Coordinate Calculi on Associative Algebras

TL;DR

This paper introduces a new framework for coordinate differentials on associative algebras, explores quadratic identities, and constructs quantum de Rham complexes, linking calculus structures with Yang--Baxter equation generalizations.

Contribution

It proposes a novel notion of optimal algebras for coordinate differentials and develops a canonical quantum de Rham complex, connecting algebraic calculus with quantum group theory.

Findings

Introduction of optimal algebras for coordinate differentials

Study of quadratic identities in these algebras

Construction of quantum de Rham complexes

Abstract

A new notion of an optimal algebra for a first order coordinate differential was introduced in \cite{BKO}. Some relevant examples are indicated. Quadratic identities in the optimal algebras and calculi on quadratic algebras are studied. Canonical construction of a quantum de Rham complex for the coordinate differential is proposed. The relations between calculi and various generalizations of the Yang--Baxter equation are established.