Quantum deformations of associative algebras and integrable systems

TL;DR

This paper explores quantum deformations of associative algebras, revealing their connection to integrable systems like the WDVV, Boussinesq, and KP equations through geometric and algebraic structures.

Contribution

It introduces a geometric framework linking quantum algebra deformations to integrable soliton equations, expanding understanding of algebraic structures in mathematical physics.

Findings

Quantum deformations governed by quantum central systems with geometric interpretation.

Isoassociative deformations described by the WDVV equation.

Weakly nonassociative deformations linked to integrable soliton equations like Boussinesq and KP.

Abstract

Quantum deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by the quantum central systems which has a geometrical meaning of vanishing Riemann curvature tensor for Christoffel symbols identified with the structure constants. A subclass of isoassociative quantum deformations is described by the oriented associativity equation and, in particular, by the WDVV equation. It is demonstrated that a wider class of weakly (non)associative quantum deformations is connected with the integrable soliton equations too. In particular, such deformations for the three-dimensional and infinite-dimensional algebras are described by the Boussinesq equation and KP hierarchy, respectively.