Quantum torus algebras and B(C) type Toda systems

TL;DR

This paper introduces a new constrained B(C) type Toda hierarchy, explores its symmetries, and generalizes it to an N-component system with symmetries linked to quantum torus algebra structures.

Contribution

It constructs a novel even constrained B(C) type Toda hierarchy and extends it to an N-component version with associated quantum torus algebra symmetries.

Findings

New B(C) type Toda hierarchy constructed

Derived B(C) type Block type additional symmetry

Generalized to N-component hierarchy with quantum torus algebra symmetries

Abstract

In this paper, we construct a new even constrained B(C) type Toda hierarchy and derive its B(C) type Block type additional symmetry. Also we generalize the B(C) type Toda hierarchy to the $N$-component B(C) type Toda hierarchy which is proved to have symmetries of a coupled $\bigotimes^NQT_+ $ algebra ( $N$-folds direct product of the positive half of the quantum torus algebra $QT$).

Equations

$${\partial }_{x_{2n+1}}{L}_1=[-(L_1^{2n+1}{)}_{-},L_1]\text{and}{\partial }_{y_{2n+1}}{L}_1=[-(L_2^{2n+1}{)}_{-},L_1],n=0,1,2,\cdots ,$$

$${\partial }_{x_{2n+1}}{L}_2=[(L_1^{2n+1}{)}_{+},L_2]\text{and}{\partial }_{y_{2n+1}}{L}_2=[(L_2^{2n+1}{)}_{+},L_2],n=0,1,2,\cdots ,$$

$$L_1=\sum _{-\infty <i\le 1}\text{diag}[a_i^{(1)}(s)]{\Lambda }^i,L_2=\sum _{-1\le i<\infty }\text{diag}[a_i^{(2)}(s)]{\Lambda }^i,$$

$$a_1^{(1)}(s)=1\text{and}{a}_{-1}^{(2)}(s)\ne 0\forall s$$

$$L_i^T=-J{L}_i{J}^{-1},L_i^T=-K{L}_i{K}^{-1},$$

$$a_i^{(k)}(s)=(-1{)}^{i+1}{a}_i^{(k)}(-s-i),k=1,2.$$

$$a_i^{(k)}(s)=(-1{)}^{i+1}{a}_i^{(k)}(-s-i-1).$$

$${\partial }_{x_{2n+1}}(L_1^{2m+1}{)}_{+}-{\partial }_{x_{2m+1}}(L_1^{2n+1}{)}_{+}+[(L_1^{2m+1}{)}_{+},(L_1^{2n+1}{)}_{+}]=0,$$

$${\partial }_{y_{2n+1}}(L_2^{2m+1}{)}_{-}+{\partial }_{y_{2m+1}}(L_2^{2n+1}{)}_{-}-[(L_2^{2m+1}{)}_{-},(L_2^{2n+1}{)}_{-}]=0,$$

$${\partial }_{y_{2n+1}}(L_1^{2m+1}{)}_{+}+{\partial }_{x_{2m+1}}(L_2^{2n+1}{)}_{-}-[(L_1^{2m+1}{)}_{+},(L_2^{2n+1}{)}_{-}]=0,$$

$$-{\partial }_{y_{2n+1}}(L_2^{2m+1}{)}_{-}-{\partial }_{x_{2m+1}}(L_1^{2n+1}{)}_{+}-[(L_2^{2m+1}{)}_{-},(L_1^{2n+1}{)}_{+}]=0.$$

$${\partial }_{x_1}{a}_{-1}^{(2)}(1)=a_{-1}^{(2)}(1)a_0^{(1)}(1),{\partial }_{x_1}{a}_{-1}^{(2)}(s)=a_{-1}^{(2)}(s)(a_0^{(1)}(s)-a_0^{(1)}(s-1))(s\ge 2),$$

$${\partial }_{y_1}{a}_0^{(1)}(s)=a_{-1}^{(2)}(s)-a_{-1}^{(2)}(s+1)(s\ge 1),$$

$${\partial }_{x_1}{a}_{-1}^{(2)}(0)=2a_{-1}^{(2)}(0)a_0^{(1)}(0),{\partial }_{x_1}{a}_{-1}^{(2)}(s)=a_{-1}^{(2)}(s)(a_0^{(1)}(s)-a_0^{(1)}(s-1))(s\ge 1),$$

$${\partial }_{y_1}{a}_0^{(1)}(s)=a_{-1}^{(2)}(s)-a_{-1}^{(2)}(s+1)(s\ge 0),$$

$$L_1$$

$$L_2$$

$$S_1(x,y)=\sum _{i\ge 0}\text{diag}[c_i(s;x,y)]{\Lambda }^{-i},$$

$$W_1(x,y)=S_1(x,y)e^{\xi (x,\Lambda )},W_2(x,y)=S_2(x,y)e^{\xi (y,{\Lambda }^{-1})}$$

$$J^{-1}{W}_i^{T}J=W_i^{-1},i=1,2.$$

$$K^{-1}{W}_i^{T}K=W_i^{-1},i=1,2.$$

$${\partial }_{x_{2n+1}}{S}_1$$

$${\partial }_{x_{2n+1}}{W}_1$$

$${\partial }_{x_{2n+1}}{S}_2$$

$${\partial }_{x_{2n+1}}{W}_2$$

$$M_1=W_1\epsilon W_1^{-1},M_2=W_2{\epsilon }^{\ast }{W}_2^{-1},$$

$$\epsilon =\mathrm{diag}[\mathrm{s}]{\mathrm{\Lambda }}^{-1},{\epsilon }^{\ast }=-\mathrm{J}\epsilon {\mathrm{J}}^{-1},$$

$$[L_i,M_i]=1,$$

$${\partial }_{x_{2n+1}}{M}_i=[(L_1^{2n+1}{)}_{+},M_i],{\partial }_{y_{2n+1}}{M}_i=[-(L_2^{2n+1}{)}_{-},M_i].$$

$${\Lambda }^{-1}\epsilon \Lambda =J^{-1}{\epsilon }^{T}J,$$

$$\epsilon =K{\epsilon }^T{K}^{-1},$$

$${M_i}^T$$

$${\partial }_{x_{m,l}}{W}_1=-A_{ml}(M_1,L_1{)}_{-}{W}_1,{\partial }_{y_{m,l}}{W}_1=-A_{ml}(M_2,L_2{)}_{-}{W}_1,$$

$${\partial }_{x_{m,l}}{W}_2=A_{ml}(M_1,L_1{)}_{+}{W}_2,{\partial }_{y_{m,l}}{W}_2=A_{ml}(M_2,L_2{)}_{+}{W}_2,$$

Full text

‡Corresponding author’s email:lichuanzhong@nbu.edu.cn

Mathematics Subject Classifications(2000): 37K05, 37K10, 37K40.

Keywords: B(C) type Toda hierarchy, additional symmetries, multicomponent B(C) type Toda hierarchy, quantum torus algebra.

1. 1 Introduction
2. 2 The B(C) type Toda hierarchy
3. 3 The even constrained BTH(CTH)
4. 4 Multicomponent B(C) type Toda hierarchy
5. 5 Symmetries of of MBTH(MCTH)

1. Introduction

The Toda lattice hierarchy as a completely integrable system has many important applications in mathematics and physics including the representation theory of Lie algebras and random matrix models [1, 2, 3]. The Toda system has many kinds of reductions or extensions, for example the B and C type Toda hierarchies[2, 4], extended Toda hierarchy (ETH)[5], bigraded Toda hierarchy (BTH)[6]-[11] and so on. There are some other generalizations called multi-component Toda systems[2, 12] which are useful in the fields of multiple orthogonal polynomials and non-intersecting Brownian motions.

The multicomponent 2D Toda hierarchy was considered from the point of view of the Gauss-Borel factorization problem, the theory of multiple matrix orthogonal polynomials, non-intersecting Brownian motions and matrix Riemann-Hilbert problem [12]-[15]. In fact the multicomponent 2D Toda hierarchy in [13] is a periodic reduction of the bi-infinite matrix-formed two dimensional Toda hierarchy. In [16], we generalize the multicomponent Toda hierarchy to an extended multicomponent Toda hierarchy including extended logarithmic flow equations. Later by a commutative algebraic reduction on the extended multicomponent Toda hierarchy, we get an extended $Z_{N}$-Toda hierarchy[17] which might be useful in Gromov-Witten theory.

This paper is organized in the following way. In Section 2, we recall some basic knowledge about the B(C) type Toda hierarchy. We construct a new even constrained B(C) type Toda hierarchy and derive its Block type additional symmetry in Section 3. Next, in Section 4 we generalize the B(C) type Toda hierarchy to a new $N$-component B(C) type Toda hierarchy. In the last section, we construct the symmetry of the $N$-component B(C) type Toda hierarchy which constitutes a coupled $\bigotimes^{N}QT_{+}$ algebra ( $N$-folds direct product of the positive half of the quantum torus algebra $QT$).

2. The B(C) type Toda hierarchy

In this section, some basic facts about the B(C) type Toda hierarchy are reviewed. One can refer to [2, 4] for more details about the B(C) type Toda hierarchy (or BTH(CTH)).

Then the BTH hierarchy is defined in the Lax forms as

[TABLE]

[TABLE]

where the Lax operator $L_{i}$ is given by a pair of infinite matrices

[TABLE]

with $\Lambda=(\delta_{j-i,1})_{i,j\in\mathbb{Z}}$, and $a_{i}^{(k)}(s)$ and $a_{i}^{(k)}(s)$ depending on $x=(x_{1},x_{3},x_{5},\cdots)$ and $y=(y_{1},y_{3},y_{5},\cdots)$, such that

[TABLE]

and satisfies the BTH(CTH) constraint[2]

[TABLE]

where $J=((-1)^{i}\delta_{i+j,0})_{i,j\in\mathbb{Z}},K=\Lambda J$ and $T$ refers to the matrix transpose. The BTH constraint is explicitly showed as

[TABLE]

The CTH constraint means

[TABLE]

The Lax equation for the BTH(CTH) can be expressed as a system of equations of the Zakharov-Shabat type:

[TABLE]

When $m=n=0$, one can get the B type Toda equation

[TABLE]

by considering the corresponding constraint (5). Also one can get the C type Toda equation

[TABLE]

The Lax operator of the BTH(CTH) (37) has the representation

[TABLE]

where

[TABLE]

and

[TABLE]

with $c_{0}(s;x,y)=1$ and $c_{o}^{\prime}(s;x,y)\neq 0$ for any $s$, and $\xi(x,\Lambda^{\pm 1})=\sum_{n\geq 0}x_{2n+1}\Lambda^{\pm 2n+1}$. For the B type Toda hierarchy, under an appropriate choice $(W_{1},W_{2})$ satisfies

[TABLE]

For the C type Toda hierarchy, under an appropriate choice $(W_{1},W_{2})$ satisfies

[TABLE]

The wave operators evolve as

[TABLE]

[TABLE]

At last, we end this section with the introduction of the additional symmetries of the BTH(CTH). The Orlov-Shulman operator[4] is defined as

[TABLE]

where

[TABLE]

satisfying

[TABLE]

To construct the Block symmetry of the BTH, the following lemma should be introduced.

Lemma 1**.**

The following identities hold true

[TABLE]

[TABLE]

For the BTH, using the above lemma, one can derive

[TABLE]

For the CTH, using the above lemma, one can derive

[TABLE]

The additional symmetry [4] of the BTH can be defined by introducing the additional independent variables $x_{m,l}$ and $y_{m,l}$,

[TABLE]

[TABLE]

where

[TABLE]

For the case of the CTH, the operator $A_{ml}$ will become

[TABLE]

These additional flows form a coupled $W_{\infty}$ Lie algebra [4].

3. The even constrained BTH(CTH)

In this section, for a new constrained BTH(CTH), the Lax operator $L$ is given by an infinite matrices $L$ as

[TABLE]

with $a_{2N}(s)=1,$ and for the BTH, it satisfies the B type constraint

[TABLE]

and for the CTH, it satisfies the C type constraint

[TABLE]

Then the constrained BTH(CTH) hierarchy is defined in the Lax forms as

[TABLE]

[TABLE]

The Lax operator of the constrained BTH(CTH) (37) has the representation

[TABLE]

where

[TABLE]

and

[TABLE]

with $c_{0}(s;x,y)=1$ and $c_{o}^{\prime}(s;x,y)\neq 0$ for any $s$, and $\xi(x,\Lambda^{\pm 1})=\sum_{n\geq 0}x_{2n+1}\Lambda^{\pm 2n+1}$. Under an appropriate choice $(W_{1},W_{2})$ of the constrained BTH(CTH) satisfies

[TABLE]

The wave operators evolve according to

[TABLE]

[TABLE]

The Orlov-Shulman operator $\bar{M}_{i}$ will be defined as as

[TABLE]

where

[TABLE]

satisfying

[TABLE]

Lemma 2**.**

The difference of two Orlov-Schulman operators $\bar{M}_{i}$ for constrained BTH hierarchy has following B type property:

[TABLE]

and for constrained CTH hierarchy has following C type property:

[TABLE]

Proof.

It is easy to find the two Orlov-Schulman operators can be expressed as

[TABLE]

Putting eq.(50) into $(\bar{M}_{1}-\bar{M}_{2})^{T}$ can lead to

[TABLE]

which can further lead to eq.(48). For the CTH, one can do the similar calculation as

[TABLE]

which can further lead to eq.(49)

In above calculation, the commutativity between $L$ and $\bar{M}_{1}-\bar{M}_{2}$ is already used. Till now, the proof is finished. ∎

For the constrained BTH(CTH), we need the following operator

[TABLE]

One can easily check that for the BTH

[TABLE]

and for the CTH

[TABLE]

That means it is reasonable to define additional flows of the constrained BTH(CTH) as

[TABLE]

Proposition 3**.**

For the BTH(CTH), the flows (60) can commute with original flows of the BTH(CTH), namely,

[TABLE]

which hold in the sense of acting on $W_{i}$ or $L.$

Theorem 4**.**

The flows in eq.(60) about additional symmetries of constrained BTH(CTH) compose following Block type Lie algebra

[TABLE]

which holds in the sense of acting on $W_{i}$ or $L.$

4. Multicomponent B(C) type Toda hierarchy

In this section we will introduce the multicomponent B type Toda hierarchy (MBTH) and multicomponent C type Toda hierarchy (MCTH). In the following, we denote $E_{\mathbb{Z}\times\mathbb{Z}}$ as the bi-infinite identity matrix and $E_{N\times N}$ as the $N\times N$ identity matrix. We also denote $E_{kk}$ as a $N\times N$ matrix which is $1$ at the position of the k-th row and k-th column and [math] for other elements. The Lax operators $\mathcal{L}_{1},\mathcal{L}_{2}$ of the MBTH(MCTH) are given by a pair of infinite matrices

[TABLE]

where $b_{i}(s),c_{i}(s)$ are matrices of size $N\times N$ and $\bar{\Lambda}^{i}=\Lambda^{i}\otimes E_{N\times N}$ and they satisfy the B type(C Type) constraint[2]

[TABLE]

where $\mathcal{J}=((-1)^{i}\delta_{i+j,0})_{i,j\in\mathbb{Z}}\otimes E_{N\times N}$, $\mathcal{K}=\Lambda J\otimes E_{N\times N}$. Here the product $\otimes$ is the Kronecker product between a matrix of size $\mathbb{Z}\times\mathbb{Z}$ and a matrix of size $N\times N$. Let us first introduce some convenient notations as $\bar{E}_{kk}=E_{\mathbb{Z}\times\mathbb{Z}}\otimes E_{kk}$.

The Lax operators of the MBTH(MCTH) (37) can have the following dressing structure

[TABLE]

where

[TABLE]

Now we define matrix operators $C_{kk},\bar{C}_{kk},B_{jk},\bar{B}_{jk}$ as follows

[TABLE]

Now we give the definition of the multicomponent B(C) type Toda hierarchy(MBTH).

Definition 1**.**

The multicomponent B(C) type Toda hierarchy is a hierarchy in which the dressing operators $\mathcal{S}_{1},\mathcal{S}_{2}$ satisfy following Sato equations

[TABLE]

Then one can easily get the following proposition about $\mathcal{W}_{1},\mathcal{W}_{2}.$

Proposition 5**.**

The matrix wave operators $\mathcal{W}_{1},\mathcal{W}_{2}$ satisfy following Sato equations

[TABLE]

From the previous proposition we can derive the following Lax equations for the Lax operators.

Proposition 6**.**

The Lax equations of the MBTH(MCTH) are as follows

[TABLE]

5. Symmetries of of MBTH(MCTH)

To introduce the additional symmetries of the MBTH(MCTH). The Orlov-Shulman operator of the MBTH(MCTH) will be defined as

[TABLE]

[TABLE]

To construct the additional quantum torus symmetry of the multicomponent BTH, firstly we define the operator $B_{mnj}^{(i)}$ as

[TABLE]

For the multicomponent CTH, we define the operator $B_{mnj}^{(i)}$ as

[TABLE]

For any matrix operator $B_{mnj}^{(i)}$ in (77), one has

[TABLE]

[TABLE]

Then we can derive the following lemma.

Lemma 7**.**

The following identities hold true

[TABLE]

[TABLE]

Then for the MBTH, by (41) and (80), we can derive

[TABLE]

Using the second equation in eq.(80), we can also derive

[TABLE]

Similarly, for the CTH, we can derive

[TABLE]

Because of the constraints (63) on the Lax operators for the MBTH(MCTH), we can have the following proposition.

Proposition 8**.**

For the MBTH, it is sufficient to ask for

[TABLE]

Proof.

From (63) and (82), we have

[TABLE]

Since $J^{T}=J^{-1}=J$. Therefore $B_{mnj}^{(i)}$ will satisfy the B type condition. ∎

Similarly, the following proposition can also be got.

Proposition 9**.**

For the MCTH, the following C type condition must hold true

[TABLE]

Now for the MBTH we will denote the matrix operator $D_{mnj}$ as

[TABLE]

which further leads to

[TABLE]

Then the following calculation will lead to the B(C) type anti-symmetry property of $D_{imnj}$ as

[TABLE]

Now for the MCTH we will denote the matrix operator $D_{mnj}$ as

[TABLE]

Therefore we get the following important B(C) type condition which the matrix operator $D_{imnj}$ satisfies

[TABLE]

Then basing on a quantum parameter $q$, the additional flows for the time variable $t_{m,n}^{ij},t_{m,n}^{*ij}$ are defined as follows

[TABLE]

[TABLE]

or equivalently rewritten as

[TABLE]

[TABLE]

Generally, one can also derive

[TABLE]

[TABLE]

This further leads to the commutativity of the additional flow $\dfrac{\partial}{\partial t^{*ij}_{m,n}}$ with the flow $\partial_{t_{jn}},\partial_{\bar{t}_{jn}}$ in the following theorem.

Theorem 10**.**

The additional flows of $\partial_{t^{*is}_{l,k}}$ are symmetries of the multicomponent BTH(CTH), i.e. they commute with all $\partial_{t_{jn}},\partial_{\bar{t}_{jn}}$ flows of the multicomponent BTH(CTH).

Comparing with the additional symmetry of the single-component BTH(CTH), the additional flows $\partial_{t_{l,k}^{s}}$ of the multicomponent BTH(CTH) form the following $N$-folds direct product of the $\mathcal{W}_{\infty}$ algebra as following

[TABLE]

Now it is time to identity the algebraic structure of the additional $t_{l,k}^{*j}$ flows of the multicomponent BTH(CTH).

Theorem 11**.**

The additional flows $\partial_{t^{*dj}_{l,k}}$ of the multicomponent BTH(CTH) form the coupled $\bigotimes^{N}QT_{+}$ algebra ( $N$-folds direct product of the positive half of the quantum torus algebra $QT$), i.e.,

[TABLE]

Proof.

One can also prove this theorem as following by rewriting the quantum torus flow in terms of a combination of $\partial_{t_{m,n}^{ij}}$ flows

[TABLE]

∎

Acknowledgments: This work is supported by the National Natural Science Foundation of China under Grant No. 11571192 and K. C. Wong Magna Fund in Ningbo University.