On the Poisson envelope of a Lie algebra. ``Noncommutative'' moment space
In the present note for an arbitrary Lie superalgebra \({\mathfrak g}\) we construct a \(\mathbb{Z}\)-graded noncommutative Poisson algebra \(E({\mathfrak g})\) which is its ``Poisson envelope'' in a natural sense (Theorem 2). The brackets in the algebra \(E({\mathfrak g})\) do not coincide with the commutator. Both the universal enveloping algebra \(U({\mathfrak g})\) and the algebra \(S({\mathfrak g})\) of polynomial functions on the space \({\mathfrak g}^*\) are quotient algebras of \(E({\mathfrak g})\), and the brackets in \(E({\mathfrak g})\) induce both the commutator in \(U({\mathfrak g})\) and the Berezin-Kirillov brackets in \(S({\mathfrak g})\). The algebra \(E({\mathfrak g})\) is interpreted as a function algebra on a ``noncommutative'' moment space of the algebra \({\mathfrak g}\) or of the corresponding supergroup \(G\). There is a ``moment map'' for a Hamiltonian action of the (super)group \(G\) on a noncommutative space. We also construct the algebra \(E({\mathfrak g}\Pi)\), the Schouten envelope of the (super)algebra \({\mathfrak g}\), playing the same role for actions of \(G\) on spaces endowed with odd brackets (antibrackets or Schouten brackets). The corresponding commutative version is formed by ``Lie-Schouten'' brackets on \({\mathfrak g}^*\Pi\) which are similar to Berezin-Kirillov brackets, but seemingly have not been discussed earlier. A slight modification of the construction makes it possible to introduce the Poisson envelope for a Lie algebra \({\mathfrak g}\) over an associative algebra \(A\), or, more precisely, for a ``small Poisson algebra'' (Theorem 3). In particular, this provides the canonical Poisson and Schouten brackets on \(T^*M\) and \(T^* M\Pi\) and their analog in noncommutative geometry. Our constructions are based on a fundamental identity connecting the commutator and the brackets in an arbitrary Poisson algebra (Theorem 1).
scientific article; zbMATH DE number 898977