Manin Matrices for Quadratic Algebras

TL;DR

This paper introduces a broad framework for Manin matrices associated with quadratic algebras, exploring their properties, categorical interpretations, and applications to Lax operators and classical Lie algebra types.

Contribution

It generalizes the concept of Manin matrices to arbitrary quadratic algebras, providing new insights into their structure, minors, and relations with algebraic objects like Brauer algebras.

Findings

Defined Manin matrices for arbitrary quadratic algebras.

Established properties and categorical interpretations.

Connected Manin matrices with Lax operators and classical Lie types.

Abstract

We give a general definition of Manin matrices for arbitrary quadratic algebras in terms of idempotents. We establish their main properties and give their interpretation in terms of the category theory. The notion of minors is generalised for a general Manin matrix. We give some examples of Manin matrices, their relations with Lax operators and obtain the formulae for some minors. In particular, we consider Manin matrices of the types $B$, $C$ and $D$ introduced by A. Molev and their relation with Brauer algebras. Infinite-dimensional Manin matrices and their connection with Lax operators are also considered.