Nilpotent representations of classical quantum groups at roots of unity

Abstract

Properly specializing the parameters in ``Schnizer modules'', for type A,B,C and D, we get its unique primitive vector. Then we show that the module generated by the primitive vector is an irreducible highest weight module of finite dimensional classical quantum groups at roots of unity.Comment: 29 pages, new section added and some proof revise

Inductive Construction of Nilpotent Modules of Quantum Groups at Roots of Unity

“…For any λ ∈ Z n l , there exists a unique (up to isomorphic) finite dimensional irreducible nilpotent U ε (g)-module L nil ε (λ) of type 1 with highest weight λ. Conversely, if L is a finite dimensional irreducible nilpotent U ε (g)-module of type 1, then there exists a λ ∈ Z n l such that L is isomorphic to L nil ε (λ). By the similar manner to the proof of Theorem 5.5(ii) in [8] or Theorem 4.10 in [1], we obtain the following proposition.…”

“…Then we can construct finite dimensional irreducible nilpotent U ε (g)-modules of type 1 with highest weight (0, · · · , 0, λ k , · · · , λ n ) as a submodule of a l (Nn−N k−1 ) -dimensional U ε (g)-module, where N n is the number of the positive roots of g and n is the rank of g. In particular, these results cover the ones of [1].…”

“…Nilpotent U ε (g)-modules of type 1 are same as U fin ε (g)-modules of type 1, where U fin ε (g) is the finite dimensional quantum algebra introduced in [6], [7] (see [1]). In general, finite dimensional irreducible U fin ε (g)-modules are divided into 2 n types according to {σ : Q −→ {±1}; homomorphism of group }.…”

“…Moreover, in [1], we discover that we can construct these modules by using the Schnizer modules introduced in [9] if g is type B n , C n or D n .…”

Evaluation representations of quantum affine algebras at roots of unity

“…showed that one can construct V nil ε (λ) as the subrepresentation of a maximal cyclic representation by specializing their parameters properly for type A. Similarly, in [1], we found that we can construct V nil ε (λ) as a submodule of a Schnizer module if g=A, B, C or D, and then we can construct V nil ε (λ) ± a as the submodule of evaluation of a Schnizer module. By using this fact, we can prove (2) (see §5 alternative proof of Proposition 5.11 (b)).…”