On Vertex Operator Algebras of Affine Type at Admissible Levels
Author
Summary, in English
We tackle the problem of describing singular vectors in Verma modules for affine Lie algebras by providing a novel way of realizing the ideas presented in an article by F. G. Malikov, B. L. Feigin and D. B. Fuchs. Our approach is based on the rigorous construction of a broader algebraic framework by means of Ore localization in the universal enveloping algebra and via the introduction of certain conjugation automorphisms. We are able to express operators corresponding to those of Malikov et al. and to partially extend to our setting their main result regarding whether or not these operators represent elements of the enveloping algebra.
Using this knowledge about singular vectors we deal with the problem of finding the irreducible modules in the category O for VOAs of affine type, when the level is admissible. Applying the theory of Zhu's algebra, the highest weights of these modules are characterized as the zeros of a polynomial ideal determined by the single singular vector generating the maximal proper submodule of the generalized Verma module. For the VOA associated to affine sl(3, C) at level -3/2, we prove that these highest weights are precisely the four admissible weights of level -3/2, and moreover that any module in the category O for this VOA is completely reducible. We also show that there are no nontrivial intertwining operators between these irreducible modules, except those deriving from the module structures. Furthermore, we demonstrate how the Sapovalov form can be employed to gain insight into the polynomial ideal, if merely the weight of the corresponding singular vector is known.
Department/s
Publishing year
2013
Language
English
Publication/Series
Doctoral Theses in Mathematical Sciences
Full text
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Document type
Dissertation
Publisher
Centre for Mathematical Sciences, Lund University
Topic
- Mathematics
Keywords
- Vertex operator algebras
- Kac-Moody algebras
- Ore localization
- Admissible levels
- Singular vectors
- Intertwining operators
Status
Published
Research group
- Algebra
Supervisor
ISBN/ISSN/Other
- ISSN: 1404-0034
- ISBN: 978-91-7473-417-1
Defence date
25 January 2013
Defence time
13:15
Defence place
Matematikcentrum, Sölvegatan 18, sal MH:C
Opponent
- Drazen Adamovic (Professor)