Non-commutative Poisson algebra structures on affine Kac-Moody algebras

Abstract

Non-commutative Poisson algebras are the algebras having an associative algebra structure and a Lie structure together with the Leibniz law. The non-commutative Poisson algebra structures on the infinite-dimensional algebras are studied. We show that these structures are standard on the poset subalgebras of the associative algebra of all endomorphisms of the countable-dimensional vector space. These structures on Kac-Moody algebras of affine type are determined. It is shown that the associative products on the derived Lie ideals are trivial, and the associative product action of the scaling elements are fully described.