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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.25867 |
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| _version_ | 1866915524523327488 |
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| author | Yin, Yangjie Han, Gang |
| author_facet | Yin, Yangjie Han, Gang |
| contents | There exists a biderivation structure on the polynomial algebra $\mathscr{A}[n] = K[x_1,\dots,x_n],$ where $K$ is a field with $\operatorname{char}(K)\ne 2$, defined by $f \circ h = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\,\frac{\partial h}{\partial x_i}.$ Let $\mathscr{A}_k[n]$ denote the subspace of homogeneous polynomials of degree $k$. Then $(\mathscr{A}_2[n],\circ)$ is a Jordan algebra, isomorphic to the special Jordan algebra $H_n(K)$ of $n\times n$ symmetric matrices. Each $\mathscr{A}_k[n]$ is a natural $\mathscr{A}_2[n]$-bimodule, which admits a weight space decomposition with respect to a complete set of mutually orthogonal idempotents. In particular, the weight space decomposition of $\mathscr{A}_2[n]$ coincides with its Peirce decomposition. $\mathscr{A}_k[n]$ is a Jordan bimodule if and only if $k=0,1,2$. Equivalently, for all $k\ge 3$, $\mathscr{A}_k[n]$ is not a Jordan bimodule. The group of algebra automorphisms of $(\mathscr{A}[n],\cdot,\circ)$ that preserve each homogeneous component $\mathscr{A}_k[n]$ is isomorphic to the orthogonal group $O(n,K)$. If $\operatorname{char}(K)=0$, then the algebra $(\mathscr{A}[n],\cdot,\circ)$ is simple, i.e., it has no nonzero proper ideals. Moreover, in this case, each $\mathscr{A}_k[n]$ is a simple $\mathscr{A}_2[n]$-bimodule. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_25867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A symmetric biderivation structure on polynomial algebras and a class of modules over the special Jordan algebra $H_n(K)$ of symmetric matrices Yin, Yangjie Han, Gang Rings and Algebras There exists a biderivation structure on the polynomial algebra $\mathscr{A}[n] = K[x_1,\dots,x_n],$ where $K$ is a field with $\operatorname{char}(K)\ne 2$, defined by $f \circ h = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\,\frac{\partial h}{\partial x_i}.$ Let $\mathscr{A}_k[n]$ denote the subspace of homogeneous polynomials of degree $k$. Then $(\mathscr{A}_2[n],\circ)$ is a Jordan algebra, isomorphic to the special Jordan algebra $H_n(K)$ of $n\times n$ symmetric matrices. Each $\mathscr{A}_k[n]$ is a natural $\mathscr{A}_2[n]$-bimodule, which admits a weight space decomposition with respect to a complete set of mutually orthogonal idempotents. In particular, the weight space decomposition of $\mathscr{A}_2[n]$ coincides with its Peirce decomposition. $\mathscr{A}_k[n]$ is a Jordan bimodule if and only if $k=0,1,2$. Equivalently, for all $k\ge 3$, $\mathscr{A}_k[n]$ is not a Jordan bimodule. The group of algebra automorphisms of $(\mathscr{A}[n],\cdot,\circ)$ that preserve each homogeneous component $\mathscr{A}_k[n]$ is isomorphic to the orthogonal group $O(n,K)$. If $\operatorname{char}(K)=0$, then the algebra $(\mathscr{A}[n],\cdot,\circ)$ is simple, i.e., it has no nonzero proper ideals. Moreover, in this case, each $\mathscr{A}_k[n]$ is a simple $\mathscr{A}_2[n]$-bimodule. |
| title | A symmetric biderivation structure on polynomial algebras and a class of modules over the special Jordan algebra $H_n(K)$ of symmetric matrices |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2509.25867 |