Lie structure of truncated symmetric Poisson algebras

Abstract

The paper naturally continues series of works on identical relations of group rings, enveloping algebras, and other related algebraic structures. Let $L$ be a Lie algebra over a field of characteristic $p>0$. Consider its symmetric algebra $S(L)=\oplus_{n=0}^\infty U_n/U_{n-1}$, which is isomorphic to a polynomial ring. It also has a structure of a Poisson algebra, where the Lie product is traditionally denoted by $\{\ ,\ \}$. This bracket naturally induces the structure of a Poisson algebra on the ring $\mathbf{s}(L)=S(L)/(x^p\,|\, x\in L)$, which we call a truncated symmetric Poisson algebra. We study Lie identical relations of $\mathbf{s}(L)$. Namely, we determine necessary and sufficient conditions for $L$ under which $\mathbf{s}(L)$ is Lie nilpotent, strongly Lie nilpotent, solvable and strongly solvable, where we assume that $p>2$ to specify the solvability. We compute the strong Lie nilpotency class of $\mathbf{s}(L)$. Also, we prove that the Lie nilpotency class coincides with the strong Lie nilpotency class in case $p>3$. Shestakov proved that the symmetric algebra $S(L)$ of an arbitrary Lie algebra $L$ satisfies the identity $\{x,\{y,z\}\}\equiv 0$ if, and only if, $L$ is abelian. We extend this result for the (strong) Lie nilpotency and the (strong) solvability of $S(L)$. We show that the solvability of $\mathbf{s}(L)$ and $S(L)$ in case $\mathrm{char} K=2$ is different to other characteristics, namely, we construct examples of such algebras which are solvable but not strongly solvable. We use delta-sets for Lie algebras and the theory of identical relations of Poisson algebras. Also, we study filtrations in Poisson algebras and prove results on products of terms of the lower central series for Poisson algebras.