Reprezentacije nekih podalgebri verteks algebre \(W_{1+\infty}\)

doctoral thesis

• Title (english): Representations of certain subalgebras of the vertex algebra \(W_{1+\infty}\)
• Author: Marijan Polić
• Mentor: Dražen Adamović (mentor)
• Mentor: Ozren Perše (komentor)
• Committee member: Mirko Primc (predsjednik povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Committee member: Miroslav Jerković (član povjerenstva)
• Committee member: Tomislav Šikić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-06-12, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

U radnji proučavamo strukturnu teoriju i teoriju reprezentacija verteks algebri \(W_{1+\infty}\) i \(W_\infty\) za neke centralne naboje. Dajemo eksplicitnu formulu za familiju singularnih vektora u univerzalnim verteks algebrama \(W_{1+\infty}\) i \(W_\infty\). Dokazano je da je netrivijalni kvocijent verteks algebre \(W_\infty\) za \(c =-2\) izomorfan \(W_{2,3}\) algebri iz članka W. Wanga [36]. U slučaju centralnog naboja \(c =-4\) prosti kvocijent od \(W_\infty\) je izomorfan parafermionskoj verteks algebri pridruženoj afinoj Liejevoj algebri \(A_1^{(1)}\) nivoa -1. U slučaju nekih drugih centralnih naboja određen je minimalan skup generatora kvocijenata univerzalne verteks–algebre za \(W_\infty\). Promatrana je teorija dualnih parova i realizirana je verteks algebra \(W_\infty\) kao podalgebra verteks superalgebre pridružene simplektičkim fermionima. Dokazano je da je verteks superalgebra pridružena simplektičkim fermionima potpuno reducibilna reprezentacija od \(W_\infty\).

Abstract (english)

In this thesis we study structure and representation theory for vertex algebras \(W_{1+\infty}\) and \(W_\infty\) for certain central charges. We give explicit formula for a family of singular vectors in universal vertex algebras \(W_{1+\infty}\) and \(W_\infty\) It is proved that a non-trivial quotient of the vertex algebra \(W_\infty\) with central charge \(c =-2\) is isomorphic to \(W_{2,3}\) algebra from the paper of W. Wang [36]. In the case of central charge \(c =-4\) simple quotient of \(W_\infty\) is isomorphic to the parafermionic vertex algebra associated to the affine Lie algebra \(A_1^{(1)}\) of the level -1. For some other central charges, the minimal set of generators for the quotient of universal vertex algebra of \(W_\infty\) is determined. The theory of dual pairs is used for realization of the vertex algebra \(W_\infty\) as a subalgebra of the vertex superalgebra associated to the symplectic fermions. It is proved that the vertex superalgebra associated to the symplectic fermions is completely reducible representation of \(W_\infty\).

• Keywords: verteks algebra; teorija reprezentacija; centralni naboj; Liejeva algebra; teorija dualnih parova; simplektički fermioni
• Keywords (english): vertex algebra; representation theory; central charge; Lie algebra; theory of dual pairs; symplectic fermions
• Language: croatian
• URN:NBN: urn:nbn:hr:217:209033
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: ii, 80 str.
• File origin: Born digital
• Access conditions: Closed access
• Created on: 2019-03-21 10:56:39