Sobre as subvariedades das variedades de crescimento quase polinomial

Abstract

Let F be a field of characteristic zero and let A be a F-algebra. In 1972, Regev proved that if A satisfies a nonzero polynomial identity then the sequence of condimensions fcn(A)gn1 is exponentially bounded. In 1978, Kemer proved that the sequence fcn(A)gn1 is polynomially bounded if and only if Id(A) contains at least one polynomial which is not an identity of G and of UT2 , i.e. cn(A) knt if and only if G; UT2 =2 var(A), where G denotes the innite dimensional Grassmann algebra and UT2 denotes the algebra of 2 2 upper triangular matrices. As a onsequence,var(G) and var(UT2) are the only two varieties of almost polynomial growth, i.e., the sequences of codimensions of those varieties grow exponentially but any proper subvariety of them has polynomial growth. The main goal of this work is to present La Mattina's results [18] which classify, up to PI-equivalence, all algebras contained in the variety generated by the algebra G or UT2.