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

TL;DR

We analyze the polynomial algebra $\mathscr{A}[n]=K[x_1,...,x_n]$ endowed with the symmetric biderivation $f\circ h=\nabla f\cdot\nabla h$, linking it to Jordan algebra theory. The subspace $\mathscr{A}_2[n]$ forms a Jordan algebra isomorphic to $H_n(K)$, and each $\mathscr{A}_k[n]$ is a natural $\mathscr{A}_2[n]$-bimodule, with a weight-space decomposition coming from an orthogonal-idempotent Cartan subalgebra; $\mathscr{A}_2[n]$ gives the Peirce decomposition and $\mathscr{A}_k[n]$ is a Jordan bimodule only for $k=0,1,2$ (char$(K)=0$ adds simplicity results). The automorphism subgroup preserving homogeneous components is $O(n,K)$, and the algebra is simple in characteristic zero, with all relevant bimodules simple. The paper thus provides a thorough structural description of $\mathscr{A}[n]$ as a bimodule over $\mathscr{A}_2[n]$ and of its automorphism group, connecting polynomial biderivations to classical Jordan algebra theory.

Abstract

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.

Theorems & Definitions (19)

• Definition 2.1
• Example 2.1
• Example 2.2
• Example 2.3
• Proposition 2.1
• proof
• Definition 3.1
• Theorem 3.1
• proof
• Proposition 3.2