Codimensões, cocaracteres, identidades e polinômios centrais Z$_2$-graduados da álgebra de Grassmann

Abstract

Let E be the in_nite-dimensional Grassmann algebra over a _eld F of characteristic zero and consider L the F-vector space spanned by all generators of E. Let 'l be an automorphism of E of order 2 such that L is an homogeneous subspace. In this work, we study the Z2-gradings (E; 'l) induced by the automorphisms 'l and we _nd their Z2-graded codimensions and cocharacter sequences, as well as their Z2-graded polynomial identities and their Z2-graded central polynomials. More precisely, we _nish the computation of the sequences of Z2-codimensions, by _nding its exact value for the unique open case left by Anisimov in 2001. Moreover, we use these sequences, as well as the graded cocharacters theory, in order to get the decomposition of the Sr _ Snr-cocharacters _r;nr(E; 'l) in irreducible characters, for any automorphism 'l. Finally, we _nd the generators of the ideal of the Z2-graded identities and we get a complet description of the graded central polynomials of the superalgebra (E; 'l). As a consequence we get the Z2-codimensions and the Z2-graded identities for a large number of superalgebras (E; ') induced by arbitrary automorphisms ' of E of order 2.