Title Takiffova verteks-algebra tipa \(A_1^{(1)}\) Title (english) Takiff vertex algebra of type \(A_1^{(1)}\) Author Iva Ćuže Mentor Gordan Radobolja (mentor)Committee member Dražen Adamović (predsjednik povjerenstva)Committee member Ozren Perše (član povjerenstva)Committee member Gordan Radobolja (član povjerenstva)Committee member Slaven Kožić (član povjerenstva)Granter University of Zagreb Defense date and country 2024-08-30, Croatia Scientific / art field, NATURAL SCIENCES Universal decimal classificationUDC ) 51 - Mathematics Abstract U ovoj disertaciji proučavamo Takiffovu verteks-algebru tipa \(A_1^{(1)}\) . Na početku konstruiramo vakuum modul \(V^{k,\overline{k}} (TA_1^{(1)})\) za Takiffovu algebru afine Liejeve algebre tipa \(A^{(1)}_1\), a zatim \(V^{k,\overline{k}} (TA_1^{(1)})\) opskrbljujemo strukturom verteks-algebre i algebre verteks operatora. Motivirani rezultatima o bijektivnoj korespondenciji restringiranih modula za afine Liejeve algebre s modulima za univerzalnu afinu verteks-algebru, dokazujemo da su restringirani moduli za Takiffovu algebru afine Liejeve algebre tipa \(A^{(1)}_1\) ujedno i moduli za verteks-algebru \(V^{k,\overline{k}} (TA_1^{(1)})\). Dodatno, definiramo generalizirani težinski modul, težinski modul i modul najveće težine, kao važne primjere restringiranih modula za Takiffovu algebru afine Liejeve algebre tipa \(A^{(1)}_1\). Algebra verteks operatora \(V^{k,\overline{k}} (TA_1^{(1)})\) je centralnog naboja neovisnog o nivou. Takvo svojstvo imaju algebre verteks operatora pridružene Heisenbergovoj algebri te je dokazano da su sve one međusobno izomorfne za netrivijalne nivoe. U ovoj disertaciji dokazujemo da ista tvrdnja vrijedi za algebre verteks operatora \(V^{k,\overline{k}} (TA_1^{(1)})\) za netrivijalne \(k,\overline{k}\). Nakon konstrukcije neke nove verteks-algebre prirodno je ispitati prostotu. Središnji dio disertacije je posvećen ispitivanju ireducibilnosti vakuum modula \(V^{k,k} (TA_1^{(1)}) : = V^k (TA_1^{(1)})\) iz čega izvodimo zaključak o prostoti. Dokaz se temelji na određivanju Kacove determinante odgovarajuće bilinearne forme, odnosno njenih korijena. Prilikom detaljnijeg proučavanja verteks-algebre od osobite je važnosti poznavanje njezine Zhuove algebre. Stoga određujemo generatore i relacije te identificiramo Zhuovu algebru \(A(V^k (TA_1^{(1)}))\). Na kraju, ispitujemo realizaciju \(W(2,2)\) algebre (ili galilejske konformne algebre) u \(V^k (TA_1^{(1)})\).
Abstract (english) In this thesis, we study the Takiff vertex algebra of type \(A_1^{(1)}\). In the first part of thesis, we construct a vacuum module \(V^{k,\overline{k}} (TA_1^{(1)})\) for the Takiff algebra of the affine Lie algebra of type \(A^{(1)}_1\), and then we endow \(V^{k,\overline{k}} (TA_1^{(1)})\) with the structure of the vertex algebra and the vertex operator algebra. Motivated by the one-to-one correspondence between the restricted modules for affine Lie algebras and modules for the universal affine vertex algebra, we prove that the restricted modules for the Takiff algebra of the affine Lie algebra \(\hat{\mathfrak{sl}}_2\) are exactly modules for the vertex algebra \(V^{k,\overline{k}} (TA_1^{(1)})\). In addition, we define generalized weight module, weight module and highest weght module as important example of restricted modules for the Takiff algebra of the affine Lie algebra of type \(A_1^(1)\). The central charge of vertex operator algebra \(V^{k,\overline{k}} (TA_1^{(1)})\) is level-independent. Vertex operator algebras associated to Heisenberg algebra have this property and all of them are isomorphic for nonzero levels. In this thesis, we prove that this is also true for vertex operator algebras \(V^{k,\overline{k}} (TA_1^{(1)})\)for nonzero complex numbers \(k,\overline{k}\). After construction, it is natural to examine simplicity of vertex algebra. The central part of the dissertation is dedicated to studying the irreducibility of the vacuum module \(V^k (TA_1^{(1)})\) from which we derive the conclusion about simplicity. The proof is based on the calculating roots of the Kac determinant of the corresponding bilinear form. For studying in detail some vertex algebra, it is important to know it’s Zhu algebra. So, we define generators and relations and identify the Zhu algebra \(A(V^k (TA_1^{(1)}))\). Finally, in the last part of thesis, we study realisation of \(W(2,2)\) algebra (or Galilean conformal algebra) in \(V^k (TA_1^{(1)})\).
Keywords
Takiffova algebra
verteks-algebra
algebra verteks operatora
restringirani moduli
Kacova determinanta
Zhuova algebra
Keywords (english)
Takiff algebra
vertex algebra
vertex operator algebra
restricted modules
Kac determinant
Zhu algebra
Language croatian URN:NBN urn:nbn:hr:217:956091 Promotion 2024 Study programme Title: Doctoral study Type of resource Text Extent v, 77 str. File origin Born digital Access conditions Open access Terms of use Created on 2024-10-16 08:47:50
