Quantum Representation Theory and Manin matrices I: finite-dimensional case

TL;DR

This paper develops Quantum Representation Theory within non-commutative linear geometry, generalizing classical concepts to quantum analogues using Manin matrices, and provides foundational examples in the finite-dimensional case.

Contribution

It introduces a new framework for quantum representations based on Manin matrices and extends the internal hom-functor to a parameterized adjunction in symmetric monoidal categories.

Findings

Constructed quantum analogues of direct sum and tensor product of representations.

Formulated the theory using Manin matrices.

Provided examples of quantum representations.

Abstract

We construct Quantum Representation Theory which describes quantum analogue of representations in frame of "non-commutative linear geometry" developed by Manin. To do it we generalise the internal hom-functor to the case of adjunction with a parameter and construct a general approach to representations of a monoid in a symmetric monoidal category with a parameter subcategory. Quantum Representation Theory is obtained by application of this approach to a monoidal category of some class of graded algebras with Manin product, where the parameter subcategory consists of connected finitely generated quadratic algebras. We formulate this theory in the language of Manin matrices and obtain quantum analogues of direct sum and tensor product of representations. Finally, we give some examples of quantum representations.