Lie structure of truncated symmetric Poisson algebras

JOURNAL OF ALGEBRA

卷:488

Article

Alves, Ilana Z. Monteiro¹  Petrogradsky, Victor² 

[1] Univ Fed Amazonas, Dept Math, Humaita, Amazonas, Brazil

[2] Univ Brasilia, Dept Math, BR-70910900 Brasilia, DF, Brazil

关键词: Poisson algebras;    Identical relations;    Solvable Lie algebras;    Nilpotent Lie algebras;    Symmetric algebras;    Truncated symmetric algebras;    Restricted Lie algebras;   

DOI  :  10.1016/j.jalgebra.2017.05.035

来源: Elsevier

【 摘 要 】

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

【 授权许可】

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