Representations of Quantum Affine Superalgebras

Abstract

We study the quantum affine superalgebra $U_q(Lsl(M,N))$ and its finite-dimensional representations. We prove a triangular decomposition and establish a system of Poincaré-Birkhoff-Witt generators for this superalgebra, both in terms of Drinfel'd currents. We define the Weyl modules in the spirit of Chari-Pressley and prove that these Weyl modules are always finite-dimensional and non-zero. In consequence, we obtain a highest weight classification of finite-dimensional simple representations when $M\ne N$. Some concrete simple representations are constructed via evaluation morphisms.

published version with minor corrections

References in corpus (4)

• Quantum Deformations of the One-Dimensional Hubbard Model
• Baxter's Q-operators for supersymmetric spin chains
• Two-parameter Quantum Affine Algebra $U_{r,s}(\widehat{{\mathfrak{sl}}_n})$, Drinfel'd Realization and Quantum Affine Lyndon Basis
• Drinfeld second realization of the quantum affine superalgebras of $D^{(1)}(2,1;x)$ via the Weyl groupoid

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• $R$ matrix for generalized quantum group of type $A$
• Shuffle algebra realization of quantum affine superalgebra $U_v(\hat{\mathfrak{D}}(2,1;\theta ))$