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Formality of the chain operad of framed little disks (2009).
| Content Provider | CiteSeerX |
|---|---|
| Author | Severa, Pavol ˇ |
| Abstract | Tamarkin proved in [4] formality of the operad of rational chains of the little disk operad. Extending his result to the framed little disk operad is a very simple exercise, and this is what we do in this note. While writing this note I learnt about existence of a work in progress [3] containing this result and possibly much more. All vector spaces in this note are supposed to be over Q, and also all chains are with rational coefficients. 1. Four operads Recall that an operad in a symmetric monoidal category C is a functor O from the groupoid of finite sets S to C together with morphisms O: A ↦ → OA ◦a: OA ⊗ OB → O (A−{a})⊔B for all a ∈ A; these morphisms are required to be functorial and associative in the obvious sense. In this section we introduce the operad BV of Batalin-Vilkovisky algebras and the framed little disk operad FLD. By a theorem of Getzer, the homology of FLD is the operad BV. The aim of this note is to show that the operad of rational chains of FLD is quasiisomorphic to BV. The quasiisomorphism is constructed via two more operads introduced in this section, namely the operad of parenthesized ribbon braids and an operad ˜t in the category of Lie algebras. This construction follows verbatim the paper of Tamarkin [4]. 1.1. Gerstenhaber and BV operad. A Gerstenhaber algebra is a graded vector space V with the following structure (1) V is a graded-commutative associative algebra (2) V [1] is a graded Lie algebra (3) [a, bc] = [a, b]c + (−1) ( a +1) b b[a, c]. A Batalin-Vilkovisky (BV) algebra is a Gerstenhaber algebra V with a linear map ∆ : V → V of degree −1, such that (1) ∆ 2 = 0 (2) [a, b] = (−1) a ∆(ab) − (−1) a (∆a)b − a∆b. |
| File Format | |
| Publisher Date | 2009-01-01 |
| Access Restriction | Open |
| Subject Keyword | Framed Little Disk Chain Operad Gerstenhaber Algebra Operad Bv Rational Chain Finite Set Little Disk Operad Linear Map Following Structure Graded-commutative Associative Algebra Operads Recall Obvious Sense Batalin-vilkovisky Algebra Symmetric Monoidal Category Parenthesized Ribbon Braid Vector Space Graded Vector Space Simple Exercise Rational Coefficient Graded Lie Algebra Oa Ob Framed Little Disk Operad Bv Operad |
| Content Type | Text |
