Classification of Associative Algebras Satisfying Quadratic Polynomial Identities

TL;DR

This paper classifies associative algebras satisfying quadratic polynomial identities, revealing structural properties and nilpotency conditions, with computational validation using GAP.

Contribution

It provides a classification of associative algebras satisfying specific quadratic polynomial identities, including nilpotency results, and employs computational methods for validation.

Findings

Algebras satisfying X^2=0 over fields of characteristic not 2 are nilpotent of index 3.

Classification results for algebras with quadratic polynomial identities.

Computational validation using GAP system.

Abstract

In quantum mechanics, associative algebras play an important role in understanding symmetries and operator algebras, providing new algebraic frameworks for describing physical systems. This work classifies associative algebras over a field K that are generated by a finite set G and satisfy a polynomial identity of the form X^{2} = aX+b, where a and b are elements of K and X varies either over all elements of the algebra or over all elements of the multiplicative semigroup S generated by G. One of the results obtained in this work shows that algebras satisfying X^{2}=0 over fields of characteristics different from 2 are nilpotent of index 3. The results were computationally validated using the GAP system.