Chevalley cohomology for aerial Kontsevich graphs

Abstract

Let T poly (R d ) denote the space of skew-symmetric polyvector fields on R d , turned into a graded Lie algebra by means of the Schouten bracket. Our aim is to explore the cohomology of this Lie algebra, with coefficients in the adjoint representation, arising from cochains defined by linear combination of aerial Kontsevich graphs. We prove that this cohomology is localized at the space of graphs without any isolated vertex, any "hand" or any "foot". As an application, we explicitly compute the cohomology of the "ascending graphs" quotient complex.

Stable cohomology of polyvector fields

“…Here the inclusion dGC → XGC appeared already in section 3. The last element S is defined as (1) is the unit element (see (1)) and U is the following series of graphs: (1). The operation U corresponds to the scaling operator, acting on multivector fields of degree (joint in x's and ξ's) k as multiplication by k. Let us prove the proposition above.…”

“…Formality, Deformation Quantization. 1 In the following it will be important that the elements of T poly are polynomials and not power series, as are often considered in physics. The power series version requires different methods, cf.…”

“…Defining a Lie{1} structure on a space V is equivalent to defining a Lie structure on the degree shifted space V [1]. The reader who likes Lie algebras better than Lie{1} algebras may replace all occurrences of Lie{1} by Lie and all occurrences of T poly by T poly [1]. We work with Lie{1} here since it is more convenient regarding signs.…”

“…The operad dGra from above acts from the right on (the S-module) XGra by "insertions". Furthermore there is a natural map of operads dGra → XGra by right action on the identity element I ∈ XGra (1).…”

“…Remark. It has been shown by the author [11] that H 0 (GC) may be identified with the Grothendieck-Teichmüller Lie algebra grt 1 . It follows that the degree 0 cohomology in Theorem 2 above is spanned by grt 1 and the additional generator S.…”