Constructions of Dgbv Algebras from Lie Algebras

We give some constructions of diierential Gerstenhaber-Batalin-Vilkovisky algebras from a class of Lie algebras. In our construction, we make use of the solutions to the classical Yang-Baxter equations, and ideas from Poisson geometry. A graded commutative algebra (A; ^) with a bracket ] of degree?1 is called a G-algebra (Gerstenhaber algebra) if: (a) (A1]; ]) is a Lie algebra, where A1] is A with grading shifted by 1; (b) For any homogeneous element a 2 A, a ] is a derivation of degree jaj + 1. A G-algebra is called a GBV algebra (Gerstenhaber-Batalin-Vilkovisky algebra) if there is a linear map of degree ?1, such that 2 = 0, and a b] = (?1) jaj (((a ^ b) ? ((a) ^ b ? (?1) jaj a ^ ((b)); for homogeneous a; b 2 A. A GBV-algebra is called a dGBV algebra if there is a derivation of degree 1, such that 2 = 0, and + = 0. G-algebras and GBV algebras have appeared in many places in Mathematics and physics, e.g. algebraic structures of cohomology and deformation theory 5, 6], quantization of guage theory ?], vertex operator theory 9, 7], Poisson geometry 14]. Very recently, dGBV algebras play an important role in some constructions in mirror symmetry 1, 10, 2, 3]. All these are preceded by 11] (see also 12]). The purpose of this paper to give more examples dGBV algebras. Our rst construction deals with (V)S(V), for any nite dimensional vector space V. We nd a structure of GBV algebra on it with a GBV subalgebra k k (V) S k (V). When V is a Lie algebra g with some additional requirement, (g)S(g) is a dGBV algebra with the Cartan-Chevalley-Eilenberg diierential 4]. Our second class of examples of dGBV algebras are (g) and (g), for nite dimensional Lie algebra. It is well-known that they are G-algebras (see e.g. 9], Appendix B). It may also be known to the experts that they GBV algebras, but the authors cannot locate a proof in the literature. Anyway, it is proved in this paper that for a class of Lie algebras, a construction similar to that in 11] and 12] provides a GBV algebra structure on (V). This point of view seems to be new. Furthermore, a solution to the classical Yang-Baxter equation (CYBE) with some additional properties provides a diierential on (g), which makes it a dGBV algebra. A solution …