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On Spineless Cacti, Deligne’s Conjecture and Connes–Kreimer’s Hopf Algebra
"... Abstract. We give a new direct proof of Deligne’s conjecture on the Hochschild cohomology. For this we use the cellular chain operad of normalized spineless cacti as a model for the chains of the little discs operad. Previously, we have shown that the operad of spineless cacti is homotopy equivalent ..."
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Abstract. We give a new direct proof of Deligne’s conjecture on the Hochschild cohomology. For this we use the cellular chain operad of normalized spineless cacti as a model for the chains of the little discs operad. Previously, we have shown that the operad of spineless cacti is homotopy equivalent to the little discs operad. Moreover, we also showed that the quasi–operad of normalized spineless cacti is homotopy equivalent to the spineless cacti operad. Now, we give a cell decomposition for the normalized spineless cacti, whose cellular chains form an operad and by our previous results a chain model for the little discs operad. The cells are indexed by bipartite black and white trees which can directly be interpreted as operations on the Hochschild cochains of an associative algebra, yielding a positive answer to Deligne’s conjecture. Furthermore, we show that the symmetric combinations of top–dimensional cells, are isomorphic to the graded pre–Lie operad. Lastly, we define the Hopf algebra of an operad which affords a direct sum. For the pre–Lie suboperad of shifted symmetric top–dimensional chains the symmetric group coinvariants of this Hopf algebra are the renormalization Hopf algebra of Connes and Kreimer.
Notes on universal algebra
 Graphs and Patterns in Mathematics and Theoretical Physics (M. Lyubich and L. Takhtajan, eds.), Proc. Sympos. Pure Math
, 2005
"... Dedicated to Dennis Sullivan on the occasion of his sixtieth birthday. Abstract. These are notes of a minicourse given at Dennisfest in June 2001. The goal of these notes is to give a selfcontained survey of deformation quantization, operad theory, and graph homology. Some new results related to “ ..."
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Cited by 14 (1 self)
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Dedicated to Dennis Sullivan on the occasion of his sixtieth birthday. Abstract. These are notes of a minicourse given at Dennisfest in June 2001. The goal of these notes is to give a selfcontained survey of deformation quantization, operad theory, and graph homology. Some new results related to “String Topology ” and cacti are announced in Section 2.7.
On several varieties of cacti and their relations
"... Abstract. Motivated by string topology and the arc operad, we introduce the notion of quasioperads and consider four (quasi)operads which are different varieties of the operad of cacti. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant si ..."
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Cited by 6 (3 self)
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Abstract. Motivated by string topology and the arc operad, we introduce the notion of quasioperads and consider four (quasi)operads which are different varieties of the operad of cacti. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. Using the recognition principle of Fiedorowicz, we prove that spineless cacti are equivalent as operads to the little discs operad. It turns out that in terms of spineless cacti Cohen’s Gerstenhaber structure and Fiedorowicz ’ braided operad structure are given by the same explicit chains. We also prove that spineless cacti and cacti are homotopy equivalent to their normalized versions as quasioperads by showing that both types of cacti are semidirect products of the quasioperad of their normalized versions with a rescaling operad based on R>0. Furthermore, we introduce the notion of bicrossed products of quasioperads and show that the cacti proper are a bicrossed product of the operad of cacti without spines and the operad based on the monoid given by the circle group S 1. We also prove that this particular bicrossed operad product is homotopy equivalent to the semidirect product of the spineless cacti with the group S 1. This implies that cacti are equivalent to the framed little discs operad. These results lead to new CW models for the little discs and the framed little discs operad.
Arc Operads and Arc Algebras
 Preprint MPI 2002114
"... Abstract. 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 identified with open subsets of a combinatorial compactification due to Penner of a space closely related t ..."
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Cited by 6 (2 self)
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Abstract. 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 identified with open subsets of a combinatorial compactification due to Penner of a space closely related to Riemann’s moduli space. Algebras over these operads are shown to be BatalinVilkovisky 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 classified and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.
ON SEVERAL VARIETIES OF CACTI AND THEIR RELATIONS
, 2002
"... Abstract. We consider four operads which are different varieties of the cacti operad. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. We show that both types of cacti are direct products as operads of ..."
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Abstract. We consider four operads which are different varieties of the cacti operad. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. We show that both types of cacti are direct products as operads of their normalized versions with a re–scaling operad based on R>0. Furthermore, we introduce the notion of bi–crossed operads and show that the cacti proper are a bi–crossed product of the operad of cacti without spines and the operad based on the monoid given by the circle group S 1. We also prove that this particular bi–crossed operad product is homotopy equivalent to the semi–direct product of the spineless cacti with the group S 1. Lastly we recall how to realize the cacti operads as suboperads of the Arc operad and show that the cacti are generated as a suboperad by the cacti without spines and a S 1 twist whose corresponding cycle represents the BV operator.
TT Arc Operads and Arc Algebras
, 2003
"... 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 identified with open subsets of a combinatorial compactification due to Penner of a space closely related to Riemann’ ..."
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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 identified with open subsets of a combinatorial compactification 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 classified and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.
ON SEVERAL VARIETIES OF CACTI AND THEIR RELATIONS
, 2002
"... Abstract. We consider four operads which are different varieties of the cacti operad. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. We show that both types of cacti are a semi–direct products as oper ..."
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Abstract. We consider four operads which are different varieties of the cacti operad. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. We show that both types of cacti are a semi–direct products as operads of their normalized versions with a re–scaling operad based on R>0 which are homotopic to direct products. Furthermore, we introduce the notion of bi–crossed operads and show that the cacti proper are a bi–crossed product of the operad of cacti without spines and the operad based on the monoid given by the circle group S 1. We also prove that this particular bi–crossed operad product is homotopy equivalent to the semi–direct product of the spineless cacti with the group S 1. Lastly we recall how to realize the cacti operads as suboperads of the Arc operad and show that the cacti are generated as a suboperad by the cacti without spines and a S 1 twist whose corresponding cycle represents the BV operator.
ON SEVERAL VARIETIES OF CACTI AND THEIR RELATIONS
, 2003
"... Abstract. We consider four (quasi)–operads which are different varieties of the cacti operad. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. We show that both types of cacti are a semi–direct products ..."
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Abstract. We consider four (quasi)–operads which are different varieties of the cacti operad. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. We show that both types of cacti are a semi–direct products of the quasi–operad of their normalized versions with a re–scaling operad based on R>0 which are homotopic to direct products. Furthermore, we introduce the notion of bi–crossed products of quasi–operads and show that the cacti proper are a bi–crossed product of the operad of cacti without spines and the operad based on the monoid given by the circle group S 1. We also prove that this particular bi–crossed operad product is homotopy equivalent to the semi–direct product of the spineless cacti with the group S 1. As a corollary we obtain that spineless cacti are homotopy equivalent to the little discs operad. Lastly, we recall how to realize the cacti operads as suboperads of the Arc operad and show that the cacti are generated as a suboperad by the cacti without spines and a S 1 twist whose corresponding cycle represents the BV operator.