## Date of Award

6-15-2024

## Document Type

Dissertation

## Degree Name

Ph.D.

## Organizational Unit

College of Natural Science and Mathematics, Mathematics

## First Advisor

Andrew R. Linshaw

## Second Advisor

Florencia Orosz

## Third Advisor

Shashank Kanade

## Fourth Advisor

Schuyler van Engelenburg

## Keywords

Vertex algebras, Mathematics

## Abstract

The universal 2-parameter vertex algebra *W*_{∞} of type *W*(2, 3, 4; . . . ) serves as a classifying object for vertex algebras of type *W*(2, 3, . . . ,*N*) for some *N* in the sense that under mild hypothesis, all such vertex algebras arise as quotients of *W*_{∞}. There is an ℕ X ℕ family of such 1-parameter vertex algebras known as *Y*-algebras. They were introduced by Gaiotto and Rapčák are expected to be building blocks for all *W*-algebras in type* A*, i.e, every *W*-(super) algebra in type *A* is an extension of a tensor product of finitely many *Y*-algebras. Similarly, the orthosymplectic *Y*-algebras are 1-parameter quotients of a universal 2-parameter vertex algebra of type *W*(2, 4, 6, . . . ), which is a classifying object for vertex algebras of type *W*(2, 4, . . . , 2*N*) for some *N*. Unlike type *A*, these algebras are not all the building blocks for *W*-algebras of types *B*, *C*, and *D*. In this thesis, we construct a new universal 2-parameter vertex algebra of type *W*(1^{3}, 2, 3^{3}, 4, 5^{3}, 6, . . . ) which we denote by *W _{∞}*

^{sp}since it contains a copy of the affine vertex algebra

*V*(sp

^{k}_{2}). We identify 8 infinite families of 1-parameter quotients

*W*

_{∞}^{sp}which are analogues of the

*Y*-algebras, and 4 infinite families with sp

_{2}-level constant. We regard

*W*

_{∞}^{sp}as a fundamental object on equal footing with

*W*

_{∞}and

*W*

_{∞}^{ev}, and we give some heuristic reasons for why we expect the 1-parameter quotients of these three objects to be the building blocks for all

*W*-algebras in classical types.

## Copyright Date

6-2024

## Copyright Statement / License for Reuse

All Rights Reserved.

## Publication Statement

Copyright is held by the author. User is responsible for all copyright compliance.

## Rights Holder

Vladimir Kovalchuk

## Provenance

Received from ProQuest

## File Format

application/pdf

## Language

English (eng)

## Extent

135 pgs

## File Size

545 KB

## Recommended Citation

Kovalchuk, Vladimir, "Building Blocks for W-Algebras of Classical Types" (2024). *Electronic Theses and Dissertations*. 2405.

https://digitalcommons.du.edu/etd/2405