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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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Nocommutative structures in quantum field theory
Peksová, Lada ; Jurčo, Branislav (advisor) ; Sachs, Ivo (referee) ; Golovko, Roman (referee)
In this thesis, structures defined via modular operads and properads are generalized to their non-commutative analogs. We define the connected sum for modular operads. This way we are able to construct the graded commutative product on the algebra over Feynman transform of the modular operad. This forms a Batalin-Vilkovisky algebra with symmetry given by the modular operad. We transfer this structure to the cohomology via the Homological perturbation lemma. In particular, we consider these constructions for Quantum closed and Quantum open modular operad. As a parallel project we introduce associative analog of Frobenius properad, called Open Frobenius properad. We construct the cobar complex over it and in the spirit of Barannikov interpret algebras over cobar complex as homological differential operators. Furthermore we present the IBA∞-algebras as analog of well-known IBL∞-algebras. 1
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Alexander polynomial
Jančová, Ľubica ; Stanovský, David (advisor) ; Peksová, Lada (referee)
Title: Alexander polynomial Author: Ľubica Jančová Department: Department of Algebra Supervisor: doc. RNDr. David Stanovský, Ph.D., Department of Algebra Abstract: The subject of interest of this thesis is the Alexander polynomial in the knot theory as a knot invariant and various methods of its computa- tion. The thesis focuses on the description of the computation of the Alexander polynomial using four different methods, namely: colouring regions of the knot diagram, colouring arcs of the knot diagram, Seifert's method and the method using the Conway polynomial. In the first chapter we introduce basic notions of the knot theory. In the following chapters we describe methods of computa- tion of the Alexander polynomial. The final chapter deals with the possibility of using the Conway polynomial to show that all of the mentioned methods result in the same polynomial. The main result of this thesis are proofs that might lead to the complete proof of equivalence of algorithms of computation of the Alexander polynomial. Keywords: knot theory, Alexander polynomial, knot invariant
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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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Quantum logic and projective spaces
Peksová, Lada ; Krýsl, Svatopluk (advisor) ; Cejnar, Pavel (referee)
A set of statements about the properties of a quantum system is looked at as at a partially ordered set of subspaces of finite or infinite dimensional Hilbert space. The operation of ordering is performed on a set of propositions comparing the truth values of these propositions and on the set of subspaces as the opera- tion of inclusion. Based on the required properties these structures are translated into operations on the lattice. The correspondence with Heisenberg uncertainty principle is shown there. Furthermore, it is shown that the lattices correspond- ing to the subspaces of infinite dimensional Hilbert space are not modular. This property is replaced with weaker property of orthomodularity, when operation of the negation is added. Following the work of G. Birkhoff and J. von Neumann, the structure of quantum logic is looked for in projective spaces, which are in- troduced either arithmetically or axiomatically. The examples of quantum logic, their physical implementation and eventual implementation in projective spaces are analysed. 1
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