# The $G$-graded identities of the Grassmann Algebra

## Lucio Centrone

Address: IMECC, Universidade Estadual de Campinas, Rua SÃ©rgio Buarque de Holanda 651, Campinas (SP), Brazil

E-mail: centrone@ime.unicamp.br

Abstract: Let $G$ be a finite abelian group with identity element $1_G$ and $L=\bigoplus _{g\in G}L^g$ be an infinite dimensional $G$-homogeneous vector space over a field of characteristic $0$. Let $E=E(L)$ be the Grassmann algebra generated by $L$. It follows that $E$ is a $G$-graded algebra. Let $|G|$ be odd, then we prove that in order to describe any ideal of $G$-graded identities of $E$ it is sufficient to deal with $G^{\prime }$-grading, where $|G^{\prime }| \le |G|$, $\dim _FL^{1_{G^{\prime }}}=\infty $ and $\dim _FL^{g^{\prime }}<\infty $ if $g^{\prime }\ne 1_{G^{\prime }}$. In the same spirit of the case $|G|$ odd, if $|G|$ is even it is sufficient to study only those $G$-gradings such that $\dim _FL^g=\infty $, where $o(g)=2$, and all the other components are finite dimensional. We also compute graded cocharacters and codimensions of $E$ in the case $\dim L^{1_G}=\infty $ and $\dim L^g<\infty $ if $g\ne 1_G$.

AMSclassification: primary 16R10; secondary 16P90, 16S10, 16W50.

Keywords: graded polynomial identities.

DOI: 10.5817/AM2016-3-141