# Representations of certain subalgebras of the vertex algebra $W_{1+\infty}$

Polić, Marijan (2015) Representations of certain subalgebras of the vertex algebra $W_{1+\infty}$. Doctoral thesis, Faculty of Science > Department of Mathematics.

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## Abstract

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 $W_\infty$.

Item Type: Thesis (Doctoral thesis) Adamović, Dražen and Perše, Ozren 2015 80 NATURAL SCIENCES > MathematicsNATURAL SCIENCES > Mathematics > Algebra Faculty of Science > Department of Mathematics Iva Prah 02 Jul 2015 09:24 02 Jul 2015 09:24 http://digre.pmf.unizg.hr/id/eprint/4070