Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type (C1∨, C1)

Assume that F is an algebraically closed field with characteristic zero. The universal Racah algebra R is a unital associative F-algebra defined by generators and relations. The generators are A, B, C, D and the relations state that [A, B] = [B, C] = [C, A] = 2D and each of [A, D] + AC - BA, [B, D] + BA - CB, [C, D] + CB - AC central in R. The universal additive DAHA (double affine Hecke algebra) H of type (C-1(boolean OR), C-1) is a unital associative F-algebra generated by t(0), t(1), t(0)(boolean OR), t(1)(boolean OR) and the relations state that t(0) + t(1) + t(0)(boolean OR) + t(1)(boolean OR) = -1 and each of t(0)(2), t(1)(2), t(0)(boolean OR 2), t(1)(boolean OR 2) is central in H. Each H-module is an R-module by pulling back via the algebra homomorphism R -> H given by A bar right arrow (t(1)(boolean OR) + t(0)(boolean OR))(t(1)(boolean OR) + t(0)(boolean OR) + 2)/4, B bar right arrow (t(1) + t(1)(boolean OR)) (t(1) + t(1)(boolean OR) + 2)/4, C bar right arrow (t(0)boolean OR + t(1)) (t(0)(boolean OR) + t(1) + 2)/4, Let V denote any finite-dimensional irreducible H-module. The set of R-submodules of V forms a lattice under the inclusion partial order. We classify the lattices that arise by this construction. As a consequence, the H-module Vis completely reducible if and only if t(0) is diagonalizable on V. (C) 2020 Elsevier B.V. All rights reserved.