Reprezentacije Virasorove algebre

master's thesis

• Author: Ante Čeperić
• Mentor: Dražen Adamović (mentor)
• Committee member: Dražen Adamović (predsjednik povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-07-10, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu proučavamo Virasorovu algebru i njene reprezentacije. Cilj prvog poglavlja je predstaviti osnovne definicije i rezultate teorije Liejevih algebri. Konstruiramo univerzalnu omotačku algebru, te definiramo centralna proširenja Liejeve algebre L i pokazujemo njihovu korespondenciju s određenim bilinearnim funkcijama na L. U drugom poglavlju definiramo tri Liejeve algebre: Wittovu, Heisenbergovu i Virasorovu algebru, te pokazujemo neka njihova svojstva. Karakteriziramo Virasorovu algebru kao univerzalno centralno proširenje Wittove algebre. U trećem poglavlju prvo dokazujemo nekoliko općenitih rezultata vezanih uz reprezentacije Virasorove algebre. Dajemo neke primjere reprezentacija Virasorove algebre te istražujemo njihova svojstva. Potom se fokusiramo na reprezentacije najveće težine. Konstruiramo univerzalnu reprezentacije najveće težine, Vermaov module, te analiziramo njihovu strukturu. Konkretnije, opisujemo kako izgledaju homomorfizmi izmedu Vermaovih modula, te kao korolar toga dobivamo i opis strukture podmodula Vermaovih modula. Četvrto poglavlje je kratko izlaganje teorije verteks algebri. Dajemo primjere verteks algebri koje su izgrađene nad Heisenbergovom i Virasorovom algebrom, te pokazujemo vezu s ranije definiranim reprezentacijama Virasorove algebre.

Abstract (english)

In this work, we study the Virasoro algebra and its representations. The goal of the first chapter is to introduce basic notions from the theory of Lie algebras. We construct the universal enveloping algebra, define central extensions of Lie algebra L and show their correspondence with certain bilinear functions on L. In second chapter three Lie algebras are defined: Witt, Heisenberg and Virasoro algebra, and some of their properties shown. Virasoro algebra is characterized as a universal central extension of Witt algebra. In third chapter we first show a few general results concerning representations of Virasoro algebra. Some examples of Virasoro algebra representations are given, and their properties analysed. From here, the focus is on the highest weight representations. We construct universal highest weight representations, the Verma modules, and analyse their structure. More concretely, homomorphisms between Verma modules are described, and as a corollary we get the description of submodules of Verma modules. Fourth chapter is a short exposition of the theory of vertex algebras. We build vertex algebras over Heisenberg and Virasoro algebra and show their relationship with representations of Virasoro algebra.

• Keywords: Virasorova algebra; Liejeve algebre; Wittova algebra; Heisenbergova algebra
• Keywords (english): Virasoro algebra; Lie algebras; Witt algebra; Heisenberg algebra
• Language: croatian
• URN:NBN: urn:nbn:hr:217:876022
• Study programme: Title: Pure Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-01-07 12:51:51