Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations

By Paul M. Terwilliger. 2001

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We define an algebra on two generators which we call the Tridiagonal algebra, and we consider its irreducible modules. The algebra is defined as follows. Let K denote a field, and let $\beta, \gamma, \gamma^*, \varrho, \varrho^*$ 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 (i) \lbrack A,A^2A^*-\beta AA^*A + A^*A^2 -\gamma (AA^*+A^*A)- \varrho A^*\rbrack = 0, (ii) \lbrack A^*,A^{*2}A-\beta A^*AA^* + AA^{*2} -\gamma^* (A^*A+AA^*)- \varrho^* A\rbrack = 0, where $\lbrack r,s\rbrack $ means $rs-sr$. We call these relations the Tridiagonal relations. For $\beta = q+q^{-1}$, $\gamma = \gamma^*=0$, $\varrho=\varrho^*=0$, the Tridiagonal relations are the $q$-Serre relations. For $\beta = 2$, $\gamma = \gamma^*=0$, $\varrho=b^2$, $\varrho^*=b^{*2}$, the Tridiagonal relations are the Dolan-Grady relations. 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, and the more general Tridiagonal pairs recently introduced by Ito, Tanabe, and the present author. In the second part of the paper, we construct an infinite dimensional irreducible $T$-module based on the Askey-Wilson polynomials.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0895a336b06f866ac67f3e2fac58320b8b73513e

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Citations in the corpus

Finite-Dimensional Irreducible Modules for the Three-Point 𝔰𝔩2 Loop Algebra (2007, background): “A Leonard pair is a pair of semisimple linear transformations on a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other (Terwilliger, 2001, Definition 1.1).”
TWO NON-NILPOTENT LINEAR TRANSFORMATIONS THAT SATISFY THE CUBIC q-SERRE RELATIONS (2005, background): “onNil ﬁnite-dimensional irreducible A q-modules. Before we state our results, we explain why this problem is of interest. A recent topic of research concerns the Leonard pairs [23], [37], [39], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [53], [54] and the closely related tridiagonal”
The augmented tridiagonal algebra (2009, methodology): “The algebra A is called the tridiagonal algebra (TD-algebra) [10] of the 1st, 2nd, 3rd kind, accordingly.”
LEONARD PAIRS AND THE ASKEY-WILSON RELATIONS (2003, background): “See [16, 26, 27, 29, 30 ] for more information on these relations.”
The shape of a tridiagonal pair (2003, unknown): “3 [13] Let β, γ, γ∗, , ∗ denote a sequence of scalars taken from K.”
Tridiagonal pairs and the quantum affine algebra Uq(sl2) (代数的組合せ論) (2005, background): “We now recall the notion of a tridiagonal pair [7], [12].”
Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair (2006, background): “7], [18], [19], [21], [22], [23], [24], [25], [26], [27], [28], [30], [31] for background on Leonard pairs. We especially recommend the survey [28]. See [1], [2], [4], [5], [6], [7], [8], [10], [11], [20], [29] for related topics. 2 Leonard systems When working with a Leonard pair, it is convenient ”
Two linear transformations each tridiagonal with respect to an eigenbasis of the other: comments on the split decomposition (2003, background): “,{E ∗ i } d i=0 ; (ii) there exists an antiautomorphism † of A such that A† = A and A∗† = A∗. 2 1 Leonard pairs and Leonard systems We begin by recalling the notion of a Leonard pair [6], [12], [13], [14], [15], [16], [17], [18]. We will use the following terms. Let X denote a square matrix. Then X ”
Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD- D canonical form and the LB- UB canonical form (2003, unknown): “For instance Leonard pairs arise naturally in the representation theory of the Lie algebra sl2 [16], the quantum algebra Uq(sl2) [18], [19], [20], [21], [22], [25, Chapter 4], [27], [36], the Askey-Wilson algebra [7], [8], [9], [37] and the Tridiagonal algebra [16], [35], [ 36 ].”
Two linear transformations each tridiagonal with respect to an eigenbasis of the other: An Algebraic approach to the Askey scheme of orthogonal polynomials (2004, unknown): “These notes are based on the papers [45], [46], [47], [79], [ 80 ], [81], [82], [83], [84], [85], [86], [87], [88].”
Tridiagonal pairs of q-Racah type (2008, background): “See [9, 35, 52, 54, 59] for results on the tridiagonal relations.”
The Universal Askey-Wilson Algebra (2011, background): “4], the theory of tridiagonal pairs [31, 33, 54, 55, 56], and integrable systems [7, 8, 9, 10, 11, 12, 13, 14, 15].”
An Algebraic Approach to the Askey Scheme of Orthogonal Polynomials (2006, unknown): “These notes are based on the papers [45], [46], [47], [79], [ 80 ], [81], [82], [83], [84], [85], [86], [87], [88].”
The Tetrahedron algebra, the Onsager algebra, and the sl2 loop algebra (2007, background): “Over the years O has been investigated many times in connection with solvable lattice models [1], [2], [3], [5], [6], [7], [8], [15], [16], [22], [26], [27], [28], [30], [39] representation theory [13], [14], [21] Kac-Moody algebras [12], [32], [33] tridiagonal pairs [23], [24], [36], [37], [38] and”