Lie structure of truncated symmetric Poisson algebras

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) = circle plus U-infinity(n=0)n/Un-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 s(L) = S(L)/(x(p) vertical bar x is an element of L), which we call a truncated symmetric Poisson algebra. We study Lie identical relations of s(L). Namely, we determine necessary and sufficient conditions for L under which 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 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}} equivalent to 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 s(L) and S(L) in case char K = 2 is different from other characteristics, namely, we construct