A dimensão de Gelfand-Kirillov em característica positiva

Abstract

The verbally prime algebras are well understood in characteristic 0 while over a field of characteristic p > 2 little is known about them. In this work we discuss some sharp differences between these two cases for the characteristic. We discuss some properties of the álgebras Aa;b and Ma;b(E)⊗E in positive characteristic and we use these properties to compute the Gelfand-Kirillov dimension of the relatively free algebras of rank m in the variety generated by Ma;b(E)⊗E. We exhibit a construction of a generic model for the algebra Um(Mn(E)⊗E). By using these models we compute the Gelfand-Kirillov dimension of the relatively free álgebras of rank m in the variety generated by Mn(E)⊗E. As a consequence we obtain the PI non equivalence of important algebras for the PI theory in positive characteristic. Finally, we launch a conjecture about the Gelfand-Kirillov dimension of the universal algebras, concerning the tensor product of verbally prime algebras by Grassmann algebra.