On infinite dimensional algebras with regular gradings
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| Format: | Preprint |
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2025
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| _version_ | 1866912673748221952 |
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| author | Centrone, Lucio Koshlukov, Plamen Pereira, Kauê |
| author_facet | Centrone, Lucio Koshlukov, Plamen Pereira, Kauê |
| contents | Let $G$ be a finite abelian group and let $K$ be an algebraically closed field of characteristic 0. We consider associative unital algebras $A$ over $K$ graded by $G$, that is $A=\oplus_{g\in G} A_g$, where the vector subspaces $A_g$ satisfy $A_gA_h\subseteq A_{g+h}$ for every $g$, $h\in G$. Such a $G$-grading is called regular whenever for every $n$-tuple $(g_1,\ldots,g_n)\in G^n$ there exist homogeneous elements $a_i\in A_{g_i}$ such that $a_1\cdots a_n\ne 0$ in $A$; furthermore, for every $g$, $h\in G$ and every $a_g\in A_g$, $a_h\in A_h$ one has $a_ga_h=β(g,h)a_ha_g$ for some $β(g,h)\in K^*$. Here $β(g,h)$ depends only on the choice of $g$ and $h$ but not on the elements $a_g$ and $a_h$. It is immediate that $β$ is a bicharacter on $G$. The regular decomposition above is minimal if for every $g\in G$ with $β(g,h)=β(g,k)$ one has $h=k$. In this paper we prove that if $G=\mathbb{Z}_2$ then every $G$-graded regular algebra whose regular decomposition is minimal, contains a copy of the infinite dimensional Grassmann algebra. By applying this result we are able to describe the generating algebras of the variety of $\mathbb{Z}_2$-graded algebras defined by the Grassmann algebra. Furthermore we describe the finitely generated subalgebras of a $\mathbb{Z}_2$-graded regular algebra having a minimal regular decomposition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_23869 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On infinite dimensional algebras with regular gradings Centrone, Lucio Koshlukov, Plamen Pereira, Kauê Rings and Algebras 16R10, 16R50, 16W55, 16T05 Let $G$ be a finite abelian group and let $K$ be an algebraically closed field of characteristic 0. We consider associative unital algebras $A$ over $K$ graded by $G$, that is $A=\oplus_{g\in G} A_g$, where the vector subspaces $A_g$ satisfy $A_gA_h\subseteq A_{g+h}$ for every $g$, $h\in G$. Such a $G$-grading is called regular whenever for every $n$-tuple $(g_1,\ldots,g_n)\in G^n$ there exist homogeneous elements $a_i\in A_{g_i}$ such that $a_1\cdots a_n\ne 0$ in $A$; furthermore, for every $g$, $h\in G$ and every $a_g\in A_g$, $a_h\in A_h$ one has $a_ga_h=β(g,h)a_ha_g$ for some $β(g,h)\in K^*$. Here $β(g,h)$ depends only on the choice of $g$ and $h$ but not on the elements $a_g$ and $a_h$. It is immediate that $β$ is a bicharacter on $G$. The regular decomposition above is minimal if for every $g\in G$ with $β(g,h)=β(g,k)$ one has $h=k$. In this paper we prove that if $G=\mathbb{Z}_2$ then every $G$-graded regular algebra whose regular decomposition is minimal, contains a copy of the infinite dimensional Grassmann algebra. By applying this result we are able to describe the generating algebras of the variety of $\mathbb{Z}_2$-graded algebras defined by the Grassmann algebra. Furthermore we describe the finitely generated subalgebras of a $\mathbb{Z}_2$-graded regular algebra having a minimal regular decomposition. |
| title | On infinite dimensional algebras with regular gradings |
| topic | Rings and Algebras 16R10, 16R50, 16W55, 16T05 |
| url | https://arxiv.org/abs/2510.23869 |