Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type $(C_1^{\vee },C_1)$

Abstract

Assume that $\mathbb{F}$ is an algebraically closed field with characteristic zero. The universal Racah algebra $\Re$ is a unital associative $\mathbb{F}$-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 \begin{gather*} [A,D]+AC-BA, \qquad [B,D]+BA-CB, \qquad [C,D]+CB-AC \end{gather*} is central in $\Re$. The universal additive DAHA (double affine Hecke algebra) $\mathfrak{H}$ of type $(C_1^{\vee },C_1)$ is a unital associative $\mathbb{F}$-algebra generated by $t_0,t_1,t_0^{\vee },t_1^{\vee }$ and the relations state that $$t_0+t_1+t_0^{\vee }+t_1^{\vee }=-1$$ and each of $t_0^2,t_1^2,t_0^{\vee 2},t_1^{\vee 2}$ is central in $\mathfrak{H}$. Each $\mathfrak{H}$-module is an $\Re$-module by pulling back via the algebra homomorphism $\Re \to \mathfrak{H}$ given by \begin{eqnarray*} A &\mapsto & \frac{(t_1^\vee+t_0^\vee)(t_1^\vee+t_0^\vee+2)}{4}, \\ B &\mapsto & \frac{(t_1+t_1^\vee)(t_1+t_1^\vee+2)}{4}, \\ C &\mapsto & \frac{(t_0^\vee+t_1)(t_0^\vee+t_1+2)}{4}. \end{eqnarray*} Let $V$ denote any finite-dimensional irreducible $\mathfrak{H}$-module. The set of $\Re$-submodules of $V$ forms a lattice under the inclusion partial order. We classify the lattices that arise by this construction. As a consequence, the $\Re$-module $V$ is completely reducible if and only if $t_0$ is diagonalizable on $V$.

30 pages

References in corpus (5)

• The Universal Askey-Wilson Algebra
• The Bannai-Ito algebra and some applications
• Embeddings of the Racah Algebra into the Bannai-Ito Algebra
• Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero
• Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero

Cited by in corpus (2)

• Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero
• The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra