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.

Khovanov-Lauda-Rouquier algebras, or KLR algebras, are a family of algebras known to categorify the upper half of the quantized enveloping algebra of a given Lie algebra. In finite type, these algebras come with a family of standard modules, which correspond to PBW bases under this categorification. In this thesis, we show that there are no non-zero homomorphisms between distinct standard modules and that all non-zero endomorphisms of standard modules are injective. We then apply this result to obtain applications to the modular representation theory of KLR algebras. Restricting our attention to finite type A, we are then able to compute explicit projective resolutions of all standard modules. Finally, in finite type A when alpha is a positive root, we let D be the direct sum of all distinct standard modules and compute the algebra structure on Ext(D, D). This dissertation includes unpublished co-authored material.

This thesis consists of two parts. In the first we prove that the Khovanov-Lauda-Rouquier algebras $R_\alpha$ of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in $R_\alpha$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\alpha$ is finite. In the second part we use the presentation of the Specht modules given by Kleshchev-Mathas-Ram to derive results about Specht modules. In particular, we determine all homomorphisms from an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook Specht module. This work generalizes a result of James. This dissertation includes previously published coauthored material.

Nėra abejonių, kad Kinijos Liaudies Respublika šiuo metu atlieka svarbų vaidmenį pasaulio politikoje, didelės įtakos turi ir tai, kas vyksta šalies viduje: jos politiniai, ekonominiai bei socialiniai pokyčiai. Paskutinius dešimt metų Kinija perėjo nuo planinės ekonomikos prie rinkos ekonomikos. Kinija susiduria su daugybę iššūkių ir problemų, kurios susijusios tiek su vidaus problemomis, tiek su spaudimu iš išorės. Kinija teigia, kad ji nesiekia dominuoti pasaulio arenoje, o tik siekia „taikiai vystytis“, t. y. pasisako prieš perdėtą kišimąsi į kitų valstybių reikalus. Darbo problematika susijusi su vienu pagrindinių dabartinės Kinijos Liaudies Respublikos politikos devizu: „harmoninga visuomenė“ (hexie shehui) ir jos atsiradimo prielaidomis. Darbo tikslas išanalizuot socialines, politines visuomenėje, kylančio nepasitenkinimo priežastis, kokių priemonių imasi KLR valdžia. Dauguma Kinijos intelektualų sutinka, kad vienas esminių valstybės politikos tikslų yra sukurti stabilią – „harmoningą visuomenę“, jis būtinas sėkmingam tolesniam šalies vystymuisi. Aptariamos istorinis politinės minties raidos kontekstas, turėjęs įtakos „harmoningos visuomenės“ idėjos susiformavimui. Šalies modernizacijos laikotarpis nuo 1979 m., kuris turėjo didelės įtakos dabartinei situacijai: socialinė politinė įtampa visuomenėje tapo sunkiai suvaldomi, vis didėja skirtumai tarp miesto ir kaimo, tarp turtingųjų ir vargšų. Dabartiniai Partijų lyderiai skiria daug dėmesio naujų politinių strategijų... [toliau žr. visą tekstą] / There is no doubt that the People's Republic of China policy now plays an important role in the world stage, and has a significant impact on what is happening in the country: the political, economic and social change. In the past ten years, China has experienced a rapid period of transition from a planned to a market economy. China is facing a number of crucial challenges and problems. Problems are related to internal as well as external pressures and processes. The task of study is to analyze the potential economic, social and political causes of the growing public discontent, and the measures taken by the government of PRC. The idea of „harmonious society“ – is the socio-political goal, which is transforming into an ideology, for the successful development of the country. China claims that it does not seek to dominate the world stage but only seeks to „peaceful development“, maintains stable relationship, not only domestically, but also with its neighbours. Current party leaders have a high attention to the new political challenges and strategies for development. „Harmonious society“ idea of an attempt to justify the need to implement the fairer social policies, the harmonization of central and local government relations, to resolve conflicts arising among the various interest groups, not only for the fair distribution of material but also for the spiritual values.

Les algèbres de Temperley–Lieb de type B, aussi appelées algèbres de Temperley–Lieb à une frontière, sont une famille d’algèbres associatives unitaires de dimension finie généralisant les algèbres de Temperley–Lieb. Elles ont été introduites en 1992 par P.P. Martin et H. Saleur pour la résolution de modèles en mécanique statistique [MS94], mais elles ont rapidement pris de l’importance en théorie de la représentation suite aux travaux de P.P. Martin et D. Woodcock [MW00] [MW03], qui montrent qu’elles s’obtiennent comme quotient d’al- gèbres de Hecke cyclotomiques et qui observent des liens profonds avec la théorie de Lie. Ces quotients sont liés aux algèbres de Khovanov–Lauda–Rouquier (KLR) par les travaux de Brundan et Kleshchev [BK09]; c’est à l’aide des algèbres KLR et de leur formulation diagrammatique que les résultats de ce mémoire seront obtenus. Elles seront maintenant appelées algèbres blob. Ce mémoire porte sur la théorie de la représentation de certains blocs des algèbres blob. Plus précisément, nous trouvons les carquois et relations décrivant les catégories de modules des blocs réguliers en caractéristique nulle. Les résultats sont obtenus par calcul diagram- matique, en utilisant la base cellulaire construite par Plaza–Ryom-Hansen [PRH14] et les idempotents primitifs de Hazi–Martin–Parker [HMP21]. Structure du mémoire: Le premier chapitre rappelle brièvement les notions algébriques qui seront utilisées. Le deuxième chapitre présente les algèbres blob de façon algébrique et diagrammatique, puis plusieurs résultats connus sur celles-ci. Les troisième et quatrième chapitres contiennent tous les résultats originaux, c’est-à-dire le calcul du carquois et relations pour les blocs réguliers. / The Temperley–Lieb algebras of type B, also known as one-boundary Temperley–Lieb al- gebras, are a family of unitary associative algebras of finite dimension that generalize the Temperley–Lieb algebras. They were introduced in 1992 by P.P Martin and H. Saleur for solving models in statistical mechanics [MS94] but they quickly became important in rep- resentation theory following the work of P.P. Martin and D. Woodcock [MW00] [MW03], who showed that they can be realized as quotients of cyclotomic Hecke algebras and observed deep connections with Lie theory. These quotients are related to Khovanov–Lauda–Rouquier (KLR) algebras through the work of Brundan and Kleshchev [BK09]; it is with the help of KLR algebras and their diagrammatic presentation that the results of this thesis will be obtained. They will now be referred to as blob algebras. This thesis focuses on the representation theory of certain blocks of blob algebras. Specif- ically, we find the quivers and relations describing the module categories of regular blocks in characteristic zero. The results are obtained through diagrammatic calculus, using the cellular basis constructed by Plaza–Ryom-Hansen [PRH14] and the primitive idempotents of Hazi–Martin–Parker [HMP21]. Structure: The first chapter briefly recalls the algebraic concepts that will be used. The second chapter presents blob algebras in both algebraic and diagrammatic ways, along with several known results about them. The third and fourth chapters contain all the original results, namely the calculation of quivers and relations for regular blocks.