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Generalized operads and their inner cohomomorphisms, arXiv:math.CT/ 0609748
, 2006
"... Abstract. In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that the ..."
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Abstract. In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that they provide an approach to symmetry and moduli objects in noncommutative geometries based upon these “ring–like ” structures. We give a unified axiomatic treatment of generalized operads as functors on categories of abstract labeled graphs. Finally, we extend inner cohomomorphism constructions to more general categorical contexts. This version differs from the previous ones by several local changes (including the title) and two extra references. 0.1. Inner cohomomorphisms of associative algebras. Let k be a field. Consider pairs A = (A, A1) consisting of an associative k–algebra A and a finite dimensional subspace A1 generating A. For two such pairs A = (A, A1) and B =
Closed/open string diagrammatics
 Nucl. Phys. B
"... Abstract. We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. All of the predicted equations of string theory are derived in this model, and new equations can be discovered as well. In particular, several ne ..."
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Cited by 8 (1 self)
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Abstract. We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. All of the predicted equations of string theory are derived in this model, and new equations can be discovered as well. In particular, several new equations together with known transformations generate the combinatorial version of open/closed duality. On the topological and chain levels, the algebraic structure discovered is new, but it specializes to a modular bioperad on the level of homology.
STRING TOPOLOGY OF CLASSIFYING SPACES
, 2007
"... Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG: = map(S 1, BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H∗(LBG) of this loop space. We ..."
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Cited by 2 (0 self)
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Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG: = map(S 1, BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H∗(LBG) of this loop space. We prove that when taken with coefficients in a field the homology of LBG is a homological conformal field theory. As a byproduct of our main theorem, we get on the cohomology H ∗ (LBG) a BValgebra structure.
Equivalenace of The Little Disk and Cacti operads
"... The subject of this thesis is to study the Little Disk operad and the Cacti operad and show that they are equivalent as operads as presented by Kaufmann in the article [Kau05]. In doing so, we go through a preliminary study of operads, what it means for them to be equivalent and the problems involve ..."
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The subject of this thesis is to study the Little Disk operad and the Cacti operad and show that they are equivalent as operads as presented by Kaufmann in the article [Kau05]. In doing so, we go through a preliminary study of operads, what it means for them to be equivalent and the problems involved. We introduce and use the Little Disk in the process. We furthermore introduce and show results about the recognition principle of Fiedorowicz that is used to compare operads up against the Little Cube operad via a “ziqzaq ” through B ∞ operads. We introduce and study the Cacti operad in detail while providing the means to finally apply the recognition principle. Throughout the thesis we will be elaborate on the graphical structures involved. This is both to fertilize the understanding of, but also to embrace the mathematical ideas and metaphors in, the subject. Resumé Emnet for nærværende speciale er at undersøge Lille Disk operaden
INTERNAL COHOMOMORPHISMS FOR OPERADS 1
, 2007
"... Abstract. In this paper we construct internal cohomomorphism objects in various categories of operads (ordinary, cyclic, modular, properads...) and algebras over operads. We argue that they provide an approach to symmetry and moduli objects in noncommutative geometries based upon these “ring–like ” ..."
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Abstract. In this paper we construct internal cohomomorphism objects in various categories of operads (ordinary, cyclic, modular, properads...) and algebras over operads. We argue that they provide an approach to symmetry and moduli objects in noncommutative geometries based upon these “ring–like ” structures. We give also a unified axiomatic treatment of operads as functors on labeled graphs. Finally, we extend internal cohomomorphism constructions to more general categorical contexts. 0.1. Internal cohomomorphisms of associative algebras. Let k be a field. Consider pairs A = (A, A1) consisting of an associative k–algebra A and a finite dimensional subspace A1 generating A. For two such pairs A = (A, A1) and
MOTIVATION AND BACKGROUND
, 904
"... Abstract. The main observable quantities in Quantum Field Theory, correlation functions, are expressed by the celebrated Feynman path integrals which are not well defined mathematical objects. Perturbation formalism interprets such an integral as a formal series of finite– dimensional but divergent ..."
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Abstract. The main observable quantities in Quantum Field Theory, correlation functions, are expressed by the celebrated Feynman path integrals which are not well defined mathematical objects. Perturbation formalism interprets such an integral as a formal series of finite– dimensional but divergent integrals, indexed by Feynman graphs, the list of which is determined by the Lagrangian of the theory. Renormalization is a prescription that allows one to systematically “subtract infinities ” from these divergent terms producing an asymptotic series for quantum correlation functions. On the other hand, graphs treated as “flowcharts”, also form a combinatorial skeleton of the abstract computation theory and various operadic formalisms in abstract algebra. In this role of descriptions of various (classes of) computable functions, such as recursive functions, functions computable by a Turing machine with oracles etc., graphs can be used to replace standard formalisms having linguistic flavor, such as Church’s λ–calculus and various programming languages. The functions in question are generally not everywhere defined due to potentially infinite loops and/or necessity to search in an infinite haystack for a needle which is not there. In this paper I argue that such infinities in classical computation theory can be addressed in the same way as Feynman divergences, and that meaningful versions of renormalization in this context can be devised. Connections with quantum computation are also touched upon.
Word symmetric functions and the RedfieldPólya theorem †
"... Abstract. We give noncommutative versions of the RedfieldPólya theorem in WSym, the algebra of word symmetric functions, and in other related combinatorial Hopf algebras. Résumé. Nous donnons des versions noncommutatives du théorème d’énumération de RedfieldPólya dans WSym, l’algèbre des fonction ..."
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Abstract. We give noncommutative versions of the RedfieldPólya theorem in WSym, the algebra of word symmetric functions, and in other related combinatorial Hopf algebras. Résumé. Nous donnons des versions noncommutatives du théorème d’énumération de RedfieldPólya dans WSym, l’algèbre des fonctions symétriques sur les mots, ainsi que dans d’autres algèbres de Hopf combinatoires.