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1
Banjo, Elizabeth. "Representation theory of algebras related to the partition algebra." Thesis, City University London, 2013. http://openaccess.city.ac.uk/2360/.
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The main objective of this thesis is to determine the complex generic representation theory of the Juyumaya algebra. We do this by showing that a certain specialization of this algebra is isomorphic to the small ramified partition algebra, introduced by Martin (the representation theory of which is computable by a combination of classical and category theoretic techniques). We then use this result and general arguments of Cline, Parshall and Scott to prove that the Juyumaya algebra En(x) over the complex field is generically semisimple for all n 2 N. The theoretical background which will facilitate an understanding of the construction process is developed in suitable detail. We also review a result of Martin on the representation theory of the small ramified partition algebra, and fill in some gaps in the proof of this result by providing proofs to results leading to it.
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Hmaida, Mufida Mohamed A. "Representation theory of algebras related to the bubble algebra." Thesis, University of Leeds, 2016. http://etheses.whiterose.ac.uk/15987/.
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In this thesis we study several algebras which are related to the bubble algebra, including the bubble algebra itself. We introduce a new class of multi-parameter algebras, called the multi-colour partition algebra $ P_{n,m} ( \breve{\delta} )$, which is a generalization of both the partition algebra and the bubble algebra. We also define the bubble algebra and the multi-colour symmetric groupoid algebra as sub-algebras of the algebra $ P_{n,m} ( \breve{\delta} ) $. We investigate the representation theory of the multi-colour symmetric groupoid algebra $ \F S_{n,m} $. We show that $ \F S_{n,m} $ is a cellular algebra and it is isomorphic to the generalized symmetric group algebra $ \F \mathbb{Z}_m \wr S_n $ when $ m $ is invertible and $ \F $ is an algebraically closed field. We then prove that the algebra $ P_{n,m} ( \breve{\delta} ) $ is also a cellular algebra and define its cell modules. We are therefore able to classify all the irreducible modules of the algebra $ P_{n,m} ( \breve{\delta} ) $. We also study the semi-simplicity of the algebra $ P_{n,m} ( \breve{\delta} ) $ and the restriction rules of the cell modules to lower rank $ n $ over the complex field. The main objective of this thesis is to solve some open problems in the representation theory of the bubble algebra $ T_{n,m} ( \breve{\delta} ) $. The algebra $ T_{n,m} ( \breve{\delta} ) $ is known to be cellular. We use many results on the representation theory of the Temperley-Lieb algebra to compute bases of the radicals of cell modules of the algebra $ T_{n,m} ( \breve{\delta} ) $ over an arbitrary field. We then restrict our attention to study representations of $ T_{n,m} ( \breve{\delta} ) $ over the complex field, and we determine the entire Loewy structure of cell modules of the algebra $ T_{n,m} ( \breve{\delta} ) $. In particular, the main theorem is Theorem 5.41.
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Speyer, Liron. "Representation theory of Khovanov-Lauda-Rouquier algebras." Thesis, Queen Mary, University of London, 2015. http://qmro.qmul.ac.uk/xmlui/handle/123456789/9114.
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This thesis concerns representation theory of the symmetric groups and related algebras. In recent years, the study of the “quiver Hecke algebras”, constructed independently by Khovanov and Lauda and by Rouquier, has become extremely popular. In this thesis, our motivation for studying these graded algebras largely stems from a result of Brundan and Kleshchev – they proved that (over a field) the KLR algebras have cyclotomic quotients which are isomorphic to the Ariki–Koike algebras, which generalise the Hecke algebras of type A, and thus the group algebras of the symmetric groups. This has allowed the study of the graded representation theory of these algebras. In particular, the Specht modules for the Ariki–Koike algebras can be graded; in this thesis we investigate graded Specht modules in the KLR setting. First, we conduct a lengthy investigation of the (graded) homomorphism spaces between Specht modules. We generalise the rowand column removal results of Lyle and Mathas, producing graded analogues which apply to KLR algebras of arbitrary level. These results are obtained by studying a class of homomorphisms we call dominated. Our study provides us with a new result regarding the indecomposability of Specht modules for the Ariki–Koike algebras. Next, we use homomorphisms to produce some decomposability results pertaining to the Hecke algebra of type A in quantum characteristic two. In the remainder of the thesis, we use homogeneous homomorphisms to study some graded decomposition numbers for the Hecke algebra of type A. We investigate graded decomposition numbers for Specht modules corresponding to two-part partitions. Our investigation also leads to the discovery of some exact sequences of homomorphisms between Specht modules.
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Carr, Andrew Nickolas. "Lie Algebras and Representation Theory." OpenSIUC, 2016. https://opensiuc.lib.siu.edu/theses/1988.
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Laking, Rosanna Davison. "String algebras in representation theory." Thesis, University of Manchester, 2016. https://www.research.manchester.ac.uk/portal/en/theses/string-algebras-in-representation-theory(c350436a-db9a-429d-a8a5-470dffc0974f).html.
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The work in this thesis is concerned with three subclasses of the string algebras: domestic string algebras, gentle algebras and derived-discrete algebras (of non-Dynkin type). The various questions we answer are linked by the theme of the Krull-Gabriel dimension of categories of functors. We calculate the Cantor-Bendixson rank of the Ziegler spectrum of the category of modules over a domestic string algebra. Since there is no superdecomposable module over a domestic string algebra, this is also the value of the Krull-Gabriel dimension of the category of finitely presented functors from the category of finitely presented modules to the category of abelian groups. We also give a description of a basis for the spaces of homomorphisms between pairs of indecomposable complexes in the bounded derived category of a gentle algebra. We then use this basis to describe the Hom-hammocks involving (possibly infinite) string objects in the homotopy category of complexes of projective modules over a derived-discrete algebra. Using this description, we prove that the Krull-Gabriel dimension of the category of coherent functors from a derived-discrete algebra (of non-Dynkin type) is equal to 2. Since the Krull-Gabriel dimension is finite, it is equal to the Cantor-Bendixson rank of the Ziegler spectrum of the homotopy category and we use this to identify the points of the Ziegler spectrum. In particular, we prove that the indecomposable pure-injective complexes in the homotopy category are exactly the string complexes. Finally, we prove that every indecomposable complex in the homotopy category is pure-injective, and hence is a string complex.
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Boddington, Paul. "No-cycle algebras and representation theory." Thesis, University of Warwick, 2004. http://wrap.warwick.ac.uk/3482/.
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In the first half of this dissertation we study certain quotient algebras of preprojective algebras called no-cycle algebras N. These are studied via one-cycle algebras, which are introduced here. Results include detailed combinatorial information on N, and in certain special cases a presentation for N as a quiver with relations. In the second half we consider deformations of coordinate algebras of Kleinian singularities. Results include an explicit presentation for the deformations of a type D singularity. These two themes are tied together at the end by some mainly speculative comments about the role the various objects studied have to play in representation theory.
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Vaso, Laertis. "Cluster Tilting for Representation-Directed Algebras." Licentiate thesis, Uppsala universitet, Algebra och geometri, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-364224.
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Nash, David A. 1982. "Graded representation theory of Hecke algebras." Thesis, University of Oregon, 2010. http://hdl.handle.net/1794/10871.
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xii, 76 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.We study the graded representation theory of the Iwahori-Hecke algebra, denoted by Hd , of the symmetric group over a field of characteristic zero at a root of unity. More specifically, we use graded Specht modules to calculate the graded decomposition numbers for Hd . The algorithm arrived at is the Lascoux-Leclerc-Thibon algorithm in disguise. Thus we interpret the algorithm in terms of graded representation theory. We then use the algorithm to compute several examples and to obtain a closed form for the graded decomposition numbers in the case of two-column partitions. In this case, we also precisely describe the 'reduction modulo p' process, which relates the graded irreducible representations of Hd over [Special characters omitted.] at a p th -root of unity to those of the group algebra of the symmetric group over a field of characteristic p.
Committee in charge: Alexander Kleshchev, Chairperson, Mathematics; Jonathan Brundan, Member, Mathematics; Boris Botvinnik, Member, Mathematics; Victor Ostrik, Member, Mathematics; William Harbaugh, Outside Member, Economics
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King, Oliver. "The representation theory of diagram algebras." Thesis, City University London, 2014. http://openaccess.city.ac.uk/5915/.
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In this thesis we study the modular representation theory of diagram algebras, in particular the Brauer and partition algebras, along with a brief consideration of the Temperley-Lieb algebra. The representation theory of these algebras in characteristic zero is well understood, and we show that it can be described through the action of a reflection group on the set of simple modules (a result previously known for the Temperley-Lieb and Brauer algebras). By considering the action of the corresponding affine reflection group, we give a characterisation of the (limiting) blocks of the Brauer and partition algebras in positive characteristic. In the case of the Brauer algebra, we then show that simple reflections give rise to non-zero decomposition numbers. We then restrict our attention to a particular family of Brauer and partition algebras, and use the block result to determine the entire decomposition matrix of the algebras therein.
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Ahmed, Chwas Abas. "Representation theory of diagram algebras : subalgebras and generalisations of the partition algebra." Thesis, University of Leeds, 2016. http://etheses.whiterose.ac.uk/15997/.
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This thesis concerns the representation theory of diagram algebras and related problems. In particular, we consider subalgebras and generalisations of the partition algebra. We study the d-tonal partition algebra and the planar d-tonal partition algebra. Regarding the d-tonal partition algebra, a complete description of the J -classes of the underlying monoid of this algebra is obtained. Furthermore, the structure of the poset of J -classes of the d-tonal partition monoid is also studied and numerous combinatorial results are presented. We observe a connection between canonical elements of the d-tonal partition monoids and some combinatorial objects which describe certain types of hydrocarbons, by using the alcove system of some reflection groups. We show that the planar d-tonal partition algebra is quasi-hereditary and generically semisimple. The standard modules of the planar d-tonal partition algebra are explicitly constructed, and the restriction rules for the standard modules are also given. The planar 2-tonal partition algebra is closely related to the two coloured Fuss-Catalan algebra. We use this relation to transfer information from one side to the other. For example, we obtain a presentation of the 2-tonal partition algebra by generators and relations. Furthermore, we present a necessary and sufficient condition for semisimplicity of the two colour Fuss-Catalan algebra, under certain known restrictions.
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Cao, Mengyuan. "Representation Theory of Lie Colour Algebras and Its Connection with the Brauer Algebras." Thesis, Université d'Ottawa / University of Ottawa, 2018. http://hdl.handle.net/10393/38125.
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In this thesis, we study the representation theory of Lie colour algebras. Our strategy follows the work of G. Benkart, C. L. Shader and A. Ram in 1998, which is to use the Brauer algebras which appear as the commutant of the orthosymplectic Lie colour algebra when they act on a k-fold tensor product of the standard representation. We give a general combinatorial construction of highest weight vectors using tableaux, and compute characters of the irreducible summands in some borderline cases. Along the way, we prove the RSK-correspondence for tableaux and the PBW theorem for Lie colour algebras.
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Nornes, Nils Melvær. "Partial Orders in Representation Theory of Algebras." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2008. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-9689.
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In this paper we investigate some partial orders used in representation theory of algebras. Let $K$ be a commutative ring, $Lambda$ a finitely generated $K$-algebra and $d$ a natural number. We then study partial orders on the set of isomorphism classes of $Lambda$-modules of length $d$. The orders degeneration, virtual degeneration and hom-order are discussed. The main purpose of the paper is to study the relation $leq_n$ constructed by considering the ranks of $ntimes n$-matrices over $Lambda$ as $K$-endomorphisms on $M^n$ for a $Lambda$-module $M$. We write $Mleq_n N$ when for any $ntimes n$-matrix the rank with respect to $M$ is greater than or equal to the rank with respect to $N$. We study these relations for various algebras and determine when $leq_n$ is a partial order.
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Paget, Rowena. "Representation theory of symmetric groups and related algebras." Thesis, University of Oxford, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.270235.
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Jacoby, Adam Michael. "ON REPRESENTATION THEORY OF FINITE-DIMENSIONAL HOPF ALGEBRAS." Diss., Temple University Libraries, 2017. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/433432.
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MathematicsPh.D.
Representation theory is a field of study within abstract algebra that originated around the turn of the 19th century in the work of Frobenius on representations of finite groups. More recently, Hopf algebras -- a class of algebras that includes group algebras, enveloping algebras of Lie algebras, and many other interesting algebras that are often referred to under the collective name of ``quantum groups'' -- have come to the fore. This dissertation will discuss generalizations of certain results from group representation theory to the setting of Hopf algebras. Specifically, our focus is on the following two areas: Frobenius divisibility and Kaplansky's sixth conjecture, and the adjoint representation and the Chevalley property.
Temple University--Theses
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Meinel, Joanna [Verfasser]. "Affine nilTemperley-Lieb algebras and generalized Weyl algebras: Combinatorics and representation theory / Joanna Meinel." Bonn : Universitäts- und Landesbibliothek Bonn, 2016. http://d-nb.info/1122193874/34.
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Pike, Jeffrey. "Quivers and Three-Dimensional Lie Algebras." Thesis, Université d'Ottawa / University of Ottawa, 2015. http://hdl.handle.net/10393/32398.
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We study a family of three-dimensional Lie algebras that depend on a continuous parameter. We introduce certain quivers and prove that idempotented versions of the enveloping algebras of the Lie algebras are isomorphic to the path algebras of these quivers modulo certain ideals in the case that the free parameter is rational and non-rational, respectively. We then show how the representation theory of the introduced quivers can be related to the representation theory of quivers of affine type A, and use this relationship to study representations of the family of Lie algebras of interest. In particular, though it is known that this particular family of Lie algebras consists of algebras of wild representation type, we show that if we impose certain restrictions on weight decompositions, we obtain full subcategories of the category of representations that are of finite or tame representation type.
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Bellamy, Gwyn. "Generalized Calogero-Moser spaces and rational Cherednik algebras." Thesis, University of Edinburgh, 2010. http://hdl.handle.net/1842/4733.
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The subject of this thesis is the interplay between the geometry and the representation theory of rational Cherednik algebras at t = 0. Exploiting this relationship, we use representation theoretic techniques to classify all complex re ection groups for which the geometric space associated to a rational Cherednik algebra, the generalized Calogero-Moser space, is singular. Applying results of Ginzburg-Kaledin and Namikawa, this classification allows us to deduce a (nearly complete) classification of those symplectic reflection groups for which there exist crepant resolutions of the corresponding symplectic quotient singularity. Then we explore a particular way of relating the representation theory and geometry of a rational Cherednik algebra associated to a group W to the representation theory and geometry of a rational Cherednik algebra associated to a parabolic subgroup of W. The key result that makes this construction possible is a recent result of Bezrukavnikov and Etingof on completions of rational Cherednik algebras. This leads to the definition of cuspidal representations and we show that it is possible to reduce the problem of studying all the simple modules of the rational Cherednik algebra to the study of these nitely many cuspidal modules. We also look at how the Etingof-Ginzburg sheaf on the generalized Calogero-Moser space can be "factored" in terms of parabolic subgroups when it is restricted to particular subvarieties. In particular, we are able to confirm a conjecture of Etingof and Ginzburg on "factorizations" of the Etingof-Ginzburg sheaf. Finally, we use Clifford theoretic techniques to show that it is possible to deduce the Calogero-Moser partition of the irreducible representations of the complex reflection groups G(m; d; n) from the corresponding partition for G(m; 1; n). This confirms, in the case W = G(m; d; n), a conjecture of Gordon and Martino relating the Calogero-Moser partition to Rouquier families for the corresponding cyclotomic Hecke algebra.
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Lemay, Joel. "Valued Graphs and the Representation Theory of Lie Algebras." Thèse, Université d'Ottawa / University of Ottawa, 2011. http://hdl.handle.net/10393/20168.
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Quivers (directed graphs) and species (a generalization of quivers) as well as their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this thesis, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K-species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles that do not appear in the literature. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K-species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.
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Chen, Bo. "The Gabriel-Roiter measure for representation-finite hereditary algebras." [S.l.] : [s.n.], 2006. http://deposit.ddb.de/cgi-bin/dokserv?idn=979949564.
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Colligan, Mark. "Some topics in the representation theory of Brauer algebras." Thesis, University of Kent, 2012. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.587556.
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This thesis is concerned with the representation theory of the Brauer algebra over an algebraically closed field of arbitrary characteristic. There are two main themes, given as follows. Firstly, we describe some classes of simple modules for the Brauer algebras with parameters ±2. Secondly, we derive a combinatorial basis of the vector space of homomorphisms from one permutation module to another. We do the same for the vector space of homomorphisms from a permutation module to a cell module. Using these bases, we show that every homomorphism from a permutation module to a cell module factors through the permutation module whose label agrees with that of the cell module.
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Hussein, Ahmed Baqer. "On the representation theory of the Fuss-Catalan algebras." Thesis, University of Leeds, 2017. http://etheses.whiterose.ac.uk/17949/.
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Throughout this thesis, we study the representation theory of the Fuss-Catalan algebras, $FC_{2,n}(a, b)$. We prove that these algebras are cellular and we define their cellular basis. In addition, we prove that they form a tower of recollement, and hence, they are quasi-hereditary. By calculating the Gram determinants of certain cell modules for the Fuss-Catalan algebras, we determine when these algebras are not semisimple. Finally, we end with defining homomorphisms between specified cell modules.
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Barnes, Gwendolyn Elizabeth. "Nonassociative geometry in representation categories of quasi-Hopf algebras." Thesis, Heriot-Watt University, 2016. http://hdl.handle.net/10399/3294.
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It has been understood that quantum spacetime may be non-geometric in the sense that its phase space algebra is noncommutative and nonassociative. It has therefore been of interest to develop a formalism to describe differential geometry on non-geometric spaces. Many of these spaces would fi t naturally as commutative algebra objects in representation categories of triangular quasi-Hopf algebras because they arise as cochain twist deformations of classical manifolds. In this thesis we develop in a systematic fashion a description of differential geometry on commutative algebra objects internal to the representation category of an arbitrary triangular quasi-Hopf algebra. We show how to express well known geometrical concepts such as tensor fields, differential calculi, connections and curvatures in terms of internal homomorphisms using universal categorical constructions in a closed braided monoidal category to capture algebraic properties such as Leibniz rules. This internal description is an invaluable perspective for physics enabling one to construct geometrical quantities with dynamical content and con guration spaces as large as those in the corresponding classical theories. We also provide morphisms which lift connections to tensor products and tensor elds. Working in the simplest setting of trivial vector bundles we obtain explicit expressions for connections and their curvatures on noncommutative and nonassociative vector bundles. We demonstrate how to apply our formalism to the construction of a physically viable action functional for Einstein-Cartan gravity on noncommutative and nonassociative spaces as a step towards understanding the effect of nonassociative deformations of spacetime geometry on models of quantum gravity.
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Bulgakova, Daria. "Some aspects of representation theory of walled Brauer algebras." Thesis, Aix-Marseille, 2020. http://www.theses.fr/2020AIXM0022.
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L'algèbre de Brauer murée est une algèbre unitaire associative. Il s’agit d’une algèbre de diagramme engendré par des diagrammes «murés» particuliers. Cette algèbre peut être définie par des générateurs et des relations. Dans le premier chapitre de la thèse, nous construisons la forme normale de l'algèbre de Brauer murée - un ensemble de monômes de base (mots) dans les générateurs. Nous introduisons une modification “ordonnée” du fameux lemme du diamant de Bergman, à savoir, nous présentons un ensemble de règles qui, étant appliquées dans un certain ordre, permet de réduire tout monôme dans les générateurs à un élément de la forme normale. Nous appliquons ensuite la forme normale pour calculer la fonction génératrice du nombre de mots avec une longueur minimale donnée.Une procédure de fusion donne une construction de la famille maximale d'idempotents orthogonaux minimaux par paire dans l'algèbre et, par conséquent, fournit un moyen de comprendre les bases dans les représentations irréductibles. Nous construisons la procédure de fusion pour l'algèbre de Brauer murée, à savoir, tous les idempotents primitifs est trouvé par les évaluations consécutives de fonction rationnelle en plusieurs variables.Dans le deuxième chapitre, nous étudions le produit tensoriel mixte des représentations fondamentales tridimensionnelles de l'algèbre de Hopf U_q sl(2|1). L'un des principaux résultats consiste à établir des formules explicites pour la décomposition des produits tensoriels de tout module de U_q sl(2|1) simple ou projectif avec les modules générateurs. Un autre résultat important consiste à décomposer le produit tensoriel mixte en un bimoduleThe walled Brauer algebra is an associative unital algebra. It is a diagram algebra spanned by particular ‘walled’ diagrams with multiplication given by concatenation. This algebra can be defined in terms of generators, obeying certain relations. In the first part of the dissertation we construct the normal form of the walled Brauer algebra - a set of basis monomials (words) in generators. This set is constructed with the aid of the so-called Bergman’s diamond lemma: we present a set of rules which allows one to reduce any monomial in generators to an element from the normal form. We then apply the normal form to calculate the generating function for the numbers of words with a given minimal length.A fusion procedure gives a construction of the maximal family of pairwise orthogonal minimal idempotents in the algebra, and therefore, provides a way to understand bases in the irreducible representations. As a main result of the second part we construct the fusion procedure for the walled Brauer algebra and show that all primitive idempotents can be found by evaluating a rational function in several variables. In the third part we study the mixed tensor product of three-dimensional fundamental representations of the Hopf algebra U_q sl(2|1). One of the main results consists in the establishing of the explicit formulae for the decomposition of tensor products of any simple or any projective U_q sl(2|1)-module with the generating modules. Another important outcome consists in decomposing the mixed tensor product as a bimodule
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O'Dell, Connor. "Non-Resonant Uniserial Representations of Vec(R)." Thesis, University of North Texas, 2018. https://digital.library.unt.edu/ark:/67531/metadc1157650/.
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The non-resonant bounded uniserial representations of Vec(R) form a certain class of extensions composed of tensor density modules, all of whose subquotients are indecomposable. The problem of classifying the extensions with a given composition series is reduced via cohomological methods to computing the solution of a certain system of polynomial equations in several variables derived from the cup equations for the extension. Using this method, we classify all non-resonant bounded uniserial extensions of Vec(R) up to length 6. Beyond this length, all such extensions appear to arise as subquotients of extensions of arbitrary length, many of which are explained by the psuedodifferential operator modules. Others are explained by a wedge construction and by the pseudodifferential operator cocycle discovered by Khesin and Kravchenko.
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edu, rojkovsk@math upenn. "Family Algebras of Representations with Simple Spectrum." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1045.ps.
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Moreira, Rodriguez Rivera Walter. "Products of representations of the symmetric group and non-commutative versions." Texas A&M University, 2008. http://hdl.handle.net/1969.1/85938.
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We construct a new operation among representations of the symmetric group that
interpolates between the classical internal and external products, which are defined in
terms of tensor product and induction of representations. Following Malvenuto and
Reutenauer, we pass from symmetric functions to non-commutative symmetric functions
and from there to the algebra of permutations in order to relate the internal and
external products to the composition and convolution of linear endomorphisms of the
tensor algebra. The new product we construct corresponds to the Heisenberg product
of endomorphisms of the tensor algebra. For symmetric functions, the Heisenberg
product is given by a construction which combines induction and restriction of representations.
For non-commutative symmetric functions, the structure constants of
the Heisenberg product are given by an explicit combinatorial rule which extends a
well-known result of Garsia, Remmel, Reutenauer, and Solomon for the descent algebra.
We describe the dual operation among quasi-symmetric functions in terms of
alphabets.
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Gordon, Iain. "Representation theory of quantised function algebras at roots of unity." Thesis, Connect to electronic version, 1998. http://hdl.handle.net/1905/177.
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Soriano, Solá Marcos. "Contributions to the integral representation theory of Iwahori-Hecke algebras." [S.l. : s.n.], 2002. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB9866651.
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Spencer, Matthew. "The representation theory of Iwahori-Hecke algebras with unequal parameters." Thesis, Queen Mary, University of London, 2014. http://qmro.qmul.ac.uk/xmlui/handle/123456789/8644.
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The Iwahori-Hecke algebras of finite Coxeter groups play an important role in many areas of mathematics. In this thesis we study the representation theory of the Iwahori-Hecke algebras of the Coxeter groups of type Bn and F4, in the unequal parameter case. We denote these algebras HQ and KQ respectively. This follows on from work done by Dipper, James, Murphy and Norton. We are interested in the Iwahori-Hecke algebras of type Bn and F4 since these are the only cases, apart from the dihedral groups, where the Coxeter generators lie in different conjugacy classes, and therefore the Iwahori-Hecke algebra can have unequal parameters. There are two parameters associated with these algebras, Q and q. Norton dealt with the case Q = q = 0, whilst Dipper, James and Murphy addressed the case q 6= 0 in type Bn. In this thesis we look at the case Q 6= 0; q = 0. We begin by constructing the simple modules for HQ, then compute the Ext quiver and find the blocks of HQ. We continue by observing that there is a natural embedding of the algebra of type n 1 in the algebra of type n, and this gives rise to the notion of an induced module. We look at the structure of the induced module associated with a given simple HQ-module. Here we are able to construct a composition series for the induced module and show that in a particular case the induced modules are self-dual. Finally, we look at KQ and find that the representation theory is related to representation theory of the Iwahori-Hecke algebra of type B3. Using this relationship we are able to construct the simple modules for KQ and begin the analysis of the Ext quiver.
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Nilsson, Jonathan. "Simple Modules over Lie Algebras." Doctoral thesis, Uppsala universitet, Algebra och geometri, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-283061.
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Simple modules are the elemental components in representation theory for Lie algebras, and numerous mathematicians have worked on their construction and classification over the last century. This thesis consists of an introduction together with four research articles on the subject of simple Lie algebra modules. In the introduction we give a light treatment of the basic structure theory for simple finite dimensional complex Lie algebras and their representations. In particular we give a brief overview of the most well-known classes of Lie algebra modules: highest weight modules, cuspidal modules, Gelfand-Zetlin modules, Whittaker modules, and parabolically induced modules. The four papers contribute to the subject by construction and classification of new classes of Lie algebra modules. The first two papers focus on U(h)-free modules of rank 1 i.e. modules which are free of rank 1 when restricted to the enveloping algebra of the Cartan subalgebra. In Paper I we classify all such modules for the special linear Lie algebras sln+1(C), and we determine which of these modules are simple. For sl2 we also obtain some additional results on tensor product decomposition. Paper II uses the theory of coherent families to obtain a similar classification for U(h)-free modules over the symplectic Lie algebras sp2n(C). We also give a proof that U(h)-free modules do not exist for any other simple finite-dimensional algebras which completes the classification. In Paper III we construct a new large family of simple generalized Whittaker modules over the general linear Lie algebra gl2n(C). This family of modules is parametrized by non-singular nxn-matrices which makes it the second largest known family of gl2n-modules after the Gelfand-Zetlin modules. In Paper IV we obtain a new class of sln+2(C)-modules by applying the techniques of parabolic induction to the U(h)-free sln+1-modules we constructed in Paper I. We determine necessary and sufficient conditions for these parabolically induced modules to be simple.
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Saleh, Ibrahim A. "Cluster automorphisms and hyperbolic cluster algebras." Diss., Kansas State University, 2012. http://hdl.handle.net/2097/14195.
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Doctor of PhilosophyDepartment of Mathematics
Zongzhu Lin
Let A[subscript]n(S) be a coefficient free commutative cluster algebra over a field K. A cluster automorphism is an element of Aut.[subscript]KK(t[subscript]1,[dot, dot, dot],t[subscript]n) which leaves the set of all cluster variables, [chi][subscript]s invariant. In Chapter 2, the group of all such automorphisms is studied in terms of the orbits of the symmetric group action on the set of all seeds of the field K(t[subscript]1,[dot,dot, dot],t[subscript]n). In Chapter 3, we set up for a new class of non-commutative algebras that carry a non-commutative cluster structure. This structure is related naturally to some hyperbolic algebras such as, Weyl Algebras, classical and quantized universal enveloping algebras of sl[subscript]2 and the quantum coordinate algebra of SL(2). The cluster structure gives rise to some combinatorial data, called cluster strings, which are used to introduce a class of representations of Weyl algebras. Irreducible and indecomposable representations are also introduced from the same data. The last section of Chapter 3 is devoted to introduce a class of categories that carry a hyperbolic cluster structure. Examples of these categories are the categories of representations of certain algebras such as Weyl algebras, the coordinate algebra of the Lie algebra sl[subscript]2, and the quantum coordinate algebra of SL(2).
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Corwin, Stephen P. "Representation theory of the diagram An over the ring k[[x]]." Diss., Virginia Polytechnic Institute and State University, 1986. http://hdl.handle.net/10919/50001.
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Fix R = k[[x]]. Let Qn be the category whose objects are ((M₁,...,Mn),(f₁,...,fn-1)) where each Mi is a free R-module and fi:Mi⟶Mi+1 for each i=1,...,n-1, and in which the morphisms are the obvious ones. Let βn be the full subcategory of Ωn in which each map fi is a monomorphism whose cokernel is a torsion module. It is shown that there is a full dense functor Ωn⟶βn. If X is an object of βn, we say that X diagonalizes if X is isomorphic to a direct sum of objects ((M₁,...,Mn),(f₁,...,fn-1)) in which each Mi is of rank one. We establish an algorithm which diagonalizes any diagonalizable object X of βn, and which fails only in case X is not diagonalizable.
Let Λ be an artin algebra of finite type. We prove that for a fixed C in mod(Λ) there are only finitely many modules A in mod(Λ) (up to isomorphism) for which a short exact sequence of the form 0⟶A⟶B⟶C⟶0 is indecomposable.Ph. D.
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Pressland, Matthew. "Frobenius categorification of cluster algebras." Thesis, University of Bath, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.678852.
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Cluster categories, introduced by Buan–Marsh–Reineke–Reiten–Todorov and later generalised by Amiot, are certain 2-Calabi–Yau triangulated categories that model the combinatorics of cluster algebras without frozen variables. When frozen variables do occur, it is natural to try to model the cluster combinatorics via a Frobenius category, with the indecomposable projective-injective objects corresponding to these special variables. Amiot–Iyama–Reiten show how Frobenius categories admitting (d-1)-cluster-tilting objects arise naturally from the data of a Noetherian bimodule d-Calabi–Yau algebra A and an idempotent e of A such that A/< e > is finite dimensional. In this work, we observe that this phenomenon still occurs under the weaker assumption that A and A^op are internally d-Calabi–Yau with respect to e; this new definition allows the d-Calabi–Yau property to fail in a way controlled by e. Under either set of assumptions, the algebra B=eAe is Iwanaga–Gorenstein, and eA is a cluster-tilting object in the Frobenius category GP(B) of Gorenstein projective B-modules. Geiß–Leclerc–Schröer define a class of cluster algebras that are, by construction, modelled by certain Frobenius subcategories Sub(Q_J) of module categories over preprojective algebras. Buan–Iyama–Reiten–Smith prove that the endomorphism algebra of a cluster-tilting object in one of these categories is a frozen Jacobian algebra. Following Keller–Reiten, we observe that such algebras are internally 3-Calabi–Yau with respect to the idempotent corresponding to the frozen vertices, thus obtaining a large class of examples of such algebras. Geiß–Leclerc–Schröer also attach, via an algebraic homogenization procedure, a second cluster algebra to each category Sub(Q_J), by adding more frozen variables. We describe how to compute the quiver of a seed in this cluster algebra via approximation theory in the category Sub(Q_J); our alternative construction has the advantage that arrows between the frozen vertices appear naturally. We write down a potential on this enlarged quiver, and conjecture that the resulting frozen Jacobian algebra A and its opposite are internally 3-Calabi–Yau. If true, the algebra may be realised as the endomorphism algebra of a cluster-tilting object in a Frobenius category GP(B) as above. We further conjecture that GP(B) is stably 2-Calabi–Yau, in which case it would provide a categorification of this second cluster algebra.
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Wackwitz, Daniel Joseph. "Versal deformation rings of modules over Brauer tree algebras." Diss., University of Iowa, 2015. https://ir.uiowa.edu/etd/1926.
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This thesis applies methods from the representation theory of finite dimensional algebras, specifically Brauer tree algebras, to the study of versal deformation rings of modules for these algebras. The main motivation for studying Brauer tree algebras is that they generalize p-modular blocks of group rings with cyclic defect groups.
We consider the case when Λ is a Brauer tree algebra over an algebraically closed field K and determine the structure of the versal deformation ring R(Λ,V) of every indecomposable Λ-module V when the Brauer tree is a star whose exceptional vertex is at the center. The ring R(Λ,V) is a complete local commutative Noetherian K-algebra with residue field K, which can be expressed as a quotient ring of a power series algebra over K in finitely many commuting variables. The defining property of R(Λ,V) is that the isomorphism class of every lift of V over a complete local commutative Noetherian K-algebra R with residue field K arises from a local ring homomorphism α : R(Λ, V )→R and that α is unique if R is the ring of dual numbers k[t]/(t2). In the case where Λ is a star Brauer tree algebra and V is an indecomposable Λ-module such that the K-dimension of Ext1Λ(V,V) is equal to R, we explicitly determine generators of an ideal J of k[[t1,....,tr]] such that R(Λ,V)≅k[[t1,....,tr]]/J.
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Muth, Robert. "Representations of Khovanov-Lauda-Rouquier algebras of affine Lie type." Thesis, University of Oregon, 2016. http://hdl.handle.net/1794/20432.
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We study representations of Khovanov-Lauda-Rouquier (KLR) algebras of affine Lie type. Associated to every convex preorder on the set of positive roots is a system of cuspidal modules for the KLR algebra. For a balanced order, we study imaginary semicuspidal modules by means of `imaginary Schur-Weyl duality'. We then generalize this theory from balanced to arbitrary convex preorders for affine ADE types. Under the assumption that the characteristic of the ground field is greater than some explicit bound, we prove that KLR algebras are properly stratified. We introduce affine zigzag algebras and prove that these are Morita equivalent to arbitrary imaginary strata if the characteristic of the ground field is greater than the bound mentioned above. Finally, working in finite or affine affine type A, we show that skew Specht modules may be defined over the KLR algebra, and real cuspidal modules associated to a balanced convex preorder are skew Specht modules for certain explicit hook shapes.
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Brown, Benjamin Charles. "Some problems in the representation theory of hyperoctahedral groups and related algebras." Thesis, Queen Mary, University of London, 2003. http://qmro.qmul.ac.uk/xmlui/handle/123456789/1819.
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We begin by using a version of Green correspondence due to Grabmeier to count the number of components of two permutation modules V®r and y®r for the hyperoctahedral group. We quantize these actions to make v®r and y®r into modules for the type B Hecke algebra 1i(r) and then show that, as 1i(r)-modules, y®r is isomorphic to a direct sum of permutation modules MA as given by Du and Scott. This enables us to use our earlier results to show that in the group case, over odd characteristic, the q-Schur2 algebra and the hyperoctahedral Schur algebra are Morita equivalent, as these algebras are respectively the centralizing algebras of the actions of the hyperoctahedral group on y®r and v®r. We then attempt to construct a bialgebra, the dual of whose rth homogeneous part is isomorphic to the q-Schur2 algebra. We show that this is not possible by the usual methods unless q = 1, and give a full description in the group case. Results of earlier chapters lead us to introduce the notion of a balanced Mackey system for a finite group G, and exhibit balanced Mackey systems for wreath products of H and the symmetric group, where H is any finite group, and a new balanced Mackey system for the symmetric group itself. We then use this as a basis for counting the number of simple modules for the partition algebra, and also derive a formula for the dimensions of these simple modules. In the final chapter we conjecture how some of our results may extend to complex reflection groups and Ariki-Koike algebras.
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Burt, William Leighton. "Homological theory of bocs representations." Thesis, University of Liverpool, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.316574.
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Miller, Janice E. "Representation theory, Borel cross-sections, and minimal measures." Diss., Virginia Tech, 1993. http://hdl.handle.net/10919/38643.
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Genra, Naoki. "Free fields realizations of W-algebras and Applications." Kyoto University, 2019. http://hdl.handle.net/2433/242577.
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Gellert, Florian Verfasser], and Henning [Akademischer Betreuer] [Krause. "Sequential structures in cluster algebras and representation theory / Florian Gellert ; Betreuer: Henning Krause." Bielefeld : Universitätsbibliothek Bielefeld, 2017. http://d-nb.info/1135724628/34.
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Sistko, Alexander Harris. "Maximal subalgebras of finite-dimensional algebras: with connections to representation theory and geometry." Diss., University of Iowa, 2019. https://ir.uiowa.edu/etd/6857.
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Let $k$ be a field and $B$ a finite-dimensional, associative, unital $k$-algebra. For each $1 \le d \le \dim_kB$, let $\operatorname{AlgGr}_d(B)$ denote the projective variety of $d$-dimensional subalgebras of $B$, and let $\operatorname{Aut}_k(B)$ denote the automorphism group of $B$. In this thesis, we are primarily concerned with understanding the relationship between $\operatorname{AlgGr}_d(B)$, the representation theory of $B$, and the representation theory of $\operatorname{Aut}_k(B)$. We begin by proving fundamental structure theorems for the maximal subalgebras of $B$. We show that maximal subalgebras of $B$ come in two flavors, which we call split type and separable type. As a consequence, we provide complete classifications for maximal subalgebras of semisimple algebras and basic algebras. We also demonstrate that the maximality of $A$ in $B$ is related to the representation theory of $B$, through the separability of functors closely associated with the extension $A \subset B$.
The rest of this document showcases applications of these results. For $k = \bar{k}$, we compute the maximal dimension of a proper subalgebra of $B$. We discuss the problem of computing the minimal number of generators for $B$ (as an algebra), and provide upper and lower bounds for basic algebras. We then study $\operatorname{AlgGr}_d(B)$ in detail, again when $B$ is basic. When $d = \dim_kB-1$, we find a projective embedding of $\operatorname{AlgGr}_d(B)$, and explicitly describe its associated homogeneous vanishing ideal. In turn, we provide a simple description of its irreducible components. We find equivalent conditions for this variety to be a finite union of $\operatorname{Aut}_k(B)$-orbits, and describe several classes of algebras which satisfy these conditions. Furthermore, we provide an algebraic description for the orbits of connected maximal subalgebras of type-$\mathbb{A}$ path algebras. Finally, we study the fixed-point variety $\operatorname{AlgGr}_d(B)^{\operatorname{Aut}_k(B)}$ (for general $d$), which connects naturally to the representation theory of $\operatorname{Aut}_k(B)$. We investigate the case where $B$ is a truncated path algebra over $\mathbb{C}$ in detail.
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Li, Zimu. "Fast Matrix Multiplication by Group Algebras." Digital WPI, 2018. https://digitalcommons.wpi.edu/etd-theses/131.
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Based on Cohn and Umans’ group-theoretic method, we embed matrix multiplication into several group algebras, including those of cyclic groups, dihedral groups, special linear groups and Frobenius groups. We prove that SL2(Fp) and PSL2(Fp) can realize the matrix tensor ⟨p, p, p⟩, i.e. it is possible to encode p × p matrix multiplication in the group algebra of such a group. We also find the lower bound for the order of an abelian group realizing ⟨n, n, n⟩ is n3. For Frobenius groups of the form Cq Cp, where p and q are primes, we find that the smallest admissible value of q must be in the range p4/3 ≤ q ≤ p2 − 2p + 3. We also develop an algorithm to find the smallest q for a given prime p.
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Bastian, Nicholas Lee. "Terwilliger Algebras for Several Finite Groups." BYU ScholarsArchive, 2021. https://scholarsarchive.byu.edu/etd/8897.
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In this thesis, we will explore the structure of Terwilliger algebras over several different types of finite groups. We will begin by discussing what a Schur ring is, as well as providing many different results and examples of them. Following our discussion on Schur rings, we will move onto discussing association schemes as well as their properties. In particular, we will show every Schur ring gives rise to an association scheme. We will then define a Terwilliger algebra for any finite set, as well as discuss basic properties that hold for all Terwilliger algebras. After specializing to the case of Terwilliger algebras resulting from the orbits of a group, we will explore bounds of the dimension of such a Terwilliger algebra. We will also discuss the Wedderburn decomposition of a Terwilliger algebra resulting from the conjugacy classes of a group for any finite abelian group and any dihedral group.
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SILVA, ALDO FERREIRA DA. "RENDALLNULLS THEOREMS AND THE UNIQUE DETERMINATION OF THE INNER PRODUCT IN REPRESENTATION OF ALGEBRAS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2004. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=5207@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROTrata-se de uma exposição do trabalho publicado por Alan D.Rendall intitulado Unique determination of an inner product by adjointness relations in the algebra of quantum observables, onde é demonstrado que dada uma representação de uma *- álgebra em um espaço pré-Hilbert V que é uma *- representação irredutível com relação ao produto interno definido em V podemos garantir, dadas algumas condições técnicas, a unicidade do produto interno a menos de uma constante multiplicativa. É feito um breve estudo sobre representações de álgebras e a Construção de Gelfand- Naymark-Segal é apresentada.
This thesis is concerned with a paper from Alan D. Rendall named Unique determination of an inner product by adjointness relations in the algebra of quantum observables, in which is proved that given a *- representation of a *-algebra in a pre-Hilbert space V irreducible with regard to the inner product defined in V we can assure the uniqueness of the inner product up to a multiplicative constant subject to some technical conditions. We also make a brief study about algebra representations and the Gelfand-Naimark- Segal Construction is presented.
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Boixeda, Alvarez Pablo. "Affine Springer fibers and the representation theory of small quantum groups and related algebras." Thesis, Massachusetts Institute of Technology, 2020. https://hdl.handle.net/1721.1/126920.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020Cataloged from the official PDF of thesis.
Includes bibliographical references (pages 125-128).
This thesis deals with the connections of Geometry and Representation Theory. In particular we study the representation theory of small quantum groups and Frobenius kernels and the geometry of an equivalued affine Springer fiber Fl[subscript ts] for s a regular semisimple element. In Chapter 2 we relate the center of the small quantum group with the cohomology of the above affine Springer fiber. This includes joint work with Bezrukavnikov, Shan and Vaserot. In Chapter 3 we study the geometry of the affine Springer fiber and in particular understand the fixed points of a torus action contained in each component. In Chapter 4 we further have a collection of algebraic results on the representation theory of Frobenius kernels. In particular we state some results pointing towards some construction of certain partial Verma functors and we compute this in the case of SL₂. We also compute the center of Frobenius kernels in the case of SL₂ and state a conjecture on a possible inductive construction of the general center.
by Pablo Boixeda Alvarez.
Ph. D.
Ph.D. Massachusetts Institute of Technology, Department of Mathematics
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Rakotoarisoa, Andriamananjara Tantely. "The Bala-Carter Classification of Nilpotent Orbits of Semisimple Lie Algebras." Thesis, Université d'Ottawa / University of Ottawa, 2017. http://hdl.handle.net/10393/36058.
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Conjugacy classes of nilpotent elements in complex semisimple Lie algebras are classified using the Bala-Carter theory. In this theory, nilpotent orbits in g are parametrized by the conjugacy classes of pairs (l,pl) of Levi subalgebras of g and distinguished parabolic subalgebras of [l,l]. In this thesis we present this theory and use it to give a list of representatives for nilpotent orbits in so(8) and from there we give a partition-type parametrization of them.
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Bogdanic, Dusko. "Graded blocks of group algebras." Thesis, University of Oxford, 2010. http://ora.ox.ac.uk/objects/uuid:faeaaeab-1fe6-46a9-8cbb-f3f633131a73.
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In this thesis we study gradings on blocks of group algebras. The motivation to study gradings on blocks of group algebras and their transfer via derived and stable equivalences originates from some of the most important open conjectures in representation theory, such as Broue’s abelian defect group conjecture. This conjecture predicts the existence of derived equivalences between categories of modules. Some attempts to prove Broue’s conjecture by lifting stable equivalences to derived equivalences highlight the importance of understanding the connection between transferring gradings via stable equivalences and transferring gradings via derived equivalences. The main idea that we use is the following. We start with an algebra which can be easily graded, and transfer this grading via derived or stable equivalence to another algebra which is not easily graded. We investigate the properties of the resulting grading. In the first chapter we list the background results that will be used in this thesis. In the second chapter we study gradings on Brauer tree algebras, a class of algebras that contains blocks of group algebras with cyclic defect groups. We show that there is a unique grading up to graded Morita equivalence and rescaling on an arbitrary basic Brauer tree algebra. The third chapter is devoted to the study of gradings on tame blocks of group algebras. We study extensively the class of blocks with dihedral defect groups. We investigate the existence, positivity and tightness of gradings, and we classify all gradings on these blocks up to graded Morita equivalence. The last chapter deals with the problem of transferring gradings via stable equivalences between blocks of group algebras. We demonstrate on three examples how such a transfer via stable equivalences is achieved between Brauer correspondents, where the group in question is a TI group.
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Diemer, Tammo. "Conformal geometry, representation theory and linear fields." Bonn : Mathematisches Institut der Universität, 2004. http://catalog.hathitrust.org/api/volumes/oclc/62770144.html.
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Callegaro, Filippo. "Cohomology of finite and affine type Artin groups over Abelian representation /." Pisa, Italy : Edizioni della normale, 2009. http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&doc_number=017728632&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA.
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Cliff, Emily Rose. "Universal D-modules, and factorisation structures on Hilbert schemes of points." Thesis, University of Oxford, 2015. https://ora.ox.ac.uk/objects/uuid:9edee0a0-f30a-4a54-baf5-c833222303ca.
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This thesis concerns the study of chiral algebras over schemes of arbitrary dimension n. In Chapter I, we construct a chiral algebra over each smooth variety X of dimension n. We do this via the Hilbert scheme of points of X, which we use to build a factorisation space over X. Linearising this space produces a factorisation algebra over X, and hence, by Koszul duality, the desired chiral algebra. We begin the chapter with an overview of the theory of factorisation and chiral algebras, before introducing our main constructions. We compute the chiral homology of our factorisation algebra, and show that the D-modules underlying the corresponding chiral algebras form a universal D-module of dimension n. In Chapter II, we discuss the theory of universal D-modules and OO- modules more generally. We show that universal modules are equivalent to sheaves on certain stacks of étale germs of n-dimensional varieties. Furthermore, we identify these stacks with the classifying stacks of groups of automorphisms of the n-dimensional disc, and hence obtain an equivalence between the categories of universal modules and the representation categories of these groups. We also define categories of convergent universal modules and study them from the perspectives of the stacks of étale germs and the representation theory of the automorphism groups.