Two relations that generalize the \(q\)-Serre relations and the Dolan-Grady relations. (English) Zbl 1061.16033
Kirillov, Anatol N. (ed.) et al., Physics and combinatorics. Proceedings of the international workshop, Nagoya, Japan, August 23–27, 1999. Singapore: World Scientific (ISBN 981-02-4578-5). 377-398 (2001).
Summary: We define an algebra on two generators which we call a tridiagonal algebra, and we consider its irreducible modules. The algebra is defined as follows. Let \(K\) denote a field, and let \(\beta,\gamma,\gamma^*,\rho,\rho^*\) denote a sequence of scalars taken from \(K\). The corresponding tridiagonal algebra \(T\) is the associative \(K\)-algebra with 1 generated by two symbols \(A\), \(A^*\) subject to the relations
\[
[A,A^2A^*-\beta AA^*A+A^*A^2-\gamma(AA^*+A^*A)-\rho A^*]=0,
\]
\[ [A^*,A^{*2}A-\beta A^*AA^*+AA^{*2}-\gamma^*(A^*A+AA^*)-\rho^*A]=0, \] where \([r,s]\) means \(rs-sr\). We call these relations the tridiagonal relations. For \(\beta=q+q^{-1}\), \(\gamma=\gamma^*=0\), \(\rho=\rho^*=0\), the tridiagonal relations are the \(q\)-Serre relations \[ A^3A^*-[3]_qA^2A^*A+[3]_qAA^*A^2-A^*A^3=0, \]
\[ A^{*3}A-[3]_qA^{*2}AA^*+[3]_qA^*AA^{*2}-AA^{*3}=0, \] where \([3]_q=q+q^{-1}+1\). For \(\beta=2\), \(\gamma=\gamma^*=0\), \(\rho=b^2\), \(\rho^*=b^{*2}\), the tridiagonal relations are the Dolan-Grady relations \[ [A,[A,[A,A^*]]]=b^2[A,A^*],\qquad [A^*,[A^*,[A^*,A]]]=b^{*2}[A^*,A]. \] In the first part of this paper, we survey what is known about irreducible finite-dimensional \(T\)-modules. We focus on how these modules are related to the Leonard pairs recently introduced by the present author [Linear Algebra Appl. 330, No. 1-3, 149-203 (2001; Zbl 0980.05054)], and the more general tridiagonal pairs recently introduced by T. Itô, K. Tanabe, and the present author [DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 56, 167–192 (2001; Zbl 0995.05148)].
In the second part of the paper, we construct an infinite dimensional irreducible \(T\)-module based on the Askey-Wilson polynomials. This module is on the vector space \(K[x]\) consisting of all polynomials in an indeterminate \(x\) that have coefficients in \(K\). Denoting by \(A\) the linear transformation on \(K[x]\) which is multiplication by \(x\), and denoting by \(A^*\) an Askey-Wilson second-order \(q\)-difference operator for \(x\), we show that \(A\) and \(A^*\) satisfy a pair of tridiagonal relations. Using this we give \(K[x]\) the structure of an irreducible \(T\)-module. The Askey-Wilson polynomials form a basis for this module, and these basis elements are eigenvectors for \(A^*\).
For the entire collection see [Zbl 0964.00054].
\[ [A^*,A^{*2}A-\beta A^*AA^*+AA^{*2}-\gamma^*(A^*A+AA^*)-\rho^*A]=0, \] where \([r,s]\) means \(rs-sr\). We call these relations the tridiagonal relations. For \(\beta=q+q^{-1}\), \(\gamma=\gamma^*=0\), \(\rho=\rho^*=0\), the tridiagonal relations are the \(q\)-Serre relations \[ A^3A^*-[3]_qA^2A^*A+[3]_qAA^*A^2-A^*A^3=0, \]
\[ A^{*3}A-[3]_qA^{*2}AA^*+[3]_qA^*AA^{*2}-AA^{*3}=0, \] where \([3]_q=q+q^{-1}+1\). For \(\beta=2\), \(\gamma=\gamma^*=0\), \(\rho=b^2\), \(\rho^*=b^{*2}\), the tridiagonal relations are the Dolan-Grady relations \[ [A,[A,[A,A^*]]]=b^2[A,A^*],\qquad [A^*,[A^*,[A^*,A]]]=b^{*2}[A^*,A]. \] In the first part of this paper, we survey what is known about irreducible finite-dimensional \(T\)-modules. We focus on how these modules are related to the Leonard pairs recently introduced by the present author [Linear Algebra Appl. 330, No. 1-3, 149-203 (2001; Zbl 0980.05054)], and the more general tridiagonal pairs recently introduced by T. Itô, K. Tanabe, and the present author [DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 56, 167–192 (2001; Zbl 0995.05148)].
In the second part of the paper, we construct an infinite dimensional irreducible \(T\)-module based on the Askey-Wilson polynomials. This module is on the vector space \(K[x]\) consisting of all polynomials in an indeterminate \(x\) that have coefficients in \(K\). Denoting by \(A\) the linear transformation on \(K[x]\) which is multiplication by \(x\), and denoting by \(A^*\) an Askey-Wilson second-order \(q\)-difference operator for \(x\), we show that \(A\) and \(A^*\) satisfy a pair of tridiagonal relations. Using this we give \(K[x]\) the structure of an irreducible \(T\)-module. The Askey-Wilson polynomials form a basis for this module, and these basis elements are eigenvectors for \(A^*\).
For the entire collection see [Zbl 0964.00054].
MSC:
| 16S15 | Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) |
| 16G30 | Representations of orders, lattices, algebras over commutative rings |
| 33D45 | Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) |
| 05E35 | Orthogonal polynomials (combinatorics) (MSC2000) |
| 05E30 | Association schemes, strongly regular graphs |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
