Classical W-algebras for Centralizers
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 fre...
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| Published in: | Communications in mathematical physics Vol. 378; no. 1; pp. 691 - 703 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01-08-2020
Springer Nature B.V |
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| Abstract | 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
. |
|---|---|
| AbstractList | 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 glN. 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. 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 . |
| Author | Molev, A. I. Ragoucy, E. |
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| Cites_doi | 10.1090/mosc/264 10.1007/s00220-013-1785-z 10.1007/s00220-018-3142-8 10.1007/s00220-014-2239-y 10.1007/BF02105860 10.1017/S1446788708000608 10.1016/j.jalgebra.2006.12.026 10.1142/S0217751X92003781 10.1007/s00220-016-2632-9 |
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| Copyright | Springer-Verlag GmbH Germany, part of Springer Nature 2020 Springer-Verlag GmbH Germany, part of Springer Nature 2020. |
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| DOI | 10.1007/s00220-020-03822-0 |
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| References | De Sole, Kac, Valeri (CR4) 2016; 348 CR10 Panyushev, Premet, Yakimova (CR12) 2007; 313 Brown, Brundan (CR2) 2009; 86 Feigin, Frenkel (CR7) 1992; 7 Arakawa, Premet (CR1) 2017; 78 De Sole, Kac, Valeri (CR5) 2018; 360 Frenkel (CR8) 2007 Drinfeld, Sokolov (CR6) 1985; 30 CR9 De Sole, Kac, Valeri (CR3) 2013; 323 Molev, Ragoucy (CR11) 2015; 336 |
| References_xml | – volume: 78 start-page: 217 year: 2017 end-page: 234 ident: CR1 article-title: Quantizing Mishchenko-Fomenko subalgebras for centralizers via affine -algebras publication-title: Trans. Mosc. Math. Soc. doi: 10.1090/mosc/264 contributor: fullname: Premet – volume: 323 start-page: 663 year: 2013 end-page: 711 ident: CR3 article-title: Classical -algebras and generalized Drinfeld-Sokolov bi–Hamiltonian systems within the theory of Poisson vertex algebras publication-title: Commun. Math. Phys. doi: 10.1007/s00220-013-1785-z contributor: fullname: Valeri – ident: CR9 – volume: 360 start-page: 851 year: 2018 end-page: 918 ident: CR5 article-title: Classical affine -algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras publication-title: Comm. Math. Phys. doi: 10.1007/s00220-018-3142-8 contributor: fullname: Valeri – year: 2007 ident: CR8 publication-title: Langlands Correspondence for Loop Groups, Cambridge Studies in Advanced Mathematics contributor: fullname: Frenkel – volume: 336 start-page: 1053 year: 2015 end-page: 1084 ident: CR11 article-title: Classical W-algebras in types and publication-title: Commun. Math. Phys. doi: 10.1007/s00220-014-2239-y contributor: fullname: Ragoucy – volume: 30 start-page: 1975 year: 1985 end-page: 2036 ident: CR6 article-title: Lie algebras and equations of Korteweg-de Vries type publication-title: J. Sov. Math. doi: 10.1007/BF02105860 contributor: fullname: Sokolov – volume: 86 start-page: 1 year: 2009 end-page: 15 ident: CR2 article-title: Elementary invariants for centralizers of nilpotent matrices publication-title: J. Aust. Math. Soc. doi: 10.1017/S1446788708000608 contributor: fullname: Brundan – volume: 313 start-page: 343 year: 2007 end-page: 391 ident: CR12 article-title: On symmetric invariants of centralisers in reductive Lie algebras publication-title: J. Algebra doi: 10.1016/j.jalgebra.2006.12.026 contributor: fullname: Yakimova – volume: 7 start-page: 197 issue: Suppl. 1A year: 1992 end-page: 215 ident: CR7 article-title: Affine Kac-Moody algebras at the critical level and Gelfand–Dikii algebras publication-title: Int. J. Mod. Phys. A doi: 10.1142/S0217751X92003781 contributor: fullname: Frenkel – volume: 348 start-page: 265 year: 2016 end-page: 319 ident: CR4 article-title: Classical -algebras for and associated integrable Hamiltonian hierarchies publication-title: Commun. Math. Phys. doi: 10.1007/s00220-016-2632-9 contributor: fullname: Valeri – ident: CR10 |
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| Snippet | 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... 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... |
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| SubjectTerms | 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 |
| Title | Classical W-algebras for Centralizers |
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