标题： 多项式代数上的对称双导子结构以及对称矩阵的特殊Jordan代数 $H_n(K)$ 的一类模 显示英文标题

标题： A symmetric biderivation structure on polynomial algebras and a class of modules over the special Jordan algebra $H_n(K)$ of symmetric matrices

摘要： 在多项式代数$\mathscr{A}[n] = K[x_1,\dots,x_n],$上存在一个双导子结构，其中$K$是一个域，具有$\operatorname{char}(K)\ne 2$，由$f \circ h = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\,\frac{\partial h}{\partial x_i}.$定义。设$\mathscr{A}_k[n]$表示次数为$k$的齐次多项式的子空间。 然后 $(\mathscr{A}_2[n],\circ)$是一个若尔当代数，与特殊若尔当代数 $H_n(K)$同构，该代数由 $n\times n$对称矩阵组成。 每个 $\mathscr{A}_k[n]$是一个自然的 $\mathscr{A}_2[n]$双模，相对于一组完全互为正交的幂等元具有权空间分解。 特别是， $\mathscr{A}_2[n]$的权空间分解与其佩尔西分解一致。 $\mathscr{A}_k[n]$是一个 Jordan 双模当且仅当$k=0,1,2$。 等价地，对于所有$k\ge 3$， $\mathscr{A}_k[n]$不是一个 Jordan 双模。 保持每个齐次分量的$(\mathscr{A}[n],\cdot,\circ)$的代数自同构群 $\mathscr{A}_k[n]$同构于正交群$O(n,K)$。 如果$\operatorname{char}(K)=0$，那么代数$(\mathscr{A}[n],\cdot,\circ)$是单的，即它没有非零真理想。 此外，在这种情况下，每个$\mathscr{A}_k[n]$是一个简单的$\mathscr{A}_2[n]$-双模。 显示英文摘要

摘要： 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.