Formal (non)-commutative symplectic geometry. (English) Zbl 0821.58018

Corwin, L. (ed.) et al., The Gelfand Seminars, 1990-1992. Basel: Birkhäuser, 173-187 (1993).
The object of investigation in this paper are three infinite-dimensional Lie algebras. We denote by \(l_ n\) the Lie subalgebra of derivations of the free Lie algebra generated by the \(2n\) elements \(p_ 1, \ldots, p_ n\), \(q_ 1, \ldots, q_ n\) consisting of derivations acting trivially on the element \(\sum [p_ i, q_ i]\). Similarly we define the Lie algebra \(a_ n\) when taking the free associative algebra without unit generated by the elements \(p_ 1, \ldots, p_ n\), \(q_ 1, \ldots, q_ n\). The last algebra \(c_ n\) is the Lie algebra of polynomials \(F \in \mathbb Q [p_ 1, \ldots, p_ n, q_ 1, \ldots, q_ n]\) satisfying \(F(0) = F'(0) = 0\) with the usual Poisson bracket. For brevity we denote by \(h_ n\) any of these three algebras. The author computes the homology \(H_ *(h_ \infty)\) of the limit \(h_ \infty\). Because \(H_ *(h_ \infty)\) carries a Hopf algebra structure which is commutative and cocommutative, \(H_ *(h_ \infty)\) is a free polynomial algebra (in the \(\mathbb Z_ 2\)-graded sense) generated by the subspace of primitive elements. The description of these subspaces in the cases \(h_ \infty = l_ \infty, a_ \infty, c_ \infty\) represent the main result of the paper. As the author mentions, this investigation was the main motivation for his interest in noncommutative geometry. Namely, he interprets the algebras \(l_ n\) and \(a_ n\) as algebras of Hamiltonian vector fields on flat symplectic manifolds in noncommutative geometries, which enables him to achieve the above results. Moreover, the paper contains various further results, important remarks and conjectures. It can be strongly recommended.
For the entire collection see [Zbl 0780.00044].


58B34 Noncommutative geometry (à la Connes)
17B56 Cohomology of Lie (super)algebras
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
17B66 Lie algebras of vector fields and related (super) algebras