Tridiagonal pairs and their use in representation theory

This thesis contains two major results in representation theory. The first major result is the following theorem about the quantum affine algebra Uq( sl&d14;2 ).;Theorem. Let V denote a finite dimensional vector space over an algebraically closed field. Let Uid i=0 denote a sequence of nonzero subspaces whose direct sum is V. Let R : V → V and L : V → V denote linear transformations with the following properties: for 0 ≤ i ≤ d, RUi ⊆ U i+1 and LUi ⊆ Ui-1 where U-1 = 0, Ud+1 = 0; for 0 ≤ i ≤ d/2, the restrictions Rd-2i&vbm0;Ui : Ui → Ud-i and Ld-2i&vbm0;Ud-1 : Ud-i → Ui are bijections; the maps R and L satisfy the cubic q-Serre relations where q is nonzero and not a root of unity. Let K : V → V denote the linear transformation such that (K - q 2i-dI) Ui = 0 for 0 ≤ i ≤ d. Then there exists a unique Uq( sl&d14;2 )-module structure on V such that each of R - e-1 , L - e-0 , K - K0, and K-1 - K1 vanish on V, where e-1,e-0 , K0, K1 are Chevalley generators for Uq( sl&d14;2 ).;We determine which Uq( sl&d14;2 )-modules arise from this construction. We also give a description of all finite dimensional Uq( sl&d14;2 )-modules in terms of the Uq( sl&d14;2 )-modules arising from this construction.;The second major result is the theorem below about the q-tetrahedron algebra. The q-tetrahedron algebra ⊠q is a generalization of Uq( sl&d14;2 ) which arose through the study of a linear algebraic object called a tridiagonal pair. The following theorem generalizes a result of Ito and Terwilliger relating finite dimensional ⊠q -modules to tridiagonal pairs.;Theorem. Let K denote an algebraically closed field and let q ∈ K be nonzero and not a root of unity. Let V denote a finite dimensional vector space over K and let A, A* denote a tridiagonal pair on V. Assume for 0 ≤ i ≤ d that q2i-d is a standard ordering of the eigenvalues of A and for 0 ≠ c ∈ K that q2i-d+ cqd-2i is a standard ordering of the eigenvalues of A*. Then the following are equivalent: (i) There exists a ⊠q -module structure on V such that x 01 acts as A and x30 + cx23 acts as A*, where x 01, x30, x23 are standard generators for ⊠q . (ii) P(q2 d-2 (q - q-1)-2) ≠ 0 where P is the Drinfeld polynomial associated to A, A*. Suppose (i),(ii) hold. Then the ⊠q -module structure on V is unique and irreducible.