The algebra Uq(sl2) in disguise

Abstract

We discuss a connection between the algebra U q (sl 2 ) and the tridiagonal pairs of q-Racah type. To describe the connection, let x, y ±1 , z denote the equitable generators for U q (sl 2 ). Let U ∨ q denote the subalgebra of U q (sl 2 ) generated by x, y −1 , z. Using a tridiagonal pair of q-Racah type we construct two finite-dimensional U ∨ q -modules. The constructions yield two nonstandard presentations of U ∨ q by generators and relations. These presentations are investigated in detail.

Totally bipartite tridiagonal pairs

“…The paper [121] shows howĤ q is related to Leonard pairs. Additional results in the literature link tridiagonal pairs and Leonard pairs with the Lie algebra sl 2 (see [3, 7-9, 26, 81, 120]), the quantum algebras U q (sl 2 ) (see [2,29,30,88,147,162]), U q ( sl 2 ) (see [5,24,41,75,76,84,152]), the tetrahedron Lie algebra (see [25,56,79,96]), and its q-deformation q (see [42,73,77,78,80,164,165]).…”

The Lusztig automorphism of Uq(𝔰𝔩2) from the equitable point of view

“…Proof. Consider the expression for Λ given in (3). Write this expression in terms of x, y, z using any identification.…”

“…The equitable presentation of U q (sl 2 ) was introduced in [8] and investigated further in [1,3,4,6,7,12,[14][15][16][17][18][19]. This presentation has generators x, y ±1 , z and relations yy −1 = 1, y −1 y = 1,…”

Tridiagonal pairs of q-Racah type, the Bockting operator ψ, and L-operators for U_q(L(sl_2))

“…In [1], [3] Pascal Baseilhac and Kozo Koizumi use L-operators for the quantum loop algebra U q (L(sl 2 )) to construct a family of finite-dimensional modules for the q-Onsager algebra O q ; see [2,[4][5][6] for related work. A finite-dimensional irreducible O q -module is essentially the same thing as a tridiagonal pair of q-Racah type [9,Section 12], [23,Section 3]. In [22,Section 9], Kei Miki uses similar L-operators to describe how U q (L(sl 2 )) is related to the q-tetrahedron algebra ⊠ q .…”

“…The known properties of ψ are described in [7][8][9]. Suppose we are given A, A * , R, K in matrix form, and wish to obtain ψ in matrix form.…”