Grothendieck rings of basic classical Lie superalgebras

Abstract

The Grothendieck rings of finite dimensional representations of the basic classical Lie superalgebras are explicitly described in terms of the corresponding generalized root systems. We show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called super Weyl groupoids.

Bernstein–Gelfand–Gelfand Reciprocity and Indecomposable Projective Modules for Classical Algebraic Supergroups

“…Denote by Λ + the set of all weights λ such that λ,β is a positive integer for any simple Hence Ch is injective. For classical Lie superalgebras the image of Ch is described in [25].…”

Orbits and Invariants of Super Weyl Groupoid

“…As we will see, in general the answers to all these questions are negative. We will demonstrate this in the case of the so called super Weyl groupoid W n,m , introduced in [16] in relation with the Grothendieck ring of the Lie superalgebra gl(n, m). We will consider a special affine action Φ κ of this groupoid depending on a non-zero parameter κ, which arosecannot from the theory of the deformed quantum Calogero-Moser systems [14].…”

“…In this paper we will consider (a particular case) of the super Weyl groupoid [16] related to any basic classical Lie superalgebra g. The roots of g form a generalised root system R ⊂ V in Serganova's sense [12], which is a certain generalisation of the root system in the presence of the isotropic roots. For isotropic roots one cannot define the reflections, which leads to a well-known problem with defining Weyl group in this case.…”

“…As it was shown in [14] the algebra of invariants of this action P [V ] Φκ is closely related to the algebra of quantum integrals of the corresponding deformed Calogero-Moser systems. In fact, the notion of the super Weyl groupoid [16] was motivated by this relation and representation theory of classical Lie superalgebras: the Grothendieck ring K(g) of finite dimensional representations of a basic classical Lie superalgebra g is isomorphic to the ring of trigonometric invariants Z[P 0 ] Φκ , where κ = −1 and P 0 is the weight lattice of the even part of g (see the details in [16]).…”

“…The corresponding algebra of invariants P [V ] Φ 1 is known as the algebra of supersymmetric polynomials. It plays an important role in geometry [8] and in the representation theory of Lie superalgebra gl(n, m), see [13,16]. It coincides with Λ n,m,−1 and is generated by the (super) power sums…”