The $G$-graded identities of the Grassmann Algebra

Centrone, Lucio. "The $G$-graded identities of the Grassmann Algebra." Archivum Mathematicum 052.3 (2016): 141-158. <http://eudml.org/doc/286707>.

@article{Centrone2016,
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 $g\ne 1_G$.},
author = {Centrone, Lucio},
journal = {Archivum Mathematicum},
keywords = {graded polynomial identities},
language = {eng},
number = {3},
pages = {141-158},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {The $G$-graded identities of the Grassmann Algebra},
url = {http://eudml.org/doc/286707},
volume = {052},
year = {2016},
}

TY - JOUR
AU - Centrone, Lucio
TI - The $G$-graded identities of the Grassmann Algebra
JO - Archivum Mathematicum
PY - 2016
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 052
IS - 3
SP - 141
EP - 158
AB - 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 $g\ne 1_G$.
LA - eng
KW - graded polynomial identities
UR - http://eudml.org/doc/286707
ER -