A Dixmier theorem for Poisson enveloping algebras
Umirbaev U. Zhelyabin V.
15 February 2021Academic Press Inc.
Journal of Algebra
2021#568576 - 600 pp.
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).
Casimir element , Malcev algebra , n-Lie algebra , Poisson algebra , Poisson enveloping algebra , Simple algebra
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Wayne State University, Detroit, 48202, MI, United States
Institute of Mathematics and Modeling, Almaty, Kazakhstan
Institute of Mathematics of the SB of RAS, Novosibirsk, 630090, Russian Federation
Wayne State University
Institute of Mathematics and Modeling
Institute of Mathematics of the SB of RAS
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