$\mathbb{Z}_2$-graded $*$-polynomial identities and cocharacteres for $M_{1,1}(E)$, $UT_{1,1}(E)$ and $UT_{(0,1,0)}(E)$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916476975316992 |
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| author | Parada, Jonatan Andres Gomez |
| author_facet | Parada, Jonatan Andres Gomez |
| contents | Let $K$ be a field of characteristic 0, and let $E$ be the infinite-dimensional Grassmann algebra over $K$. We consider $E$ as a $\mathbb{Z}_2$-graded algebra, where the grading is given by the vector subspaces $E_0$ and $E_1$, consisting of monomials of even and odd lengths, respectively. Thus, if $A = A_0 \oplus A_1$ is an associative $\mathbb{Z}_2$-graded algebra, we can consider the $\mathbb{Z}_2$-graded algebra $A \hat{\otimes} E = (A_0 \otimes E_0) \oplus (A_1 \otimes E_1)$. In case both $E$ and $A$ are endowed with superinvolutions, we can define a $\mathbb{Z}_2$-graded involution on $A \hat{\otimes} E$ induced by the respective superinvolutions. In this paper, we consider the $\mathbb{Z}_2$-graded matrix algebras $M_{1,1}(K)$, $UT_{1,1}(K)$, and $UT_{(0,1,0)}(K)$ endowed with superinvolutions. We shall provide a description of the polynomial identities and the cocharacter sequences of $M_{1,1}(K)\hat{\otimes} E$, $UT_{1,1}(K)\hat{\otimes} E$, and $UT_{(0,1,0)}(K)\hat{\otimes} E$, considering these resulting algebras as $\mathbb{Z}_2$-graded algebras with graded involution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06942 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathbb{Z}_2$-graded $*$-polynomial identities and cocharacteres for $M_{1,1}(E)$, $UT_{1,1}(E)$ and $UT_{(0,1,0)}(E)$ Parada, Jonatan Andres Gomez Rings and Algebras 16Rxx, 16Wxx Let $K$ be a field of characteristic 0, and let $E$ be the infinite-dimensional Grassmann algebra over $K$. We consider $E$ as a $\mathbb{Z}_2$-graded algebra, where the grading is given by the vector subspaces $E_0$ and $E_1$, consisting of monomials of even and odd lengths, respectively. Thus, if $A = A_0 \oplus A_1$ is an associative $\mathbb{Z}_2$-graded algebra, we can consider the $\mathbb{Z}_2$-graded algebra $A \hat{\otimes} E = (A_0 \otimes E_0) \oplus (A_1 \otimes E_1)$. In case both $E$ and $A$ are endowed with superinvolutions, we can define a $\mathbb{Z}_2$-graded involution on $A \hat{\otimes} E$ induced by the respective superinvolutions. In this paper, we consider the $\mathbb{Z}_2$-graded matrix algebras $M_{1,1}(K)$, $UT_{1,1}(K)$, and $UT_{(0,1,0)}(K)$ endowed with superinvolutions. We shall provide a description of the polynomial identities and the cocharacter sequences of $M_{1,1}(K)\hat{\otimes} E$, $UT_{1,1}(K)\hat{\otimes} E$, and $UT_{(0,1,0)}(K)\hat{\otimes} E$, considering these resulting algebras as $\mathbb{Z}_2$-graded algebras with graded involution. |
| title | $\mathbb{Z}_2$-graded $*$-polynomial identities and cocharacteres for $M_{1,1}(E)$, $UT_{1,1}(E)$ and $UT_{(0,1,0)}(E)$ |
| topic | Rings and Algebras 16Rxx, 16Wxx |
| url | https://arxiv.org/abs/2411.06942 |