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 This dissertation includes unpublished co-authored material.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.