Raising/Lowering Maps and Modules for the Quantum Affine Algebra

Abstract

Let V denote a finite dimensional vector space over an algebraically closed field. Let U ₀, U ₁,…, U _d 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, R U _i  ⊆ U _i+1 and L U _i  ⊆ U _i−1 where U ₋₁ = 0, U _d+1 = 0; for 0 ≤ i ≤ d/2, the restrictions R ^d−2i | _{U i} : U _i  → U _d−i and L ^d−2i | _{U d−i} : U _d−i  → U _i 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−d I)U _i  = 0 for 0 ≤ i ≤ d. We show that there exists a unique -module structure on V such that each of , , K − K ₀, and K ⁻¹ − K ₁ vanish on V, where are Chevalley generators for . We determine which -modules arise from our construction.

Key Words:

• Affine Lie algebra∘₂;
• Quantum affine algebra
• Quantum group
• Raising and lowering maps
• Tridiagonal pair

2000 Mathematics Subject Classification:

• Primary 17B37
• Secondary 16W35
• 20G42
• 81R50

ACKNOWLEDGMENT

I would like to express my gratitude to my thesis advisor Paul Terwilliger for introducing me to this subject and for his many useful suggestions.

Notes

Communicated by K. C. Misra.