Representations of quantum affine superalgebras

Abstract

We study the quantum affine superalgebra U q (Lsl(M, N )) and its finitedimensional 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 = N . Some concrete simple representations are constructed via evaluation morphisms.

Integrable representations of the quantum affine special linear superalgebra

“…(2) ⇒ (3). This was established in [34,Theorem 4.5], which is a key result in the classification of finite dimensional simple U ′ q -modules. Note that f is a formal Laurent series in general.…”

“…In a very recent paper [34], Huafeng Zhang gave a classification of the finite dimensional simple modules for U q ( sl(M|N)) (more precisely the subalgebra U ′ q ( sl(M|N)) without the degree operator) at generic q, providing a parametrisation of such simple modules in terms of highest weight polynomials. This has much similarity to the classification [40] of finite dimensional simple modules for the gl(M|N) super Yangian, as explained in [34].…”

“…(3.10)). This way we recover all the finite dimensional simple U ′ q ( sl(M|N))-modules, which were classified in [34].…”

Fundamental Representations of Quantum Affine Superalgebras and R-Matrices

“…In this section M = N = 1 and g = gl(1, 1). We discuss Weyl modules over U q ( g), which were previously defined in [Zh1]. 4 Let R 0 be the set of rational functions f (z) ∈ C(z) which are products of the c 1−za…”

Weyl modules for Lie superalgebras

“…In the more general setting where the coordinate ring of X is finitely generated, g is a basic Lie superalgebra, and where we also consider maps equivariant with respect to a finite abelian group acting freely on the rational points of X, the irreducible finite-dimensional modules were classified in [Sav14]. However, Weyl modules have not been defined in the super setting, except for a quantum analogue in the loop case for g = sl(m, n) considered in [Zha14].…”