Gradings, graded identities, $*$-identities and graded $*$-identities of an algebra of upper triangular matrices
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917833743532032 |
|---|---|
| author | Parada, Jonatan Andres Gomez Koshlukov, Plamen |
| author_facet | Parada, Jonatan Andres Gomez Koshlukov, Plamen |
| contents | Let $K \langle X\rangle$ be the free associative algebra freely generated over the field $K$ by the countable set $X = \{x_1, x_2, \ldots\}$. If $A$ is an associative $K$-algebra, we say that a polynomial $f(x_1,\ldots, x_n) \in K \langle X\rangle$ is a polynomial identity, or simply an identity in $A$ if $f(a_1,\ldots, a_n) = 0$ for every $a_1, \ldots, a_n \in A$.
Consider $\mathcal{A}$ the subalgebra of $UT_3(K)$ given by: \[ \mathcal{A} = K(e_{1,1} + e_{3,3}) \oplus Ke_{2,2} \oplus Ke_{2,3} \oplus Ke_{3,2} \oplus Ke_{1,3} , \] where $e_{i,j}$ denote the matrix units. We investigate the gradings on the algebra $\mathcal{A}$, determined by an abelian group, and prove that these gradings are elementary. Furthermore, we compute a basis for the $\mathbb{Z}_2$-graded identities of $\mathcal{A}$, and also for the $\mathbb{Z}_2$-graded identities with graded involution. Moreover, we describe the cocharacters of this algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06964 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gradings, graded identities, $*$-identities and graded $*$-identities of an algebra of upper triangular matrices Parada, Jonatan Andres Gomez Koshlukov, Plamen Rings and Algebras 16R10, 16W10, 16R50, 16W50 Let $K \langle X\rangle$ be the free associative algebra freely generated over the field $K$ by the countable set $X = \{x_1, x_2, \ldots\}$. If $A$ is an associative $K$-algebra, we say that a polynomial $f(x_1,\ldots, x_n) \in K \langle X\rangle$ is a polynomial identity, or simply an identity in $A$ if $f(a_1,\ldots, a_n) = 0$ for every $a_1, \ldots, a_n \in A$. Consider $\mathcal{A}$ the subalgebra of $UT_3(K)$ given by: \[ \mathcal{A} = K(e_{1,1} + e_{3,3}) \oplus Ke_{2,2} \oplus Ke_{2,3} \oplus Ke_{3,2} \oplus Ke_{1,3} , \] where $e_{i,j}$ denote the matrix units. We investigate the gradings on the algebra $\mathcal{A}$, determined by an abelian group, and prove that these gradings are elementary. Furthermore, we compute a basis for the $\mathbb{Z}_2$-graded identities of $\mathcal{A}$, and also for the $\mathbb{Z}_2$-graded identities with graded involution. Moreover, we describe the cocharacters of this algebra. |
| title | Gradings, graded identities, $*$-identities and graded $*$-identities of an algebra of upper triangular matrices |
| topic | Rings and Algebras 16R10, 16W10, 16R50, 16W50 |
| url | https://arxiv.org/abs/2411.06964 |