Tridiagonal pairs and the q-tetrahedron algebra

Abstract

The q-tetrahedron algebra ⊠ q was recently introduced and has been studied in connection with tridiagonal pairs. In this paper we further develop this connection. Let K denote an algebraically closed field and let q denote a nonzero scalar in K that is not a root of unity. Let V denote a vector space over K with finite positive dimension and let A, A * denote a tridiagonal pair on V . Let) denote a standard ordering of the eigenvalues of A (resp. A * ). T. Ito and P. Terwilliger have shown that when θ i = q 2i−d and θ * i = q d−2i (0 ≤ i ≤ d) there exists an irreducible ⊠ qmodule structure on V such that the ⊠ q generators x 01 , x 23 act as A, A * respectively. In this paper we examine the case in which there exists a nonzero scalar c in K such that θ i = q 2i−d and θ * i = q 2i−d + cq d−2i for 0 ≤ i ≤ d. In this case we associate to A, A * a polynomial P in one variable and prove the following theorem as our main result. Theorem The following are equivalent:(i) There exists a ⊠ q -module structure on V such that x 01 acts as A and x 30 + cx 23 acts as A * , where x 01 , x 30 , x 23 are standard generators for ⊠ q .(ii) P (q 2d−2 (q − q −1 ) −2 ) = 0.Suppose (i),(ii) hold. Then the ⊠ q -module structure on V is unique and irreducible.

A classification of sharp tridiagonal pairs

“…Some notable papers on the topic are [6, 13, 28-31, 35-37, 71]. There are connections to representation theory [2,9,21,26,29,31,39,41,42,64,65,75], partially ordered sets [73], the bispectral problem [5,6,[22][23][24]82], statistical mechanical models [8-14, 17-20, 62], and other areas of physics [61,81,83].…”

How to Sharpen a Tridiagonal Pair

“…We refer the reader to [1,2,[4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein for background information on TD pairs.…”

“…It is an open problem to classify the sharp TD pairs up to isomorphism, but progress is being made [4,8,11,12,14,16,18]. Concerning the nonsharp TD pairs, relatively little research has been done.…”

Tridiagonal pairs of $q$-Racah type, the double lowering operator $ψ$, and the quantum algebra $U_q(\mathfrak{sl}_2)$

“…Over time, connections were found between TD pairs and other areas of mathematics and physics. For instance, there are connections between TD pairs and representation theory [1,6,12,14,16,19,20,33], orthogonal polynomials [28,32], partially ordered sets [29], statistical mechanical models [2,5,27], and other areas of physics [21,26]. Among the above papers on representation theory, there are several works that connect TD pairs to quantum groups [1,6,12,16].…”

“…They showed that these TD pairs induce a U q ( sl 2 )-module structure on their underlying vector space. In [6], Funk-Neubauer extended this construction to TD pairs of q-Hahn type. In [12], Ito and Terwilliger extended the construction to the entire q-Serre class.…”