Tridiagonal pairs of $q$-Racah type and the $q$-tetrahedron algebra

Abstract

Let F denote a field, and let V denote a vector space over F with finite positive dimension. We consider an ordered pair of F-linear maps A :, where V * −1 = 0 and V * δ+1 = 0; (iv) there does not exist a subspace U of V such that AU ⊆ U , A * U ⊆ U , U = 0, U = V . We call such a pair a tridiagonal pair on V . We assume that A, A * belongs to a family of tridiagonal pairs said to have q-Racah type. There is an infinite-dimensional algebra ⊠ q called the q-tetrahedron algebra; it is generated by four copies of U q (sl 2 ) that are related in a certain way. Using A, A * we construct two ⊠ q -module structures on V . In this construction the two main ingredients are the double lowering map ψ : V → V due to Sarah Bockting-Conrad, and a certain invertible map W : V → V motivated by the spin model concept due to V. F. R. Jones.

Leonard pairs, spin models, and distance-regular graphs

“…Next, we consider the case of equality in (49). If p i 2,i = 0 then equality holds in (49), according to the previous proof. We now consider the case in which p i 2,i = 0.…”

Freidel-Maillet type presentations of $U_q(sl_2)$

“…However, in recent years a different presentation for U q (g) has been introduced, called equitable, with generators {X ±1 i , Y i , Z i , i = 1, ..., n} [ ITW05,T05]. Since [ITW05], the equitable presentation of U q (sl 2 ) has been studied from various perspectives: algebra [T11, T15b,GVZ15,FJ16,LXJ19], combinatorics and representation theory [A11, MR12, Fu13, HG13, IRT13, BCT14, T14, Y15, N15, T15a, SG16,T20] for instance. Also, note that irreducible finite dimensional representations have been studied in details in [ ITW05,T09].…”