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Institute: search.filters.UBSInstitut.Fakultät Mathematik und Physik (Institutsübergreifend)Publication Type: search.filters.UBSTyp.Abschlussarbeit (Master)

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Now showing 1 - 7 of 7
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    Descriptions of some double Burnside rings
    (2017) Krauß, Nora
    The double Burnside R-algebra B_R(G,G) of a finite group G with coefficients in a commutative ring R has been introduced by S. Bouc. It is R-linearly generated by finite (G,G)-bisets, modulo a relation identifying disjoint union and sum. Its multiplication is induced by the tensor product. It contains the bifree double Burnside R-algebra B_R^Delta(G,G) generated by bifree finite (G,G)-bisets. Let S_n denote the symmetric group on n letters. For R in {Q, Z, Z_(2), F_2, Z_(3), F_3}, we calculate B_R(S_3,S_3) and B_R^Delta(S_4,S_4).
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    Basic representation theory of crossed modules
    (2018) Truong, Monika
    A group corresponds to a topological space with one nontrivial homotopy group. A crossed module corresponds to a topological space with two nontrivial homotopy groups. In classical group theory, Cayley's Theorem constructs for every group G an injective group morphism to the symmetric group S_G. For a crossed module V, we have a similar statement. For every category C, we have the symmetric crossed module S_C. For every crossed module V, we construct an injective crossed module morphism to the symmetric crossed module S_VCat. Suppose given an R-linear category M. On the one hand, we obtain the invertible monoidal category Aut_R(M) by means of category theory. On the other hand, we have the symmetric crossed module S_M as in the Cayley context. In S_M, we have the crossed submodule Aut^CM_R(M) containing only the R-linear elements of S_M. We consider the corresponding invertible monoidal category (Aut^CM_R(M))Cat. We show that there exists a monoidal isofunctor Real_M : (Aut^CM_R(M))Cat -~-> Aut_R(M). This means that starting with M, we obtain essentially the same object via crossed module theory as via category theory. A representation of a group G on an R-module N is given by a group morphism G -> Aut_R(N). Analogously, a representation of a crossed module V on an R-linear category M is given by a crossed module morphism V -> Aut^CM_R(M). We begin to study the representation theory of crossed modules.
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    The minimal projective resolution of Z_(2) over Z_(2)S_4
    (2017) Nitsche, Sebastian
    We consider the trivial module Z_(2) over the group ring Z_(2)S_4. Using the Wedderburn image of Z_(2)S_4, a minimal projective resolution of Z_(2) over Z_(2)S_4 with regular behavior is constructed. We give a closed formula for the projective terms and the differentials. The minimal projective resolution is used to calculate the cohomology groups of S_4 over Z_(2). In 1974, Thomas gave a description of the cohomology ring of S_4 as a factor ring of a polynomial ring. As far as we were able to compare both using Magma, our calculation is in accordance with his result.
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    The resolution equivalence for n-complexes
    (2022) Klein, Veronika
    An injective resolution equivalence is constructed which generalizes the construction from classical homological algebra to n-complexes. The construction proceeds by showing that a functor in the converse direction is an equivalence. The injective resolution functor is defined to be its inverse.
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    Universelle Konstruktionen für Relationen
    (2024) Bechtel, Jonas
    In der Arbeit wird die Kategorie der Mengen und Relationen betrachtet, deren Objekte Mengen und deren Morphismen Relationen sind. Gegeben seien zwei Relationen, welche als Ursprungsmenge die gleiche Menge aufweisen. Für diese Relationen können je nach Situation die Eigenschaften linkstotal, rechtstotal, linkseindeutig und rechtseindeutig gefordert sein. Für die sich hieraus ergebenden Ausgangssituationen wurde untersucht, wann stets ein Pushout konstruiert werden kann und für welche Situationen es Gegenbeispiele zur Existenz eines Pushouts gibt. Existiert ein Pushout, so ist in der Arbeit auch die Pushout-Konstruktion angegeben. Des Weiteren wurde die Faktorisierung einer Relation über ihren Graphen betrachtet und hierfür eine universelle Konstruktion angegeben.
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    On the Bisson-Tsemo model category of graphs
    (2022) Hess, Jannik
    Gezeigt wird, auf Grundlage von Bisson und Tsemo, dass die Kategorie der Graphen eine Modellkategorie darstellt. Es wird ein hinreichendes Kriterium dafür gezeigt, dass ein Graphmorphismus ein Quasiisomorphismus ist. Des Weiteren werden einige Beispiele und Gegenbeispiele konstruiert.
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    Operads in the sense of Mac Lane
    (2017) Eggert, Vanda
    Operads determine algebra structures. We develop versions of the theory of operads that closely resemble the PROPs as defined by S. Mac Lane. Over sets, we construct the operads ASS_0 and COM_0, which determine monoids and commutative monoids, respectively. Over R-modules, we construct the operads ASS, COM, LIE, BIALG, which determine associative algebras, commutative algebras, Lie algebras and Bialgebras, respectively.