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Structure of classical affine and classical affine fractional

-algebras

### Abstract

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

© 2015 AIP Publishing LLC

Received 23 February 2014
Accepted 07 January 2015
Published online 29 January 2015

Acknowledgments:
I would like to thank my Ph.D. thesis advisor, Victor Kac, for valuable discussions.

Article outline:

I. INTRODUCTION
II. POISSON VERTEX ALGEBRAS AND INTEGRABLE SYSTEMS
A. Lie conformal superalgebras and Poisson vertex algebras
B. Integrable systems
III. TWO EQUIVALENT DEFINITIONS OF CLASSICAL AFFINE -ALGEBRAS
A. The first definition of classical affine -algebras
B. The second definition of classical affine -algebras
C. Equivalence of the two definitions of classical affine -algebras
IV. TWO EQUIVALENT DEFINITIONS OF CLASSICAL AFFINE FRACTIONAL -ALGEBRAS
A. First definition of classical affine fractional -algebras
B. Second definition of classical affine fractional -algebras
V. GENERATING ELEMENTS OF A CLASSICAL AFFINE FRACTIONAL -ALGEBRA AND POISSON *λ*-BRACKETS BETWEEN THEM
A. Generating elements of classical affine fractional -algebras
B. Examples
C. More results on generating elements of a classical affine fractional -algebra associated to a minimal nilpotent
VI. INTEGRABLE SYSTEMS RELATED TO CLASSICAL AFFINE FRACTIONAL -ALGEBRAS

/content/aip/journal/jmp/56/1/10.1063/1.4906144

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2015-01-29

2016-10-27

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