Moyal quantization and stable homology of necklace Lie algebras

We compute the stable homology of necklace Lie algebras associated with quivers and give a construction of stable homology classes from certain A∞-categories. Our construction is a generalization of the construction of homology classes of moduli spaces of curves due to M. Kontsevich. In the second part of the paper we produce a Moyal-type quantization of the symmetric algebra of a necklace Lie algebra. The resulting quantized algebra has natural representations in the usual Moyal quantization of polynomial algebras. This paper consists of two parts, essentially independent of each other. The results of the first part, to be outlined in Sect. 1, are concerned with stable homology of necklace Lie algebras. In the second part, we will be concerned with a natural quantization of necklace Lie algebras. The results of this part are outlined in Sect. 2 below. 1 Stable homology of necklace Lie algebras. 1.1 The graph complex and Lie algebra homology. In [Kon93], Kontsevich gives a construction for the stable homology of Lie algebras associated to commutative, Lie, and associative operads (generalized to arbitrary operads in [CV03]). A key idea of Kontsevich was to interpret the chain complex involved in the computation of Lie algebra homology in question as a certain graph complex. Furthermore, in the associative and Lie cases, Kontsevich related the homology of the graph complex with the cohomology of the coarse moduli space of smooth algebraic curves of genus g with n punctures, and the space of outer automorphisms of a free group with n punctures, respectively. The Lie algebra corresponding to the associative operad is defined in [Kon93] as follows. Throughout, fix a field k of characteristic zero. For any n, let Pn be the free associative (noncommutative) k-algebra with generators x1, x2, . . . , xn, y1, . . . , yn. Let Ln be the sub-Lie algebra of derivations of Pn which kill the element ∑n i=1[xi, yi] ∈ Pn, which can be interpreted as “Hamiltonian vector fields”, and let Ln,+ ⊂ Ln be the subspace spanned by derivations of nonnegative degree (where a derivation has degree d if it sends homogeneous polynomials