A proof of a cyclic version of Deligne’s conjecture via Cacti

By R. Kaufmann. 2004

68 citations recorded by the source corpus.

We generalize our results on Deligne's conjecture to prove the statement that the normalized Hochschild co--chains of a finite--dimensional associative algebra with a non--degenerate, symmetric, invariant inner product are an algebra over a chain model of the framed little discs operad which is given by cacti. In particular, in characteristic zero they are a BV algebra up to homotopy and the Hochschild cohomology of such an algebra is a BV algebra whose induced bracket coincides with Gerstenhaber's bracket. To show this, we use a cellular chain model for the framed little disc operad in terms of normalized cacti. This model is given by tensoring our chain model for the little discs operad in terms of spineless cacti with natural chain models for $(S^1)^{\times n}$ adapted to cacti.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f3b697d978261e9366ac24c81718ba6bd885e350

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Citations in the corpus

The Batalin-Vilkovisky Algebra on Hochschild Cohomology Induced by Infinity Inner Products (2008, background): “For instance, theorem 1 can be lifted to the chain level, which was described by Costello [5], Kaufmann [11], Menichi [16], and Tradler and Zeinalian [21].”
On the cyclic Deligne conjecture (2004, unknown): “Q A ] 2 7 O ct 2 00 4”
Batalin-Vilkovisky algebra structures on Hochschild Cohomology (2007, background): “ollowing Corollary is implicit in [26] and was for the ﬁrst time explicited in [24, Theorem 1.6]. Corollary 18. [26,24] Let A be a symmetric algebra. Then HH∗(A,A) is a Batalin-Vilkovisky algebra. In [20] or [28, Corollary 3.4] or [5, Section 1.4] or [21, Theorem B] or [22, Section 11.6], this Batal”
Topological conformal field theories and Calabi–Yau categories (2004, result): “This has been proved by Kaufmann [ Kau04 ] and Tradler-Zeinalian [TZ04].”
On spineless cacti, Deligne's conjecture and Connes-Kreimer's Hopf algebra (2003, methodology): “perad of Arc which is the natural topological operad model for these cells. 50 RALPH M. KAUFMANN 7.1.2. The cyclic case. The results and methods of this paper have been extended to the cyclic case in [K3] in which it is proven that the Hochschild cochains of a Frobenius algebra carry an action of a ”
Hochschild homology of structured algebras (2011, unknown): “) In particular, in genus 0, this gives the cyclic Deligne conjecture first proved in [30], see also [54].”
Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads (2006, methodology): “Finally, the BV operator, which is given by the action of the sub–PROP equivalent to the framed little discs operad, acts as in [ K3 ].”