Classical W-algebras for Centralizers

• Published in: Communications in mathematical physics Vol. 378; no. 1; pp. 691 - 703
• Main Authors: Molev, A. I., Ragoucy, E.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-08-2020; Springer Nature B.V
• Subjects: Algebra; Classical and Quantum Gravitation; Complex Systems; Mathematical analysis; Mathematical and Computational Physics; Mathematical Physics; Oil field equipment; Physics; Physics and Astronomy; Polynomials; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-020-03822-0

We introduce a new family of Poisson vertex algebras
W
(
a
)
analogous to the classical
W
-algebras. The algebra
W
(
a
)
is associated with the centralizer
a
of an arbitrary nilpotent element in
gl
N
. We show that
W
(
a
)
is an algebra of polynomials in infinitely many variables and produce its free generators in an explicit form. This implies that
W
(
a
)
is isomorphic to the center at the critical level of the affine vertex algebra associated with
a
.