# Structure of classical affine and classical affine fractional W-algebras

## Abstract

We introduce a classical BRST complex (See Definition 3.2.) and show that one can construct a classical affine W-algebra via the complex. This definition clarifies that classical affine W-algebras can be considered as quasi-classical limits of quantum affine W-algebras. We also give a definition of a classical affine fractional W-algebra as a Poisson vertex algebra. As in the classical affine case, a classical affine fractional W-algebra has two compatible λ-brackets and is isomorphic to an algebra of differential polynomials as a differential algebra. When a classical affine fractional W-algebra is associated to a minimal nilpotent, we describe explicit forms of free generators and compute λ-brackets between them. Provided some assumptions on a classical affine fractional W-algebra, we find an infinite sequence of integrable systems related to the algebra, using the generalized Drinfel’d and Sokolov reduction.

- Authors:

- Department of Mathematical Sciences, Seoul National University, GwanAkRo 1, Gwanak-Gu, Seoul 151-747 (Korea, Republic of)

- Publication Date:

- OSTI Identifier:
- 22403077

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 1; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; INTEGRAL CALCULUS; POLYNOMIALS; REDUCTION

### Citation Formats

```
Suh, Uhi Rinn, E-mail: uhrisu1@math.snu.ac.kr.
```*Structure of classical affine and classical affine fractional W-algebras*. United States: N. p., 2015.
Web. doi:10.1063/1.4906144.

```
Suh, Uhi Rinn, E-mail: uhrisu1@math.snu.ac.kr.
```*Structure of classical affine and classical affine fractional W-algebras*. United States. doi:10.1063/1.4906144.

```
Suh, Uhi Rinn, E-mail: uhrisu1@math.snu.ac.kr. Thu .
"Structure of classical affine and classical affine fractional W-algebras". United States.
doi:10.1063/1.4906144.
```

```
@article{osti_22403077,
```

title = {Structure of classical affine and classical affine fractional W-algebras},

author = {Suh, Uhi Rinn, E-mail: uhrisu1@math.snu.ac.kr},

abstractNote = {We introduce a classical BRST complex (See Definition 3.2.) and show that one can construct a classical affine W-algebra via the complex. This definition clarifies that classical affine W-algebras can be considered as quasi-classical limits of quantum affine W-algebras. We also give a definition of a classical affine fractional W-algebra as a Poisson vertex algebra. As in the classical affine case, a classical affine fractional W-algebra has two compatible λ-brackets and is isomorphic to an algebra of differential polynomials as a differential algebra. When a classical affine fractional W-algebra is associated to a minimal nilpotent, we describe explicit forms of free generators and compute λ-brackets between them. Provided some assumptions on a classical affine fractional W-algebra, we find an infinite sequence of integrable systems related to the algebra, using the generalized Drinfel’d and Sokolov reduction.},

doi = {10.1063/1.4906144},

journal = {Journal of Mathematical Physics},

number = 1,

volume = 56,

place = {United States},

year = {Thu Jan 15 00:00:00 EST 2015},

month = {Thu Jan 15 00:00:00 EST 2015}

}