Quantum Representation Theory and Manin matrices II: super case

TL;DR

This paper develops a super-version of Quantum Representation Theory, introducing super-Manin matrices and quantum representations, and explores their categorical properties and relations to classical representations.

Contribution

It constructs the super-version of Quantum Representation Theory, including super-Manin matrices and monoidal functors, extending classical concepts to the super algebra setting.

Findings

Super-Manin matrices relate to quadratic super-algebras.

The monoidal category of graded super-algebras is coclosed.

Classical representations lift to the quantum super setting.

Abstract

We construct super-version of Quantum Representation Theory. The quadratic super-algebras and operations on them are described. We also describe some important monoidal functors. We proved that the monoidal category of graded super-algebras with Manin product is coclosed relative to the subcategory of finitely generated quadratic super-algebras. The super-version of the $(A,B)$-Manin matrices is introduced and related with the quadratic super-algebras. We define a super-version of quantum representations and of quantum linear actions, relate them to each other and describe them by the super-Manin matrices. Some operations on quantum representations/quantum linear actions are described. We show how the classical representations lift to the quantum level.