Weyl modules for equivariant map Lie superalgebras

Abstract

Equivariant map superalgebras are Lie superalgebras of algebraic maps from a scheme to a target finite dimensional Lie superalgebra that are equivariant with respect to the action of a (cyclic) group. In this paper, we extend the notion of Weyl modules, previously defined for the untwisted case, to the case of equivariant(twisted) map superalgebras. Consider $\mathbb{K}$ be an algebraically closed field of characteristic $0$. We define global Weyl modules, and Weyl functors for equivariant map Lie superalgebras $(\g\otimes A)^Γ$, where $\g$ is a basic classical Lie $\mathbb{K}$-superalgebra and $A$ is an associative commutative unital $\mathbb{K}$-algebra. Under certain assumption on the triangular decomposition of $\g$, we prove that global Weyl modules are universal objects in certain category. We introduce a commutative algebra ${\mathbf{A}}_{\hat{I}}{»}^{\hat{I}}“$ and further prove that global Weyl modules are finitely generated ${\mathbf{A}}_{\hat{I}}{»}^{\hat{I}}“$-modules when $A$ is finitely generated. Finally we define the local Weyl modules for $(\g\otimes A)^Γ$, where $\g$ is basic classical, using Weyl functors. We show that they are finite dimensional irrespective of the triangular decomposition of $\g^Γ$. Finally it has been shown that twisted local Weyl modules of $(\g\otimes A)^Γ$ are precisely the image of the untwisted local Weyl modules under the twisting functor ${\text{T}}_{\mathbf{x}}$.

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