Invariant functions on supermatrices

TL;DR

This paper characterizes invariant functions under supergroup actions on supermanifolds related to supermatrices, extending classical invariant theory to the superalgebra setting.

Contribution

It provides a description of invariant functions for supergroups GQ(n) and GL(n|n) on associated supermanifolds, generalizing classical invariants to the super context.

Findings

Invariant functions characterized for GQ(n; C) and GL(n|C) actions

Results interpreted in terms of classical invariants on supermanifolds

Invariant functions described functorially, independent of superalgebra C

Abstract

There are two superanalogs of the general linear group: GL(m|n) and GQ(n). For any supercommutative superalgebra C let G(C) be the set of C-points of the supermanifold G. Here there are described the GQ(n; C)-invariant functions on Q(n; C) and GL(n; C)-invariant functions on the odd subspace of Mat (n|n; C), more exactly, the invariants of the action of Lie supergroups GQ(n) and GL(n|n) on the supermanifolds corresponding to Q(n; C) and the odd subspace of Mat(n|n; C). The obtained answer is interpreted in terms of GL(n; C)-invariants on Q(n; C) and GL(n|n; C)-invariants on the odd subspace of Mat (n|n; C) described functorially in (i.e., independently of) C.