Weyl modules for Equivariant map Lie superalgebras
Lakshmi S K, Saudamini Nayak
TL;DR
This work extends the Weyl-module framework to equivariant map Lie superalgebras $($\mathfrak{g}\otimes A)^{\Gamma}$ by introducing twisted global Weyl modules $W^{\Gamma}(\lambda)$ and Weyl functors. A compatible triangular decomposition satisfying condition $\mathbf{C}$ yields a universal highest-weight object in the twisted setting, and with finitely generated $A$ the global Weyl modules are finitely generated as modules over a commutative algebra $\mathbf{A}_{\lambda}^{\Gamma}$. The authors define the category $\mathcal{I}^{\Gamma}$ and develop a twisted global Weyl module theory, showing $W^{\Gamma}(\lambda)$ can be realized via a PBW-type construction and that the associated Weyl functors $\mathbf{W}^{\lambda}$ behave well with respect to $\mathbf{A}_{\lambda}^{\Gamma}$. The results unify and extend prior work on Weyl modules for map algebras to the equivariant superalgebra setting, providing a robust toolkit for studying highest-weight representations in twisted contexts. Overall, the paper establishes finite generation, universality, and functoriality for twisted Weyl modules, with potential implications for representation theory in mathematical physics and related areas.
Abstract
We define Weyl functors, global modules for equivariant map Lie superalgebras $(\g \otimes A)^Γ$, where $\g$ is basic classical $\mathbb{C}$- Lie superalgebra and $A$ is an associative commutative unital $\mathbb{C}$-algebra. Under certain condition on the triangular decomposition of $\g$ we prove that global Weyl modules are universal highest weight objects in certain category. Then with the assumption that $A$ is finitely generated, it is shown that the global Weyl modules are finitely generated.