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Algebras over operads and properads
Peksová, Lada ; Jurčo, Branislav (advisor) ; Vysoký, Jan (referee)
Operads are objects that model operations with several inputs and one output. We define such structures in the context of graphs, namely oriented trees. Then we generalize operads to properads and modular operads by taking general graphs with, or without, orientation. Further we construct the cobar complex of operads and properads and illustrate the construction on the examples of the associative operad Ass and the Frobenius properad Frob. Algebras over the cobar complex of operads correspond to certain homotopy algebras, for our example of Ass it is A1. We find its Maurer-Cartan equation and convert it from coderivations to derivations. Similarly we find the Maurer-Cartan equation for cobar complex of Frobenius properad. Powered by TCPDF (www.tcpdf.org)
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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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Algebras over operads and properads
Peksová, Lada ; Jurčo, Branislav (advisor) ; Vysoký, Jan (referee)
Operads are objects that model operations with several inputs and one output. We define such structures in the context of graphs, namely oriented trees. Then we generalize operads to properads and modular operads by taking general graphs with, or without, orientation. Further we construct the cobar complex of operads and properads and illustrate the construction on the examples of the associative operad Ass and the Frobenius properad Frob. Algebras over the cobar complex of operads correspond to certain homotopy algebras, for our example of Ass it is A1. We find its Maurer-Cartan equation and convert it from coderivations to derivations. Similarly we find the Maurer-Cartan equation for cobar complex of Frobenius properad. Powered by TCPDF (www.tcpdf.org)
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