Equivariant map superalgebras

Abstract

Suppose a group $\Gamma$ acts on a scheme $X$ and a Lie superalgebra $\mathfrak{g}$. The corresponding equivariant map superalgebra is the Lie superalgebra of equivariant regular maps from $X$ to $\mathfrak{g}$. We classify the irreducible finite dimensional modules for these superalgebras under the assumptions that the coordinate ring of $X$ is finitely generated, $\Gamma$ is finite abelian and acts freely on the rational points of $X$, and $\mathfrak{g}$ is a basic classical Lie superalgebra (or $\mathfrak{sl}(n,n)$, $n>0$, if $\Gamma$ is trivial). We show that they are all (tensor products of) generalized evaluation modules and are parameterized by a certain set of equivariant finitely supported maps defined on $X$. Furthermore, in the case that the even part of $\mathfrak{g}$ is semisimple, we show that all such modules are in fact (tensor products of) evaluation modules. On the other hand, if the even part of $\mathfrak{g}$ is not semisimple (more generally, if $\mathfrak{g}$ is of type I), we introduce a natural generalization of Kac modules and show that all irreducible finite dimensional modules are quotients of these. As a special case, our results give the first classification of the irreducible finite dimensional modules for twisted loop superalgebras.

27 pages. v2: Section numbering changed to match published version. Other minor corrections. v3: Minor corrections (see change log at end of introduction)

References in corpus (3)

• Irreducible finite-dimensional representations of equivariant map algebras
• Local Weyl modules for equivariant map algebras with free abelian group actions
• Classification of irreducible quasifinite modules over map Virasoro algebras

Cited by in corpus (6)

• Invariant polynomials on truncated multicurrent algebras
• Weyl modules for Lie superalgebras
• Equivariant map queer Lie superalgebras
• Irreducible modules for equivariant map superalgebras and their extensions
• Weyl modules and Weyl functors for Lie superalgebras
• Generalized Casimir Operators for Lie Superalgebras