Quiver Representations in the Super-Category and Gabriel's Theorem for A(m,n)

TL;DR

This paper extends Gabriel's Theorem and quiver representation theory to Lie superalgebras, introducing super-representations and reflection functors to construct root systems categorically and combinatorially.

Contribution

It introduces super-representations and reflection functors for quivers, establishing a categorical and combinatorial framework for the root system A(n,m) in Lie superalgebras.

Findings

Established a super-category version of Gabriel's Theorem.

Constructed the root system A(n,m) categorically.

Provided a combinatorial model for roots using a quiver mmahat.

Abstract

Gabriel's Theorem, and the work of Bernstein, Gelfand and Ponomarev established a connection between the theory of quiver representations and the theory of simple Lie algebras. Lie superalgebras have been studied from many perspectives, and many results about Lie algebras have analogues for Lie superalgebras. In this paper, the notion of a super-representation of a quiver is introduced, as well as the notion of reflection functors for odd roots. These ideas are then used to give a categorical construction of the root system A(n,m) by establishing a version of Gabriel's Theorem and modifying the Bernstein, Gelfand, Ponomarev construction to the super-category. This is then used to give a combinatorial construction of the root system A(n,m) where roots correspond to vertices of a canonically defined quiver $\Gammahat$.