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2022•Communications in AlgebraRequires access

Poincaré–Birkhoff–Witt theorem for Poisson enveloping algebra

Y. Van Huynh, Sei‐Qwon Oh, Hanna Sim

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Abstract

Let A be a Poisson polynomial algebra and let I be a Poisson ideal of A. It is established that the Poisson algebra B:=A/I satisfies the Poincaré-Birkhoff-Witt theorem. That is, the symmetric algebra Sym(ΩB) of the Kähler differential ΩB of B is isomorphic as a graded B-algebra to the associated graded algebra Be with respect to a canonical filtration of Be, where Be is the Poisson enveloping algebra of B.

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Let A be a Poisson polynomial algebra and let I be a Poisson ideal of A. It is established that the Poisson algebra B:=A/I satisfies the Poincaré-Birkhoff-Witt theorem. That is, the symmetric algebra Sym(ΩB) of the Kähler differential ΩB of B is isomorphic as a graded B-algebra to the associated graded algebra Be with respect to a canonical filtration of Be, where Be is the Poisson enveloping algebra of B.

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Available abstract

Let A be a Poisson polynomial algebra and let I be a Poisson ideal of A. It is established that the Poisson algebra B:=A/I satisfies the Poincaré-Birkhoff-Witt theorem. That is, the symmetric algebra Sym(ΩB) of the Kähler differential ΩB of B is isomorphic as a graded B-algebra to the associated graded algebra Be with respect to a canonical filtration of Be, where Be is the Poisson enveloping algebra of B.

Key concepts: Poisson algebra, Mathematics, Filtered algebra, Symmetric algebra, Differential graded algebra, Witt algebra, Universal enveloping algebra, Algebra over a field

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