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 Poincar\'{e}-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 $M \neq N$. Some concrete simple representations are constructed via evaluation morphisms.
Highlights
In this paper q ∈ C\{0} is not a root of unity and our ground field is always C
We study a quantized version of the enveloping algebra of the affine Lie superalgebra Lsl(M, N ), which we denote by Uq (Lsl(M, N ))
For the quantum affine superalgebra Uq (Lsl(M, N )) defined in terms of Drinfel’d generators, the coproduct structure is highly non-trivial, and we do not have the analogue of the automorphisms φs
Summary
In this paper q ∈ C\{0} is not a root of unity and our ground field is always C. For the quantum affine superalgebra Uq (Lsl(M, N )) defined in terms of Drinfel’d generators, the coproduct structure is highly non-trivial (its existence is not clear a priori), and we do not have the analogue of the automorphisms φs. To overcome such difficulties we propose the PBW argument in this paper, which is independent of coproduct structures. The three subalgebras above admit presentations as superalgebras With respect to this triangular decomposition, we can define the Verma modules M( ), which are parametrised by the linear characters on Uq0(Lsl(M, N )), and are isomorphic to Uq−(Lsl(M, N )) as vector superspaces. We include in the two appendixes the related calculations that are needed in the triangular decomposition and the coproduct formulae for some Drinfel’d currents
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