Reprezentacije nekih iracionalnih W-algebri

doctoral thesis

• Title (english): Representations of certain irrational W-algebras
• Author: Ana Kontrec
• Mentor: Dražen Adamović (mentor)
• Committee member: Ozren Perše (predsjednik povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Tomislav Šikić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-02-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

U disertaciji proučavamo neke iracionalne \(\mathcal{W}\)-algebre i njihove reprezentacije. Glavni primjer koji istražujemo je prosta Bershadsky-Polyakov verteks algebra \(\mathcal{W}_k(=\mathcal{W}_k(sl_3, f_\theta))\), tj. minimalna afina \(\mathcal{W}\)-algebra pridružena \(sl_3\). T. Arakawa je dokazao u članku [9] da je \(\mathcal{W}_k\) racionalna ako je \(k\) polucijeli broj veći od \(-3/2\). Mi proučavamo slučajeve kad je \(\mathcal{W}_k\) iracionalna, te klasificiramo \(\mathcal{W}_k\)-module za neke \(k\). Klasifikacija ireducibilnih modula koristi Zhuovu teoriju i formule za singularne vektore. Istaknimo da je Zhuova algebra realizirana kao kvocijent Smithove algebre iz [44]. Klasifikacija ireducibilnih jakih modula za algebru \(\mathcal{W}_k\) (tj. modula s konačno-dimenzionalnim težinskim potprostorima za operator \(L(0)\)) je zato povezana s klasifikacijom konačno-dimenzionalnih reprezentacija Smithove algebre. U slučaju \(k=-5/3\) pokazujemo da je \(\mathcal{W}_k\) realizirana kao verteks podalgebra Weylove verteks algebre. Dokazujemo da tada \(\mathcal{W}_k\) ima točno 6 ireducibilnih jakih modula. U slučaju \(k=-9/4\), \(\mathcal{W}_k\) je važan primjer logaritamske verteks algebre. Klasificiramo jake \(\mathcal{W}_k\) module za \(k=-9/4\) i dokazujemo da \(\mathcal{W}_k\) ima točno 3 ireducibilna jaka modula. Da bi pokazali da je \(\mathcal{W}_k\) iracionalna, konstruiramo neprebrojivu familiju težinskih modula za \(\mathcal{W}_k\) izvan kategorije \(\mathcal{O}\). Konstruiramo familiju singularnih vektora koja generalizira Arakawine formule za singularne vektore za \(k\) polucijeli broj iz [9]. U slučajevima \(k=-1\) i \(k=0\) klasificiramo sve module u kategoriji \(\mathcal{O}\). U slučaju \(k=0\), dajemo eksplicitnu realizaciju verteks algebre \(\mathcal{W}_k\) i njenih modula kao određene iracionalne podalgebre verteks algebri pridruženih rešetkama. Također proučavamo algebru \(\mathcal{W}_k\) u slučaju kad je \(k\) cijeli broj veći od -1.

Abstract (english)

In this thesis we study certain irrational \(\mathcal{W}\)-algebras and their representations. Main example that we investigate is the simple Bershadsky-Polyakov vertex algebra \(\mathcal{W}_k(=\mathcal{W}_k(sl_3, f_\theta))\), i.e. the minimal affine \(\mathcal{W}\)-algebra associated to \(sl_3\). T. Arakawa proved in [9] that \(\mathcal{W}_k\) is rational if \(k\) is a half integer greater that \(-3/2\). We study cases where \(\mathcal{W}_k\) is irrational, and classify \(\mathcal{W}_k\)-modules for certain \(k\). Main tools that we use to classify irreducible modules are Zhu's theory and formulas for singular vectors. Zhu algebra is realized as a quotient of the Smith algebra from [44]. Classification of irreducible strong modules for \(\mathcal{W}_k\) (i.e. modules with finite dimensional weight subspaces for the the operator \(L(0)\)) is hence connected to the classification of finite dimensional representations of the Smith algebra. In the case \(k=-5/3\), we show that \(\mathcal{W}_k\) is realized as a vertex subalgebra of the Weyl vertex algebra. We prove that in this case \(\mathcal{W}_k\) has exactly 6 irreducible strong modules. In the case \(k=-9/4\) , \(\mathcal{W}_k\) is an important example of a logarithmic vertex algebra. We classify strong \(\mathcal{W}_k\)-modules for \(k=-9/4\) and prove that \(\mathcal{W}_k\) has exactly 3 irreducible strong modules. In order to show that \(\mathcal{W}_k\) is irrational, we construct an uncountable family of weight modules for \(\mathcal{W}_k\) outside of category \(\mathcal{O}\). We construct a family of singular vectors which generalizes Arakawa’s formulas for singular vectors for \(k\) half integer from [9]. For \(k=-1\) and \(k=0\) we classify all modules in the category \(\mathcal{O}\). In the case \(k=0\), we give an explicit realization of the vertex algebra \(\mathcal{W}_k\) and its modules as certain irrational subalgebras of lattice VOAs. We also study the algebra \(\mathcal{W}_k\) when \(k\) is an integer greater than -1.

• Keywords: verteks algebre; afine \(\mathcal{W}\)-algebre; Bershadsky-Polaykov algebra; Smithova algebra; Zhuova algebra
• Keywords (english): vertex algebras; affine \(\mathcal{W}\)-algebras; Bershadsky-Polaykov algebra; Smith algebra; Zhu algebra
• Language: croatian
• URN:NBN: urn:nbn:hr:217:617049
• 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: vi, 91 str.
• File origin: Born digital
• Access conditions: Open access; Embargo expiration date: 2019-09-22
• Created on: 2019-03-22 13:12:26