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

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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 : V → V and A ⁎ : V → V such that (i) each of A , A ⁎ is diagonalizable; (ii) there exists an ordering { V i } i = 0 d of the eigenspaces of A such that A ⁎ V i ⊆ V i − 1 + V i + V i + 1 for 0 ≤ i ≤ d , where V − 1 = 0 and V d + 1 = 0 ; (iii) there exists an ordering { V i ⁎ } i = 0 δ of the eigenspaces of A ⁎ such that A V i ⁎ ⊆ V i − 1 ⁎ + V i ⁎ + V i + 1 ⁎ for 0 ≤ i ≤ δ , where V − 1 ⁎ = 0 and V δ + 1 ⁎ = 0 ; (iv) there does not exist a subspace U of V such that A U ⊆ 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.