Classification of Associative Algebras Satisfying Quadratic Polynomial Identities
Josimar da Silva Rocha *
Department of Mathematics, Universidade Tecnol´ogica Federal do Paran´a, Corn´elio Proc´opio Campus, Corn´elio Proc´opio, Paran´a, Brazil.
*Author to whom correspondence should be addressed.
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 X2 = 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 X2 = 0 over fields of characteristics different from 2 are nilpotent of index 3.
The results obtained were validated computationally using the GAP system.
Keywords: Associative algebra, Polynomial identity, Nilpotency index, Nagata- Higman theorem