Let K denote an algebraically closed field. Let V denote a vector space over K with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations in End(V ) such that for each of these transformations there exists a basis of V with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. There is a family of Leonard triples said to have QRacah type. This is the most general type of Leonard triple. We classify the Leonard triples of QRacah type up to isomorphism. We show that any Leonard triple of QRacah type satisfies the Z 3 -symmetric Askey-Wilson relations.

Let F denote an algebraically closed field and assume that q ∈ F is a primitive d th root of unity with d = 1, 2, 4. The universal Askey-Wilson algebra △ q is a unital associative F-algebra defined by generators and relations. The generators are A, B, C and the relations assert that each ofWe show that every finite-dimensional irreducible △ q -module is of dimension less than or equal toMoreover we provide an example to show that the bound is tight.

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Let F denote a field with char F = 2. The Racah algebra ℜ is the unital associative F-algebra defined by generators and relations in the following way. The generators are A, B, C, D. The relations assert thatand each of the elementsIn this paper we explore the relationship between the Racah algebra ℜ and the universal enveloping algebra U (sl 2 ). Let a, b, c denote mutually commuting indeterminates. We show that there exists a unique F-algebra homomorphism ♮ :where x, y, z are the equitable generators for U (sl 2 ). We additionally give the images of α, β, γ, δ, and certain Casimir elements of ℜ under ♮. We also show that the map ♮ is an injection and thus provides an embedding of ℜ into F[a, b, c] ⊗ U (sl 2 ). We use the injection to show that ℜ contains no zero divisors.

Assume that F is an algebraically closed field with characteristic zero. The Racah algebra ℜ is the unital associative F-algebra defined by generators and relations in the following way. The generators are A, B, C, D and the relations assert thatand that each ofIn this paper we discuss the finite-dimensional irreducible ℜ-modules in detail and classify them up to isomorphism. To do this, we apply an infinite-dimensional ℜ-module and its universal property. We additionally give the necessary and sufficient conditions for A, B, C to be diagonalizable on finite-dimensional irreducible ℜ-modules.