Finite-dimensional irreducible □ -modules and their Drinfel'd polynomials

An Infinite-Dimensional q-Module Obtained from the q-Shuffle Algebra for Affine sl₂

“…The finite-dimensional q -modules are investigated in [33,34]. By [34,Proposition 5.2], each q -generator is invertible on every nonzero finite-dimensional q -module.…”

“…This result gets used in [34,Sections 8 and 9] to obtain some remarkable q-exponential formulas. In [33], the finite-dimensional irreducible q -modules are classified up to isomorphism. This classification is summarized as follows.…”

“…There is a family of finite-dimensional irreducible q -modules, said to have type 1 [33,Definition 6.8]. Any finite-dimensional irreducible q -module can be normalized to have type 1, by twisting it via an appropriate automorphism of q [33,Note 6.9]. Let V denote a finite-dimensional irreducible q -module of type 1.…”

“…Let V denote a finite-dimensional irreducible q -module of type 1. In [33,Definition 8.6] V gets attached to a polynomial P V in one variable z that has constant coefficient 1; this P V is called the Drinfel'd polynomial of V . By [33,Proposition 1.4], the map V → P V induces a bijection between the following two sets: (i) the isomorphism classes of finite-dimensional irreducible q -modules of type 1; (ii) the polynomials in the variable z that have constant coefficient 1 and do not vanish at z = 1.…”

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]).…”