Arc Operads and Arc Algebras
By R. Kaufmann, M. Livernet, R. Penner. 2002
56 citations recorded by the source corpus.
Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identied with open subsets of a combinatorial compactication due to Penner of a space closely related to Riemann’s moduli space. Algebras over these operads are shown to be Batalin{Vilkovisky algebras, where the entire BV structure is realized simplicially. Furthermore, our basic operad contains the cacti operad up to homotopy. New operad structures on the circle are classied and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.
Subfields: Algebra and Number Theory, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/a1f1222797ea8260bff00cdc381b4f16d8125346
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In Algebra and Number Theory
In Mathematical Physics
Citations in the corpus
- A proof of a cyclic version of Deligne’s conjecture via Cacti (2004, methodology): “In order to translate the proofs in the arc complex given in [ KLP ] into statements about the Hochschild complex, we will need a slightly finer cell structure then the one above.”
- On spineless cacti, Deligne's conjecture and Connes-Kreimer's Hopf algebra (2003, methodology): “Equivalently, using the map Loop of [26] one arrives at a description of the set of cacti as marked treelike ribbon graphs with a metric.”
- Decorated Teichmüller theory of bordered surfaces (2002, background): “previously described a related convex hull construction in [3], and Chekhov and Fock [1] have previously given analogous global coordinates on Teichmüller space.”
- Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads (2006, methodology): “In this section, we start by giving a brief review of the salient features of the Arc operad of [ KLP ] which is reasonably self-contained.”
- Hochschild homology of structured algebras (2011, unknown): “The idea of the gluing used by Kaufmann and Penner (first introduced in [32]) is to think of a weighted arc family, a point in the space of arc families, as a collection of ribbons whose widths are given by the weights, and then gluing two such families by scaling so that the total weights match, po”