| Title:
|
The $G$-graded identities of the Grassmann Algebra (English) |
| Author:
|
Centrone, Lucio |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
52 |
| Issue:
|
3 |
| Year:
|
2016 |
| Pages:
|
141-158 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $g\ne 1_G$. (English) |
| Keyword:
|
graded polynomial identities |
| MSC:
|
16P90 |
| MSC:
|
16R10 |
| MSC:
|
16S10 |
| MSC:
|
16W50 |
| idZBL:
|
Zbl 06644064 |
| idMR:
|
MR3553173 |
| DOI:
|
10.5817/AM2016-3-141 |
| . |
| Date available:
|
2016-09-20T11:56:17Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145829 |
| . |
| Reference:
|
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| Reference:
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