lemma the monomials (2.14) form an ${\mathbb{F}}$-basis for $\Re$. ∎

Recall that the dihedral group $D_{6}$ has a presentation with generators
$\sigma$, $\tau$ and relations

$\displaystyle\sigma^{2}=1,\qquad\tau^{6}=1,\qquad(\sigma\tau)^{2}=1.$ (2.15)

######  Proposition 2.5 ([22, Propositions 4.1 and 4.3]).

There exists a unique $D_{6}$-action on $\Re$ such that $(i)$, $(ii)$ hold:

  1. $(i)$

$\sigma$ acts on $\Re$ as an ${\mathbb{F}}$-algebra antiautomorphism of $\Re$
given in the following way:

$u$ | $A$ | $B$ | $C$ | $D$ | $\alpha$ | $\beta$ | $\gamma$ | $\delta$
---|---|---|---|---|---|---|---|---
$\sigma(u)$ | $B$ | $A$ | $C$ | $D$ | $-\beta$ | $-\alpha$ | $-\gamma$ | $\delta$
  2. $(ii)$

$\tau$ acts on $\Re$ as an ${\mathbb{F}}$-algebra antiautomorphism of $\Re$
given in the following way:

$u$ | $A$ | $B$ | $C$ | $D$ | $\alpha$ | $\beta$ | $\gamma$ | $\delta$
---|---|---|---|---|---|---|---|---
$\tau(u)$ | $B$ | $C$ | $A$ | $-D$ | $\beta$ | $\gamma$ | $\alpha$ | $\delta$

Moreover the $D_{6}$-action on $\Re$ is faithful.

Let $\mathfrak{C}$ denote the ${\mathbb{F}}$-subalgebra of $\Re$ generated by
$\alpha$, $\beta$, $\gamma$, $\delta$. It follows from Lemma 2.2(ii) that
$\mathfrak{C}$ is commutative.

######  Definition 2.6 ([22, Definition 5.2]).

The coset

$\displaystyle
D^{2}+A^{2}+B^{2}+\frac{(\delta+2)\\{A,B\\}-\big{\\{}A^{2},B\big{\\}}-\big{\\{}A,B^{2}\big{\\}}}{2}+A(\beta-\delta)+B(\delta-\alpha)+\mathfrak{C}$

is called the Casimir class of $\Re$. Each element of the Casimir class of
$\Re$ is called a Casimir element of $\Re$.

Define

$\displaystyle\Omega_{A}=D^{2}+\frac{BAC+CAB}{2}+A^{2}+B\gamma-C\beta-A\delta,$
(2.16)
$\displaystyle\Omega_{B}=D^{2}+\frac{CBA+ABC}{2}+B^{2}+C\alpha-A\gamma-B\delta,$
(2.17)
$\displaystyle\Omega_{C}=D^{2}+\frac{ACB+BCA}{2}+C^{2}+A\beta-B\alpha-C\delta.$
(2.18)

Note that $\Omega_{A}$, $\Omega_{B}$, $\Omega_{C}$ are mutually distinct [22,
Corollary 6.5].

######  Lemma 2.7 ([22, Proposition 3.7]).

Each of $\Omega_{A}$, $\Omega_{B}$, $\Omega_{C}$ is a Casimir element.

######  Lemma 2.8 ([22, Lemma 3.6]).

The set $\\{\Omega_{A},\Omega_{B},\Omega_{C}\\}$ is invariant under the
$D_{6}$-action on $\Re$. Moreover the restrictions of $\sigma$ and $\tau$ to
$\\{\Omega_{A},\Omega_{B},\Omega_{C}\\}$ are as follows:

$u$ | $\Omega_{A}$ | $\Omega_{B}$ | $\Omega_{C}$
---|---|---|---
$\sigma(u)$ | $\Omega_{B}$ | $\Omega_{A}$ | $\Omega_{C}$
$\tau(u)$ | $\Omega_{B}$ | $\Omega_{C}$ | $\Omega_{A}$

######  Definition 2.9 ([22, Section 5]).

The elements $\Omega_{A}$, $\Omega_{B}$, $\Omega_{C}$ are called the
$D_{6}$-symmetric Casimir elements of $\Re$.

##  3 The Bannai–Ito algebra $\boldsymbol{\mathfrak{BI}}$

######  Definition 3.1.

The Bannai–Ito algebra $\mathfrak{BI}$ is an ${\mathbb{F}}$-algebra defined by
generators and relations. The generators are $X$, $Y$, $Z$ and the relations
assert that each of $\\{X,Y\\}-Z$, $\\{Y,Z\\}-X$, $\\{Z,X\\}-Y$ is central in
$\mathfrak{BI}$.

We define $\iota$, $\kappa$, $\lambda$, $\mu$ as the following elements of
$\mathfrak{BI}$:

$\displaystyle\iota=X+Y+Z,$ (3.1) $\displaystyle\kappa=\\{X,Y\\}-Z,$ (3.2)
$\displaystyle\lambda=\\{Y,Z\\}-X,$ (3.3) $\displaystyle\mu=\\{Z,X\\}-Y.$
(3.4)

######  Proposition 3.2.

There exists a unique $D_{6}$-action on $\mathfrak{BI}$ such that $(i)$,
$(ii)$ hold:

  1. $(i)$

$\sigma$ acts on $\mathfrak{BI}$ as an ${\mathbb{F}}$-algebra antiautomorphism
of $\mathfrak{BI}$ given in the following way:

$u$ | $X$ | $Y$ | $Z$ | $\iota$ | $\kappa$ | $\lambda$ | $\mu$
---|---|---|---|---|---|---|---
$\sigma(u)$ | $Y$ | $X$ | $Z$ | $\iota$ | $\kappa$ | $\mu$ | $\lambda$
  2. $(ii)$

$\tau$ acts on $\mathfrak{BI}$ as an ${\mathbb{F}}$-algebra antiautomorphism
of $\mathfrak{BI}$ given in the following way:

$u$ | $X$ | $Y$ | $Z$ | $\iota$ | $\kappa$ | $\lambda$ | $\mu$
---|---|---|---|---|---|---|---
$\tau(u)$ | $Y$ | $Z$ | $X$ | $\iota$ | $\lambda$ | $\mu$ | $\kappa$

Moreover the $D_{6}$-action on $\mathfrak{BI}$ is faithful.

###### Proof.

It is straightforward to verify the existence of the $D_{6}$-action on
$\mathfrak{BI}$ by using (2.15) and Definition 3.1. Since $D_{6}$ is generated
by $\sigma$ and $\tau$ the uniqueness follows. The ${\mathbb{F}}$-algebra
antiautomorphism of $\mathfrak{BI}$ given in (ii) is of order $6$. It follows
from [22, Lemma 4.2] that the $D_{6}$-action on $\mathfrak{BI}$ is faithful. ∎

######  Proposition 3.3.

The ${\mathbb{F}}$-algebra $\mathfrak{BI}$ has a presentation with generators
$X$, $Y$, $Z$, $\kappa$, $\lambda$, $\mu$ and relations

$\displaystyle YX=-XY+Z+\kappa,\qquad ZY=-YZ+X+\lambda,\qquad ZX=-XZ+Y+\mu,$
$\displaystyle\kappa X=X\kappa,\qquad\kappa Y=Y\kappa,\qquad\kappa Z=Z\kappa,$
$\displaystyle\lambda X=X\lambda,\qquad\lambda Y=Y\lambda,\qquad\lambda
Z=Z\lambda,\qquad\lambda\kappa=\kappa\lambda,$ $\displaystyle\mu
X=X\mu,\qquad\mu Y=Y\mu,\qquad\mu
Z=Z\mu,\qquad\mu\kappa=\kappa\mu,\qquad\mu\lambda=\lambda\mu.$

###### Proof.

Immediate from Definition 3.1. ∎

Applying the diamond lemma to Proposition 3.3, we obtain the following
Poincaré–Birkhoff–Witt basis for $\mathfrak{BI}$. Since the argument is
similar to the proof of Theorem 2.4, we omit the proof here.

######  Theorem 3.4.

The elements

$\displaystyle X^{i}Y^{j}Z^{k}\kappa^{r}\lambda^{s}\mu^{t}\qquad\text{for all
$i,j,k,r,s,t\in{\mathbb{N}}$}$ (3.5)

form an ${\mathbb{F}}$-basis for $\mathfrak{BI}$.

Let $\mathcal{A}$ denote an ${\mathbb{F}}$-algebra and let
$\mathcal{H},\mathcal{K}$ denote two ${\mathbb{F}}$-subspaces of
$\mathcal{A}$. The product $\mathcal{H}\cdot\mathcal{K}$ is meant to be the
${\mathbb{F}}$-subspace of $\mathcal{A}$ spanned by $h\cdot k$ for all
$h\in\mathcal{H}$ and all $k\in\mathcal{K}$. Recall that an
${\mathbb{N}}$-filtration of $\mathcal{A}$ is a sequence
$\\{\mathcal{A}_{n}\\}_{n\in{\mathbb{N}}}$ of ${\mathbb{F}}$-subspaces of
$\mathcal{A}$ satisfies the following conditions:

  1. (N1)

$\bigcup_{n\in{\mathbb{N}}}\mathcal{A}_{n}=\mathcal{A}$.

  2. (N2)

$\mathcal{A}_{n}\subseteq\mathcal{A}_{n+1}$ for all $n\in{\mathbb{N}}$.

  3. (N3)

$\mathcal{A}_{m}\cdot\mathcal{A}_{n}\subseteq\mathcal{A}_{m+n}$ for all
$m,n\in{\mathbb{N}}$.

For convenience we always let $\mathcal{A}_{-1}$ denote the zero subspace of
$\mathcal{A}$.

We consider the following ${\mathbb{F}}$-subspaces of $\mathfrak{BI}$ induced
from Theorem 3.4: Let $w_{X}$, $w_{Y}$, $w_{Z}$, $w_{\kappa}$, $w_{\lambda}$,
$w_{\mu}$ denote the nonnegative integers. For each $n\in{\mathbb{N}}$ let
$\mathfrak{BI}_{n}$ denote the ${\mathbb{F}}$-subspace of $\mathfrak{BI}$
spanned by $X^{i}Y^{j}Z^{k}\kappa^{r}\lambda^{s}\mu^{t}$ for all
$i,j,k,r,s,t\in{\mathbb{N}}$ with

$\displaystyle w_{X}i+w_{Y}j+w_{Z}k+w_{\kappa}r+w_{\lambda}s+w_{\mu}t\leq n.$

We call $\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$ the
${\mathbb{F}}$-subspaces of $\mathfrak{BI}$ associated with
$(w_{X},w_{Y},w_{Z},w_{\kappa},w_{\lambda},w_{\mu})$. In what follows we give
a simple criterion for the above ${\mathbb{F}}$-subspaces of $\mathfrak{BI}$
to be an ${\mathbb{N}}$-filtration of $\mathfrak{BI}$.

######  Theorem 3.5.

Let $w_{X},w_{Y},w_{Z},w_{\kappa},w_{\lambda},w_{\mu}\in{\mathbb{N}}$. Let
$\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$ denote the
${\mathbb{F}}$-subspaces of $\mathfrak{BI}$ associated with
$(w_{X},w_{Y},w_{Z},w_{\kappa},w_{\lambda},w_{\mu})$. Then
$\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$ is an ${\mathbb{N}}$-filtration
of $\mathfrak{BI}$ if and only if

$\displaystyle\max\\{w_{Z},w_{\kappa}\\}\leq w_{X}+w_{Y},$ (3.6)
$\displaystyle\max\\{w_{X},w_{\lambda}\\}\leq w_{Y}+w_{Z},$ (3.7)
$\displaystyle\max\\{w_{Y},w_{\mu}\\}\leq w_{Z}+w_{X}.$ (3.8)

###### Proof.

($\Rightarrow$) By the construction of
$\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$ and Theorem 3.4 the element

$\displaystyle Z+\kappa\notin\mathfrak{BI}_{\max\\{w_{Z},w_{\kappa}\\}-1}.$

On the other hand, by (N3) we have $\\{X,Y\\}\in\mathfrak{BI}_{w_{X}+w_{Y}}$.
The equation (3.2) implies

$\displaystyle Z+\kappa=\\{X,Y\\}.$

By the above comments we see that $\mathfrak{BI}_{w_{X}+w_{Y}}$ contains
$Z+\kappa$ which is not in $\mathfrak{BI}_{\max\\{w_{Z},w_{\kappa}\\}-1}$.
Combined with (N2) the inequality (3.6) follows. The inequalities (3.7) and
(3.8) follow by similar arguments.

($\Leftarrow$) Condition (N1) is immediate from Theorem 3.4. Condition (N2) is
immediate from the construction of
$\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$. Set
$S=\\{X,Y,Z,\kappa,\lambda,\mu\\}$. For all $n\in{\mathbb{N}}$, let $I_{n}$
denote the set of all $(i,j,k,r,s,t)\in{\mathbb{N}}^{6}$ with
$w_{X}i+w_{Y}j+w_{Z}k+w_{\kappa}r+w_{\lambda}s+w_{\mu}t\leq n$. Let $M$ denote
the free monoid with the alphabet set $S$. There exists a unique monoid
homomorphism $\tilde{w}\colon M\to{\mathbb{N}}$ such that

$\displaystyle\tilde{w}(u)=w_{u}\qquad\text{for all $u\in S$}.$

By (3.6)–(3.8), for each relation of Proposition 3.3, the value of $\tilde{w}$
on the monomial in the left-hand side is greater than or equal to those in the
right-hand side. Thus, for all $m,n\in{\mathbb{N}}$ and for all
$(i^{\prime},j^{\prime},k^{\prime},r^{\prime},s^{\prime},t^{\prime})\in I_{m}$
and
$(i^{\prime\prime},j^{\prime\prime},k^{\prime\prime},r^{\prime\prime},s^{\prime\prime},t^{\prime\prime})\in
I_{n}$ the product

$\displaystyle
X^{i^{\prime}}Y^{j^{\prime}}Z^{k^{\prime}}\kappa^{r^{\prime}}\lambda^{s^{\prime}}\mu^{t^{\prime}}\cdot
X^{i^{\prime\prime}}Y^{j^{\prime\prime}}Z^{k^{\prime\prime}}\kappa^{r^{\prime\prime}}\lambda^{s^{\prime\prime}}\mu^{t^{\prime\prime}}$

is equal to an ${\mathbb{F}}$-linear combination of
$X^{i}Y^{j}Z^{k}\kappa^{r}\lambda^{s}\mu^{t}$ for all $(i,j,k,r,s,t)\in
I_{m+n}$. In other words (N3) holds. The theorem follows. ∎

##  4 The homomorphism $\boldsymbol{\zeta\colon\Re\to\mathfrak{BI}}$

According to [16, Section 2] there exists an ${\mathbb{F}}$-algebra
homomorphism $\zeta\colon\Re\to\mathfrak{BI}$ and the images of $A$, $B$, $C$,
$\alpha$, $\beta$, $\gamma$, $\delta$ under $\zeta$ are as follows:

######  Theorem 4.1 ([16]).

There exists a unique ${\mathbb{F}}$-algebra homomorphism
$\zeta\colon\Re\to\mathfrak{BI}$ that sends

$\displaystyle A\mapsto\frac{(2X-3)(2X+1)}{16},\qquad
B\mapsto\frac{(2Y-3)(2Y+1)}{16},\qquad C\mapsto\frac{(2Z-3)(2Z+1)}{16},$
$\displaystyle\alpha\mapsto\frac{(2\iota-\kappa-\mu-3)(\kappa-\mu)}{64},\qquad\beta\mapsto\frac{(2\iota-\lambda-\kappa-3)(\lambda-\kappa)}{64},$
$\displaystyle\gamma\mapsto\frac{(2\iota-\mu-\lambda-3)(\mu-\lambda)}{64},\qquad\delta\mapsto\frac{\iota^{2}-2\iota-\kappa-\lambda-\mu}{4}-\frac{9}{16}.$

We are now going to evaluate the image of $D$ under $\zeta$.

######  Lemma 4.2.

  1. $(i)$

The following equations hold in $\mathfrak{BI}$:

$\displaystyle\big{[}X^{2},Y\big{]}=[X,Z],\qquad\big{[}Y^{2},Z\big{]}=[Y,X],\qquad\big{[}Z^{2},X\big{]}=[Z,Y],$
$\displaystyle\big{[}Y^{2},X\big{]}=[Y,Z],\qquad\big{[}Z^{2},Y\big{]}=[Z,X],\qquad\big{[}X^{2},Z\big{]}=[X,Y].$

  2. $(ii)$

The following elements of $\mathfrak{BI}$ are equal:

$\displaystyle\\{X,[Z,Y]\\},\qquad\\{Y,[X,Z]\\},\qquad\\{Z,[Y,X]\\},$
$\displaystyle\big{[}X^{2},Y^{2}\big{]},\qquad\big{[}Y^{2},Z^{2}\big{]},\qquad\big{[}Z^{2},X^{2}\big{]}.$

###### Proof.

(i) Since $\kappa$ is central in $\mathfrak{BI}$ and by (3.2) it follows that

$\displaystyle[X,Z]=[X,\\{X,Y\\}].$

Observe that $[X,\\{X,Y\\}]=\big{[}X^{2},Y\big{]}$. Therefore

$\displaystyle\big{[}X^{2},Y\big{]}=[X,Z].$ (4.1)

Applying Proposition 3.2 to (4.1) yields the remaining equations in (i).

(ii) By Proposition 3.2(ii) it suffices to show that

$\displaystyle\\{X,[Z,Y]\\}=\\{Y,[X,Z]\\},$ (4.2)
$\displaystyle\\{X,[Z,Y]\\}=\big{[}X^{2},Y^{2}\big{]}.$ (4.3)

With trivial cancellations we obtain

$\displaystyle\\{X,[Z,Y]\\}-\\{Y,[X,Z]\\}=[Z,\\{Y,X\\}].$ (4.4)

Since $\kappa$ is central in $\mathfrak{BI}$ and by (3.2) the element $Z$
commutes with $\\{Y,X\\}$. Hence the right-hand side of (4.4) is zero.
Therefore (4.2) follows. Using (3.2) twice we find that

$\displaystyle X^{2}Y^{2}=XY^{2}X+XZY-XYZ.$ (4.5)

By Proposition 3.2(ii), $\tau^{3}$ is an ${\mathbb{F}}$-algebra
antiautomorphism of $\mathfrak{BI}$ that fixes $X$, $Y$, $Z$. Thus, applying
$\tau^{3}$ to (4.5) yields that

$\displaystyle Y^{2}X^{2}=XY^{2}X+YZX-ZYX.$ (4.6)

Subtracting (4.6) from (4.5) yields (4.3). Hence (ii) follows. ∎

For convenience we let $L$ denote the common element of $\mathfrak{BI}$ from
Lemma 4.2(ii).

######  Proposition 4.3.

The image of $D$ under $\zeta$ is equal to

$\displaystyle\frac{[X,Y]+[Y,Z]+[Z,X]+L}{32}.$

###### Proof.

By (2.1) we have $2D^{\zeta}=\big{[}A^{\zeta},B^{\zeta}\big{]}$. A direct
calculation yields that $[A^{\zeta},B^{\zeta}]$ is equal to

$\displaystyle\frac{\big{[}X^{2},Y^{2}\big{]}+[X,Y]+\big{[}Y^{2},X\big{]}+\big{[}Y,X^{2}\big{]}}{16}.$

By Lemma 4.2(i), $\big{[}Y^{2},X\big{]}=[Y,Z]$ and
$\big{[}Y,X^{2}\big{]}=[Z,X]$. By Lemma 4.2(ii),
$\big{[}X^{2},Y^{2}\big{]}=L$. The proposition follows. ∎

######  Corollary 4.4.

For each $g\in D_{6}$ the following diagram commutes:

${\Re}$${\mathfrak{BI}}$${\Re}$${\mathfrak{BI}.}$$\zeta$$g$$\zeta$$g$

###### Proof.

It is routine to verify the corollary by using Propositions 2.5, 3.2 and
Theorem 4.1. ∎

We end this section with a comment: Recall from [18, 21] that a universal
analogue of the additive DAHA (double affine Hecke algebra) of type
$\big{(}C_{1}^{\vee},C_{1}\big{)}$, denoted by $\mathfrak{H}$ here, is an
${\mathbb{F}}$-algebra generated by $t_{0}$, $t_{1}$, $t_{0}^{\vee}$,
$t_{1}^{\vee}$ and the relations assert that

$\displaystyle t_{0}+t_{1}+t_{0}^{\vee}+t_{1}^{\vee}=-1$

and each of $t_{0}^{2}$, $t_{1}^{2}$, $t_{0}^{\vee 2}$, $t_{1}^{\vee 2}$ is
central in $\mathfrak{H}$. By [18, Proposition 2] there exists an
${\mathbb{F}}$-algebra isomorphism
$\natural\colon\mathfrak{BI}\to\mathfrak{H}$ that sends

$\displaystyle X\mapsto t_{0}+t_{1}+\frac{1}{2},\qquad Y\mapsto
t_{0}+t_{0}^{\vee}+\frac{1}{2},\qquad Z\mapsto
t_{0}+t_{1}^{\vee}+\frac{1}{2},\qquad\iota\mapsto 2t_{0}+\frac{1}{2},$
$\displaystyle\kappa\mapsto t_{0}^{2}-t_{1}^{2}-t_{0}^{\vee 2}+t_{1}^{\vee
2},\qquad\lambda\mapsto t_{0}^{2}-t_{0}^{\vee 2}-t_{1}^{\vee
2}+t_{1}^{2},\qquad\mu\mapsto t_{0}^{2}-t_{1}^{\vee 2}-t_{1}^{2}+t_{0}^{\vee
2}.$

The universal Askey–Wilson algebra [24] and the universal DAHA of type
$\big{(}C_{1}^{\vee},C_{1}\big{)}$ [25] are the $q$-analogues of $\Re$ and
$\mathfrak{H}$, respectively. Therefore [25, Theorem 4.1] is a $q$-analogue of
the homomorphism $\natural\circ\zeta\colon\Re\to\mathfrak{H}$. Note that
$\natural\circ\zeta$ sends

$\displaystyle
A\mapsto\frac{\big{(}t_{1}^{\vee}+t_{0}^{\vee}\big{)}\big{(}t_{1}^{\vee}+t_{0}^{\vee}+2\big{)}}{4},\qquad
B\mapsto\frac{\big{(}t_{1}+t_{1}^{\vee}\big{)}\big{(}t_{1}+t_{1}^{\vee}+2\big{)}}{4},$
$\displaystyle
C\mapsto\frac{\big{(}t_{0}^{\vee}+t_{1}\big{)}\big{(}t_{0}^{\vee}+t_{1}+2\big{)}}{4},$
$\displaystyle\alpha\mapsto\frac{\big{(}t_{1}^{\vee 2}-t_{0}^{\vee
2}\big{)}\big{(}t_{1}^{2}-t_{0}^{2}+2t_{0}-1\big{)}}{16},\qquad\beta\mapsto\frac{\big{(}t_{1}^{2}-t_{1}^{\vee
2}\big{)}\big{(}t_{0}^{\vee 2}-t_{0}^{2}+2t_{0}-1\big{)}}{16},$
$\displaystyle\gamma\mapsto\frac{\big{(}t_{0}^{\vee
2}-t_{1}^{2}\big{)}\big{(}t_{1}^{\vee
2}-t_{0}^{2}+2t_{0}-1\big{)}}{16},\qquad\delta\mapsto\frac{t_{0}^{2}+t_{1}^{2}+t_{0}^{\vee
2}+t_{1}^{\vee 2}}{4}-\frac{t_{0}}{2}-\frac{3}{4}.$

##  5 The injectivity of $\boldsymbol{\zeta}$

Throughout this section, we let $\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$
denote the ${\mathbb{F}}$-subspaces of $\mathfrak{BI}$ associated with

$\displaystyle(w_{X},w_{Y},w_{Z},w_{\kappa},w_{\lambda},w_{\mu})=(4,4,6,8,9,9).$
(5.1)

Since the number sequence (5.1) satisfies (3.6)–(3.8), it follows from Theorem
3.5 that $\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$ is an
${\mathbb{N}}$-filtration of $\mathfrak{BI}$.

######  Lemma 5.1.

  1. $(i)$

For any even integer $n\geq 0$ the following equations hold:

$\displaystyle Y^{n}X=XY^{n}\pmod{\mathfrak{BI}_{4n+3}},$ $\displaystyle
X^{n}Y=YX^{n}\pmod{\mathfrak{BI}_{4n+3}},$ $\displaystyle
Z^{n}Y=YZ^{n}\pmod{\mathfrak{BI}_{6n+3}},$ $\displaystyle
Y^{n}Z=ZY^{n}\pmod{\mathfrak{BI}_{4n+5}},$ $\displaystyle
X^{n}Z=ZX^{n}\pmod{\mathfrak{BI}_{4n+5}},$ $\displaystyle
Z^{n}X=XZ^{n}\pmod{\mathfrak{BI}_{6n+3}}.$

  2. $(ii)$

For any odd integer $n\geq 1$ the following equations hold:

$\displaystyle Y^{n}X=-XY^{n}+\kappa Y^{n-1}\pmod{\mathfrak{BI}_{4n+3}},$
$\displaystyle X^{n}Y=-YX^{n}+\kappa X^{n-1}\pmod{\mathfrak{BI}_{4n+3}},$
$\displaystyle Y^{n}Z=-ZY^{n}\pmod{\mathfrak{BI}_{4n+5}},$ $\displaystyle
Z^{n}Y=-YZ^{n}\pmod{\mathfrak{BI}_{6n+3}},$ $\displaystyle
X^{n}Z=-ZX^{n}\pmod{\mathfrak{BI}_{4n+5}},$ $\displaystyle
Z^{n}X=-XZ^{n}\pmod{\mathfrak{BI}_{6n+3}}.$

###### Proof.

All equations are established by routine inductions and using (3.2)–(3.4). ∎

######  Lemma 5.2.

  1. $(i)$

For any integer $n\geq 0$ the following equations hold:

$\displaystyle\big{(}A^{\zeta}\big{)}^{n}=\left(\frac{X}{2}\right)^{2n}\pmod{\mathfrak{BI}_{8n-1}},$
$\displaystyle\big{(}B^{\zeta}\big{)}^{n}=\left(\frac{Y}{2}\right)^{2n}\pmod{\mathfrak{BI}_{8n-1}},$
$\displaystyle\big{(}C^{\zeta}\big{)}^{n}=\left(\frac{Z}{2}\right)^{2n}\pmod{\mathfrak{BI}_{12n-1}},$
$\displaystyle\big{(}\alpha^{\zeta}\big{)}^{n}=\left(\frac{\mu}{8}\right)^{2n}\pmod{\mathfrak{BI}_{18n-1}},$
$\displaystyle\big{(}\beta^{\zeta}\big{)}^{n}=(-1)^{n}\left(\frac{\lambda}{8}\right)^{2n}\pmod{\mathfrak{BI}_{18n-1}}.$

  2. $(ii)$

For any even integer $n\geq 0$ the following equation holds:

$\displaystyle\big{(}D^{\zeta}\big{)}^{n}=\frac{1}{16^{n}}\sum_{i=0}^{\frac{n}{2}}(-4)^{i}\binom{\frac{n}{2}}{i}X^{2i}Y^{2i}\kappa^{n-2i}Z^{n}\pmod{\mathfrak{BI}_{14n-1}}.$

  3. $(iii)$

For any odd integer $n\geq 1$ the following equation holds:

$\displaystyle\big{(}D^{\zeta}\big{)}^{n}=\frac{1}{16^{n}}\sum_{i=0}^{\frac{n-1}{2}}(-4)^{i}\binom{\frac{n-1}{2}}{i}\big{(}X^{2i}Y^{2i}\kappa^{n-2i}-2X^{2i+1}Y^{2i+1}\kappa^{n-2i-1}\big{)}Z^{n}$
$\displaystyle\pmod{\mathfrak{BI}_{14n-1}}.$

###### Proof.

(i) Immediate from Theorem 4.1 and the construction of
$\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$.

(ii) It follows from Proposition 4.3 that

$\displaystyle D^{\zeta}=\frac{L}{32}\pmod{\mathfrak{BI}_{13}}.$

Evaluating $L\bmod{\mathfrak{BI}_{13}}$ by using Lemma 4.2(ii) and Lemma
5.1(ii) yields that

$\displaystyle
D^{\zeta}=\frac{Z\kappa}{16}-\frac{XYZ}{8}\pmod{\mathfrak{BI}_{13}}.$ (5.2)

Squaring the equation (5.2) a direct calculation shows that

$\displaystyle\big{(}D^{\zeta}\big{)}^{2}=\frac{Z^{2}\kappa^{2}}{256}-\frac{X^{2}Y^{2}Z^{2}}{64}\pmod{\mathfrak{BI}_{27}}.$
(5.3)

It follows from Lemma 5.1(i) that

$\displaystyle Z^{2}\cdot X^{2}Y^{2}Z^{2}=X^{2}Y^{2}Z^{2}\cdot
Z^{2}\pmod{\mathfrak{BI}_{39}}.$ (5.4)

Now it is routine to derive (ii) by using (5.3) and (5.4).

(iii) To get (iii), one may multiply (5.2) by the equation from (ii) and
simplify the resulting equation by using Lemma 5.1(i). ∎

######  Lemma 5.3.

Let $i,j,k,\ell,n,r,s\in{\mathbb{N}}$ with $8i+8j+12k+14\ell+18r+18s=n$. Then
the following $(i)$–$(iii)$ hold:

  1. $(i)$

For all
$i^{\prime},j^{\prime},k^{\prime},r^{\prime},s^{\prime},t^{\prime}\in{\mathbb{N}}$
with
$4i^{\prime}+4j^{\prime}+6k^{\prime}+8r^{\prime}+9s^{\prime}+9t^{\prime}=n$
and $r^{\prime}>\ell$, the coefficient of

$\displaystyle
X^{i^{\prime}}Y^{j^{\prime}}Z^{k^{\prime}}\kappa^{r^{\prime}}\lambda^{s^{\prime}}\mu^{t^{\prime}}$

in
$\big{(}A^{\zeta}\big{)}^{i}\big{(}B^{\zeta}\big{)}^{j}\big{(}C^{\zeta}\big{)}^{k}\big{(}D^{\zeta}\big{)}^{\ell}\big{(}\alpha^{\zeta}\big{)}^{r}\big{(}\beta^{\zeta}\big{)}^{s}$
with respect to the ${\mathbb{F}}$-basis (3.5) for $\mathfrak{BI}$ is zero.

  2. $(ii)$

For all
$i^{\prime},j^{\prime},k^{\prime},r^{\prime},s^{\prime},t^{\prime}\in{\mathbb{N}}$
with
$4i^{\prime}+4j^{\prime}+6k^{\prime}+8r^{\prime}+9s^{\prime}+9t^{\prime}=n$
and $r^{\prime}=\ell$, the coefficient of

$\displaystyle
X^{i^{\prime}}Y^{j^{\prime}}Z^{k^{\prime}}\kappa^{r^{\prime}}\lambda^{s^{\prime}}\mu^{t^{\prime}}$

in
$\big{(}A^{\zeta}\big{)}^{i}\big{(}B^{\zeta}\big{)}^{j}\big{(}C^{\zeta}\big{)}^{k}\big{(}D^{\zeta}\big{)}^{\ell}\big{(}\alpha^{\zeta}\big{)}^{r}\big{(}\beta^{\zeta}\big{)}^{s}$
with respect to the ${\mathbb{F}}$-basis (3.5) for $\mathfrak{BI}$ is nonzero
if and only if

$\displaystyle(i^{\prime},j^{\prime},k^{\prime},s^{\prime},t^{\prime})=(2i,2j,2k+\ell,2s,2r).$

  3. $(iii)$

The coefficient of

$\displaystyle X^{2i}Y^{2j}Z^{2k+\ell}\kappa^{\ell}\lambda^{2s}\mu^{2r}$

in
$\big{(}A^{\zeta}\big{)}^{i}\big{(}B^{\zeta}\big{)}^{j}\big{(}C^{\zeta}\big{)}^{k}\big{(}D^{\zeta}\big{)}^{\ell}\big{(}\alpha^{\zeta}\big{)}^{r}\big{(}\beta^{\zeta}\big{)}^{s}$
with respect to the ${\mathbb{F}}$-basis (3.5) for $\mathfrak{BI}$ is

$\displaystyle(-1)^{s}4^{-i-j-k-2\ell-3r-3s}.$

###### Proof.

Using Lemmas 5.1(i) and 5.2 one may express

$\displaystyle\big{(}A^{\zeta}\big{)}^{i}\big{(}B^{\zeta}\big{)}^{j}\big{(}C^{\zeta}\big{)}^{k}\big{(}D^{\zeta}\big{)}^{\ell}\big{(}\alpha^{\zeta}\big{)}^{r}\big{(}\beta^{\zeta}\big{)}^{s}+\mathfrak{BI}_{n-1}$

as an ${\mathbb{F}}$-linear combination of
$X^{i^{\prime}}Y^{j^{\prime}}Z^{k^{\prime}}\kappa^{r^{\prime}}\lambda^{s^{\prime}}\mu^{t^{\prime}}+\mathfrak{BI}_{n-1}$
for all
$i^{\prime},j^{\prime},k^{\prime},r^{\prime},s^{\prime},t^{\prime}\in{\mathbb{N}}$
with
$4i^{\prime}+4j^{\prime}+6k^{\prime}+8r^{\prime}+9s^{\prime}+9t^{\prime}=n$.
The lemma follows from the expression. ∎

######  Theorem 5.4.

The homomorphism $\zeta\colon\Re\to\mathfrak{BI}$ is injective.

###### Proof.

Suppose on the contrary that there exists a nonzero element $I$ in the kernel
of $\zeta$. For all $i,j,k,\ell,r,s\in{\mathbb{N}}$ let $c(i,j,k,\ell,r,s)$
denote the coefficient of

$\displaystyle A^{i}B^{j}C^{k}D^{\ell}\alpha^{r}\beta^{s}$

in $I$ with respect to the ${\mathbb{F}}$-basis (2.14) for $\Re$. Let $S$
denote the set of all $(i,j,k,\ell,r,s)\in{\mathbb{N}}^{6}$ with
$c(i,j,k,\ell,r,s)\not=0$. For each $n\in{\mathbb{N}}$ we let $S(n)$ denote
the set of all $(i,j,k,\ell,r,s)\in S$ with $8i+8j+12k+14\ell+18r+18s=n$. We
may write

$\displaystyle I=\sum_{n\in{\mathbb{N}}}\sum_{(i,j,k,\ell,r,s)\in
S(n)}c(i,j,k,\ell,r,s)A^{i}B^{j}C^{k}D^{\ell}\alpha^{r}\beta^{s}.$ (5.5)

Applying $\zeta$ to (5.5) we have

$\displaystyle 0=\sum_{n\in{\mathbb{N}}}\sum_{(i,j,k,\ell,r,s)\in
S(n)}c(i,j,k,\ell,r,s)\big{(}A^{\zeta}\big{)}^{i}\big{(}B^{\zeta}\big{)}^{j}\big{(}C^{\zeta}\big{)}^{k}\big{(}D^{\zeta}\big{)}^{\ell}\big{(}\alpha^{\zeta}\big{)}^{r}\big{(}\beta^{\zeta}\big{)}^{s}.$
(5.6)

Since $I\not=0$ there exists at least one $n\in{\mathbb{N}}$ with
$S(n)\not=\varnothing$. Set

$\displaystyle N=\max\\{n\,|\,S(n)\not=\varnothing\\}.$

Among the elements in $S(N)$ we choose a $6$-tuple $(i,j,k,\ell,r,s)$ that has
the maximum value at $\ell$. In what follows we evaluate the coefficient of

$\displaystyle X^{2i}Y^{2j}Z^{2k+\ell}\kappa^{\ell}\lambda^{2s}\mu^{2r}$ (5.7)

in the right-hand side of (5.6) with respect to the ${\mathbb{F}}$-basis (3.5)
for $\mathfrak{BI}$. Denote by $c$ the coefficient. Suppose that
$(i^{\prime},j^{\prime},k^{\prime},\ell^{\prime},r^{\prime},s^{\prime})$ is a
$6$-tuple in $S(n)$ for some $n\in{\mathbb{N}}$ such that

$\displaystyle\big{(}A^{\zeta}\big{)}^{i^{\prime}}\big{(}B^{\zeta}\big{)}^{j^{\prime}}\big{(}C^{\zeta}\big{)}^{k^{\prime}}\big{(}D^{\zeta}\big{)}^{\ell^{\prime}}\big{(}\alpha^{\zeta}\big{)}^{r^{\prime}}\big{(}\beta^{\zeta}\big{)}^{s^{\prime}}$
(5.8)

contributes to the coefficient $c$. By Theorem 3.4 the monomial (5.7) lies in
$\mathfrak{BI}_{N}$ not in $\mathfrak{BI}_{N-1}$. By Lemma 5.2 the term (5.8)
lies in $\mathfrak{BI}_{n}$. It follows from (N2) that $n\geq N$ and the
maximality of $N$ implies $n=N$. By Lemma 5.3(i) we have
$\ell^{\prime}\geq\ell$ and the maximality of $\ell$ forces that
$\ell^{\prime}=\ell$. Combined with Lemma 5.3(ii) this yields that
$(i^{\prime},j^{\prime},k^{\prime},r^{\prime},s^{\prime})=(i,j,k,r,s)$.
Therefore

$\displaystyle\big{(}A^{\zeta}\big{)}^{i}\big{(}B^{\zeta}\big{)}^{j}\big{(}C^{\zeta}\big{)}^{k}\big{(}D^{\zeta}\big{)}^{\ell}\big{(}\alpha^{\zeta}\big{)}^{r}\big{(}\beta^{\zeta}\big{)}^{s}$

is the only summand in the right-hand side of (5.6) contributes to the
coefficient $c$. By Lemma 5.3(iii) the coefficient $c$ is the nonzero scalar

$\displaystyle(-1)^{s}\cdot 4^{-i-j-k-2\ell-3r-3s}\cdot c(i,j,k,\ell,r,s).$

It follows from Theorem 3.4 that the right-hand side of (5.6) is nonzero, a
contradiction. The theorem follows. ∎

As a consequence of Theorem 5.4 the ${\mathbb{F}}$-algebra homomorphism
$\natural\circ\zeta\colon\Re\to\mathfrak{H}$ described in Section 4 is
injective. Note that [25, Theorem 4.5] is a $q$-analogue of the injectivity
for $\natural\circ\zeta$.

##  6 The images of the Casimir elments of $\boldsymbol{\Re}$ under
$\boldsymbol{\zeta}$

In light of Theorem 5.4 the Racah algebra $\Re$ can be viewed as an
${\mathbb{F}}$-subalgebra of the Bannai–Ito algebra $\mathfrak{BI}$ via
$\zeta$.

######  Lemma 6.1.

The element $\iota$ is in the centralizer of $\Re$ in $\mathfrak{BI}$.

###### Proof.

By Theorem 4.1 and (3.1) the commutator $[\iota,A]$ is equal to $\frac{1}{4}$
times

$\displaystyle\big{[}Y+Z,X^{2}\big{]}-[Y+Z,X].$ (6.1)

Simplifying (6.1) by using Lemma 4.2(i) yields that (6.1) is zero. Therefore
$\iota$ commutes with $A$. Similarly $\iota$ commutes with $B$ and $C$.
Combined with Lemma 2.2(i) the lemma follows. ∎

By Lemma 6.1 each of $\iota$, $\kappa$, $\lambda$, $\mu$ lies in the
centralizer of $\Re$ in $\mathfrak{BI}$. The intention of the final section is
to show that each Casimir element of $\Re$ can be uniquely expressed as a
polynomial in $\iota$, $\kappa$, $\lambda$, $\mu$ with coefficients in
${\mathbb{F}}$.

Throughout this section, let $\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$
denote the ${\mathbb{F}}$-subspaces of $\mathfrak{BI}$ associated with

$\displaystyle(w_{X},w_{Y},w_{Z},w_{\kappa},w_{\lambda},w_{\mu})=(1,1,2,0,0,0).$
(6.2)

Since the sequence (6.2) satisfies (3.6)–(3.8), it follows from Theorem 3.5
that $\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$ is an
${\mathbb{N}}$-filtration of $\mathfrak{BI}$.

######  Lemma 6.2.

$Z^{n}=\iota^{n}\bmod{\mathfrak{BI}_{2n-1}}$ for all $n\in{\mathbb{N}}$.

###### Proof.

Proceed by induction on $n$. It is trivial for $n=0$. By (3.1) we have

$\displaystyle\iota-Z=X+Y\in\mathfrak{BI}_{1}.$ (6.3)

Hence the lemma holds for $n=1$. Suppose that $n\geq 2$. We divide
$\iota^{n}-Z^{n}$ into

$\displaystyle Z\big{(}\iota^{n-1}-Z^{n-1}\big{)}+(\iota-Z)\iota^{n-1}.$ (6.4)

Since $Z\in\mathfrak{BI}_{2}$ and by induction hypothesis, the first summand
of (6.4) is in $\mathfrak{BI}_{2n-1}$. By (3.1) the element
$\iota\in\mathfrak{BI}_{2}$ and hence $\iota^{n-1}\in\mathfrak{BI}_{2n-2}$.
Combined with (6.3) the second summand of (6.4) is in $\mathfrak{BI}_{2n-1}$.
The lemma follows. ∎

######  Lemma 6.3.

  1. $(i)$

For all $n\in{\mathbb{N}}$ the elements

$\displaystyle
X^{i}Y^{j}Z^{k}\kappa^{r}\lambda^{s}\mu^{t}+\mathfrak{BI}_{n-1}\qquad\text{for
all $i,j,k,r,s,t\in{\mathbb{N}}$ with $i+j+2k=n$}$

are an ${\mathbb{F}}$-basis for $\mathfrak{BI}_{n}/\mathfrak{BI}_{n-1}$.

  2. $(ii)$

For all $n\in{\mathbb{N}}$ the elements

$\displaystyle
X^{i}Y^{j}\iota^{k}\kappa^{r}\lambda^{s}\mu^{t}+\mathfrak{BI}_{n-1}\qquad\text{for
all $i,j,k,r,s,t\in{\mathbb{N}}$ with $i+j+2k=n$}$

are an ${\mathbb{F}}$-basis for $\mathfrak{BI}_{n}/\mathfrak{BI}_{n-1}$.

  3. $(iii)$

For all $n\in{\mathbb{N}}$ the elements

$\displaystyle X^{i}Y^{j}\iota^{k}\kappa^{r}\lambda^{s}\mu^{t}\qquad\text{for
all $i,j,k,r,s,t\in{\mathbb{N}}$ with $i+j+2k\leq n$}$

are an ${\mathbb{F}}$-basis for $\mathfrak{BI}_{n}$.

###### Proof.

(i) Immediate from Theorem 3.4 and the construction of
$\\{\mathfrak{BI}_{n}\\}_{n\in{\mathbb{N}}}$.

(ii) Immediate from Lemma 6.2 and (i).

(iii) Using (ii) the statement (iii) follows by a routine induction on $n$. ∎

######  Theorem 6.4.

The elements

$\displaystyle X^{i}Y^{j}\iota^{k}\kappa^{r}\lambda^{s}\mu^{t}\qquad\text{for
all $i,j,k,r,s,t\in{\mathbb{N}}$}$ (6.5)

are an ${\mathbb{F}}$-basis for $\mathfrak{BI}$.

###### Proof.

Immediate from (N1) and Lemma 6.3(iii). ∎

######  Corollary 6.5.

The elements $\iota$, $\kappa$, $\lambda$, $\mu$ of $\mathfrak{BI}$ are
algebraically independent over ${\mathbb{F}}$.

###### Proof.

Immediate from Theorem 6.4. ∎

######  Lemma 6.6.

The ${\mathbb{F}}$-algebra $\mathfrak{BI}$ has a presentation with generators
$X$, $Y$, $\iota$, $\kappa$, $\lambda$, $\mu$ and relations

$\displaystyle YX=-XY-X-Y+\iota+\kappa,$ $\displaystyle\iota
Y=2Y^{2}-Y\iota-Y+\iota+\kappa+\lambda,$ $\displaystyle\iota
X=2X^{2}-X\iota-X+\iota+\kappa+\mu,$ $\displaystyle\kappa
X=X\kappa,\qquad\kappa Y=Y\kappa,\qquad\kappa\iota=\iota\kappa,$
$\displaystyle\lambda X=X\lambda,\qquad\lambda
Y=Y\lambda,\qquad\lambda\iota=\iota\lambda,\qquad\lambda\kappa=\kappa\lambda,$
$\displaystyle\mu X=X\mu,\qquad\mu
Y=Y\mu,\qquad\mu\iota=\iota\mu,\qquad\mu\kappa=\kappa\mu,\qquad\mu\lambda=\lambda\mu.$

###### Proof.

This is a reformulation of Proposition 3.3 by using (3.1). ∎

Recall the $D_{6}$-symmetric Casimir elements $\Omega_{A}$, $\Omega_{B}$,
$\Omega_{C}$ of $\Re$ from (2.16)–(2.18).

######  Proposition 6.7.

The $D_{6}$-symmetric Casimir elements $\Omega_{A}$, $\Omega_{B}$,
$\Omega_{C}$ of $\Re$ have the following expressions:

$\displaystyle\Omega_{A}=\frac{\lambda(\lambda-2\iota+3)\big{(}4\iota^{2}-8\iota-4\kappa-4\lambda-4\mu+7\big{)}}{1024}-\Gamma,$
(6.6)
$\displaystyle\Omega_{B}=\frac{\mu(\mu-2\iota+3)\big{(}4\iota^{2}-8\iota-4\kappa-4\lambda-4\mu+7\big{)}}{1024}-\Gamma,$
(6.7)
$\displaystyle\Omega_{C}=\frac{\kappa(\kappa-2\iota+3)\big{(}4\iota^{2}-8\iota-4\kappa-4\lambda-4\mu+7\big{)}}{1024}-\Gamma,$
(6.8)

where

$\displaystyle\Gamma=\frac{3(2\iota+3)(2\iota+1)(2\iota-5)(2\iota-7)}{4096}-\frac{(2\iota+1)(6\iota-13)(\kappa+\lambda+\mu)}{512}$
$\displaystyle\hphantom{\Gamma=}{}+\frac{(\kappa+\lambda+\mu)(\kappa+\lambda+\mu+4)}{64}-\frac{(2\iota-3)(\kappa\lambda+\lambda\mu+\mu\kappa)}{512}+\frac{\kappa\lambda\mu}{256}.$

###### Proof.

Applying Theorem 4.1 and Proposition 4.3 to (2.16) and replacing $Z$ by
$\iota-X-Y$, we may express $\Omega_{A}$ in terms of $X$, $Y$, $\iota$,
$\kappa$, $\lambda$, $\mu$. To get (6.6) we apply Lemma 6.6 to express the
resulting expression as an ${\mathbb{F}}$-linear combination of (6.5).
Combined with Lemma 2.8 and Proposition 3.2 we obtain (6.7) and (6.8). ∎

######  Theorem 6.8.

For each Casimir element $\Omega$ of $\Re$ there exists a unique four-variable
polynomial $P(x_{1},x_{2},x_{3},x_{4})$ over ${\mathbb{F}}$ such that

$\displaystyle\Omega=P(\iota,\kappa,\lambda,\mu).$

###### Proof.

By Definition 2.6 there exists a four-variable polynomial
$Q(y_{1},y_{2},y_{3},y_{4})$ over ${\mathbb{F}}$ such that
$\Omega=\Omega_{A}+Q(\alpha,\beta,\gamma,\delta)$. Set
$\widehat{Q}(x_{1},x_{2},x_{3},x_{4})=Q(y_{1},y_{2},y_{3},y_{4})$ by
substituting

$\displaystyle y_{1}=\frac{(2x_{1}-x_{2}-x_{4}-3)(x_{2}-x_{4})}{64},\qquad
y_{2}=\frac{(2x_{1}-x_{3}-x_{2}-3)(x_{3}-x_{2})}{64},$ $\displaystyle
y_{3}=\frac{(2x_{1}-x_{4}-x_{3}-3)(x_{4}-x_{3})}{64},\qquad
y_{4}=\frac{x_{1}^{2}-2x_{1}-x_{2}-x_{3}-x_{4}}{4}-\frac{9}{16}.$

It follows from Theorem 4.1 that
$\Omega=\Omega_{A}+\widehat{Q}(\iota,\kappa,\lambda,\mu)$. Combined with
Proposition 6.7 the existence follows. The uniqueness is immediate from
Corollary 6.5. ∎

### Acknowledgements

The research is supported by the Ministry of Science and Technology of Taiwan
under the project MOST 106-2628-M-008-001-MY4.

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