Tsygan Formality and Duflo Formula

We prove the 0-(co)homology part of the conjecture on the cup-products on tangent cohomology in the Tsygan formality [Sh2]. We discuss its applications to the Duflo formula. A short introduction The Tsygan formality conjecture for chains [Ts] was proven in the author’s work [Sh2] by an explicit construction of suitable Kontsevich-type integrals. This paper is a further development of ideas of [Sh2]. We will freely use the notations and results of [Sh2]. In [Sh2] we formulated a conjecture on “the cup-products on tangent cohomology”, which is a version of the analogous Kontsevich theorem from Section 8 of [K]. Here we prove this conjecture for 0-(co)homology. 1. The classical Duflo formula and the generalized Duflo formula 1.1. Let g be a finite-dimensional Lie algebra, S(g) and U(g) be its symmetric and universal enveloping algebra. They are not isomorphic as algebras S(g) is a commutative algebra and U(g) is a non-commutative algebra. We can consider both spaces S(g) and U(g) as g-modules with the adjoint action for S(g) and the action g ·ω = g⊗ω− ω⊗ g for U(g) (here g ∈ g and ω ∈ U(g)). It is clear that these g-modules are isomorphic, the isomorphism is the classical Poincaré–Birkhoff–Witt map: φPBW (g1 · · · · · gk) = 1 k! ∑ σ∈Σk gσ(1) ⊗ · · · ⊗ gσ(k) (1) (g1, . . . , gk ∈ g). The Duflo theorem [D] states that the invariants [S(g)] and [U(g)] are isomorphic as algebras. The Duflo formula is a canonical formula for this isomorphism. We recall it here. For any k ≥ 1, there exists a canonical element in [S(g)]. It is the symmetrization of the map g 7→ Tr|g ad k g (g ∈ g). We denote this element in [S(g)] by Trk. (It was a conjecture of M. Duflo that these elements are zero for odd k and any finite-dimensional Lie algebra g; this conjecture was proven recently in [AB]). We can consider an element from S(g) as a differential operator of the k-th order with constant coefficients, acting on S(g). (Thus, an element from g is a derivation of S(g)).