The $G$-graded identities of the Grassmann Algebra

Centrone, Lucio

Archivum mathematicum, Tome 52 (2016) no. 3, pp. 141-158.

Résumé

Let $G$ be a finite abelian group with identity element $1_G$ and $L=\underset{g\in G}{⨁}{L}^g$ 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 $|G^{\prime }|\le |G|$, ${\dim }_F{L}^{1_{G^{\prime }}}=\mathrm{\infty }$ and ${\dim }_F{L}^{g^{\prime }}\mathrm{\infty }$ 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 ${\dim }_F{L}^g=\mathrm{\infty }$, where $o(g)=2$, and all the other components are finite dimensional. We also compute graded cocharacters and codimensions of $E$ in the case $\dim L^{1_G}=\mathrm{\infty }$ and $\dim L^g\mathrm{\infty }$ if $g\ne 1_G$.

Détail

DOI : 10.5817/AM2016-3-141

Classification : 16P90, 16R10, 16S10, 16W50
Mots-clés : graded polynomial identities