Raising/lowering maps and modules for the quantum affine algebra $U_q(sl_2hat)$
Abstract
Description
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 \to V and L:V \to V denote linear maps with the following properties: for 0 \leq i \leq d, R U_i \subseteq U_{i+1} and L U_i \subseteq U_{i-1} where U_{-1}=0, U_{d+1}=0; for 0 \leq i \leq d/2, the restrictions R^{d-2i}: U_i \to U_{d-i} and L^{d-2i}: U_{d-i} \to 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 \to V be the linear map such that (K-q^{2i-d}I)U_i=0 for 0 \leq i \leq d. We show that there exists a unique U_q(sl_2hat)-module structure on V such that each of R-e^{-}_1, L-e^{-}_0, K-K_0, and K^{-1}-K_1 vanish on V, where e^{-}_1, e^{-}_0, K_0, K_1 are Chevalley generators for U_q(sl_2hat). We determine which U_q(sl_2hat)-modules arise from our construction.
21 pages, to be published in Communications in Algebra, longer introduction added, some proofs shortened, and some notation changes
21 pages, to be published in Communications in Algebra, longer introduction added, some proofs shortened, and some notation changes