Universal Poisson Envelope for Binary-Lie Algebras

Abstract

In this article the universal Poisson enveloping algebra for a binary-Lie algebra is constructed. Taking a basis 𝔹 of a binary-Lie algebra B, we consider the symmetric algebra S(B) of polynomials in the elements of 𝔹. We consider two products in S(B), the usual product of polynomials fg and the braces {f, g}, defined by the product in B and the Leibniz rule. This algebra is a general Poisson algebra. We find an ideal I of S(B) such that the factor algebra S(B)/I is the universal Poisson envelope of B. We provide some examples of this construction for known binary-Lie algebras.

Key Words:

• Binary-Lie algebra
• Poisson algebra
• Universal Poisson envelope

2010 Mathematic Subject Classification:

• 17B63
• 17B35

ACKNOWLEDGMENTS

The first author was supported by Fondecyt proyect 11100092.

The second author was supported by Fondecyt proyect 1100127.

Notes

Communicated by I. Shestakov.