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Homotopické struktury v algebře, geometrii a matematické fyzice
Černohorská, Eva ; Markl, Martin (advisor) ; Somberg, Petr (referee)
Title: Homotopic structures in algebra, geometry and mathematical physics Author: Eva Černohorská Department: Mathematical Institute of Charles University Supervisor: RNDr. Martin Markl, DrSc., Institute of Mathematics of the Academy of Sciences of the Czech Republic, Mathematical Institute of Charles University Abstract: The aim of this thesis was to generalize the result that associative algebras on finite dimensional vector spaces can be described using differentials on free algebras. This result is limited by the duality of vector spaces. If we assume that the underlying space has a linear topology, then we can use the duality between discrete and linearly compact (profinite) vector spaces. To generalize the notion of an algebra, we need to recall the completed tensor product on linear vector spaces. Since this topics does not seem to be sufficiently covered by the literature, this thesis could serve also as a comprehensive text on linear vector spaces and their completed tensor products. We prove that also A∞ structures on linearly compact vector spaces could be represented by differentials on a free algebra. Keywords: Strongly homotopy associative algebra, linear topological vector space, Pontryagin duality, completed tensor product, differential
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Operadic resolutions of diagrams
Doubek, Martin ; Markl, Martin (advisor) ; Somberg, Petr (referee) ; Čadek, Martin (referee)
of the Doctoral Thesis Operadic Resolutions of Diagrams by Martin Doubek We study resolutions of the operad AC describing diagrams of a given shape C in the category of algebras of a given type A. We prove the conjecture by Markl on constructing the resolution out of resolutions of A and C, at least in a certain restricted setting. For associative algebras, we make explicit the cohomology theory for the diagrams and recover Gerstenhaber-Schack diagram cohomology. In general, we show that the operadic cohomology is Ext in the category of operadic modules. 1
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Homotopy transfer for A-infinity algebras
Kopřiva, Jakub ; Doubek, Martin (advisor) ; Markl, Martin (referee)
Homotopický přenos A∞ algeber Jakub Kopřiva Abstract This bachelor's thesis deals with the problem of homotopy transfer for A∞ algebras. It strives to give an account of the problem as complete and as self- contained as possible. At first, it presents the correspondence with codiffe- rentials on reduced tensor coalgebras and A∞ algebras, which is colloquially know as the bar construction. The thesis is, however, mainly concerned with the homotopy transfer for A∞ algebras accordning to Markl (2006). We de- duce the formulæ published by Markl and we give proof of their correctness. We also demonstrate that, under additional requirements, Markl's formulas coincide with formulas derived using the homological perturbation lemma.
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Homotopické struktury v algebře, geometrii a matematické fyzice
Černohorská, Eva ; Markl, Martin (advisor) ; Somberg, Petr (referee)
Title: Homotopic structures in algebra, geometry and mathematical physics Author: Eva Černohorská Department: Mathematical Institute of Charles University Supervisor: RNDr. Martin Markl, DrSc., Institute of Mathematics of the Academy of Sciences of the Czech Republic, Mathematical Institute of Charles University Abstract: The aim of this thesis was to generalize the result that associative algebras on finite dimensional vector spaces can be described using differentials on free algebras. This result is limited by the duality of vector spaces. If we assume that the underlying space has a linear topology, then we can use the duality between discrete and linearly compact (profinite) vector spaces. To generalize the notion of an algebra, we need to recall the completed tensor product on linear vector spaces. Since this topics does not seem to be sufficiently covered by the literature, this thesis could serve also as a comprehensive text on linear vector spaces and their completed tensor products. We prove that also A∞ structures on linearly compact vector spaces could be represented by differentials on a free algebra. Keywords: Strongly homotopy associative algebra, linear topological vector space, Pontryagin duality, completed tensor product, differential
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Operadic resolutions of diagrams
Doubek, Martin ; Markl, Martin (advisor) ; Somberg, Petr (referee) ; Čadek, Martin (referee)
of the Doctoral Thesis Operadic Resolutions of Diagrams by Martin Doubek We study resolutions of the operad AC describing diagrams of a given shape C in the category of algebras of a given type A. We prove the conjecture by Markl on constructing the resolution out of resolutions of A and C, at least in a certain restricted setting. For associative algebras, we make explicit the cohomology theory for the diagrams and recover Gerstenhaber-Schack diagram cohomology. In general, we show that the operadic cohomology is Ext in the category of operadic modules. 1
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