Abstract: We consider a skew-symmetric n-ary bracket on the polynomial algebra K[x1, ..., xn, xn+1] (n ≥ 2) over a field K of characteristic zero defined by {a1, ..., an} = Jac(a1, ...,an, C), where C is a fixed element of K[x1, ..., xn, xn+1] and Jac is the Jacobian. If n = 2 then this bracket is a Poisson bracket and if n ≥ 3 then it is an n-Lie-Poisson bracket on K[x1, ..., xn, xn+1]. We describe the center of the corresponding n-Lie-Poisson algebra and show that the quotient algebra K[x1, ..., xn, xn+1]/(C − λ), where (C − λ) is the ideal generated by C − λ, 0 = λ ∈ K, is a simple central n-Lie-Poisson algebra if C is a homogeneous polynomial that is not a proper power of any nonzero polynomial. This construction includes the quotients P(sl2(K))/(C − λ) of the Poisson enveloping algebra P(sl2(K)) of the simple Lie algebra sl2(K), where C is the standard Casimir element of sl2(K) in P(sl2(K)). It is also proven that the quotients P(M)/(CM − λ) of the Poisson enveloping algebra P(M) of the exceptional simple seven dimensional Malcev algebra M are central simple, where CM is the standard Casimir element of M in P(M).

Cite: Umirbaev U. , Zhelyabin V.
A Dixmier theorem for Poisson enveloping algebras, Journal of algebra
Journal of Algebra. 2021. V.568. P.576-600. DOI: 10.1016/j.jalgebra.2020.11.001 WOS Scopus OpenAlex

Dates:

Submitted:	Mar 16, 2020
Accepted:	Nov 4, 2020
Published online:	Nov 4, 2020

Identifiers:

Web of science:	WOS:000594258900023
Scopus:	2-s2.0-85095439923
OpenAlex:	W3095922677

Citing:

DB	Citing
Scopus	2
OpenAlex	2
Web of science	2

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