Involuções da álgebra de Grassmann: um estudo sobre *- polinômios centrais, *-identidades e *-isomorfismos

Abstract

Let K be a field of characteristic distinct from two, E be the Grassmann algebra of a K-vector space with infinite and countable base L and let φ be any involution. Based on the article (CENTRONE; GONCALVES; SILVA, 2020), we present generating sets for the ∗-polynomial identities and ∗-central polynomials of the Grassmann algebra, with a strong distinction between the cases char(K) = 0 and char(K) > 2. According to (DINIZ; GUIMARÃES; ROCHA, 2024), when φ satisfies certain conditions, we show that there is homogeneous involution φl (i.e., φl(L) = L) such that (E, φ) and (E, φl) are ∗-isomorphic. Additionally, we present certain necessary and sufficient condition for two homogeneous involutions to produce ∗-isomorphic structures. As a consequence, we obtain examples of algebras that are ∗-PI-equivalent and that are not ∗-isomorphic algebras.