New quiver-like varieties and Lie superalgebras

TL;DR

This paper introduces new geometric varieties related to Lie superalgebras to extend the geometrization of Yangian R-matrices from classical Lie algebras, providing new tools for superalgebra representation theory.

Contribution

It develops superquiver varieties and super stable envelopes, generalizing existing constructions to the superalgebra setting, and connects them with Yangian R-matrices.

Findings

Defined superquiver-like varieties for $gl(M|N)$

Constructed super stable envelopes and weight functions

Demonstrated transformation properties under Yangian R-matrices

Abstract

In order to extend the geometrization of Yangian $R$-matrices from Lie algebras $gl(n)$ to superalgebras $gl(M|N)$, we introduce new quiver-related varieties which are associated with representations of $gl(M|N)$. In order to define them similarly to the Nakajima-Cherkis varieties, we reformulate the construction of the latter by replacing the Hamiltonian reduction with the intersection of generalized Lagrangian subvarieties in the cotangent bundles of Lie algebras sitting at the vertices of the quiver. The new varieties come from replacing some Lagrangian subvarieties with their Legendre transforms. We present superalgerba versions of stable envelopes for the new quiver-like varieties that generalize the cotangent bundle of a Grassmannian. We define superalgebra generalizations of the Tarasov-Varchenko weight functions, and show that they represent the super stable envelopes. Both super stable envelopes and super weight functions transform according to Yangian $\check{R}$-matrices of $gl(M|N)$ with $M+N=2$.