Quantization of the shift of argument subalgebras in type A

Futorny, Vyacheslav; Molev, Alexander

doi:10.1016/j.aim.2015.07.038

Quantization of the shift of argument subalgebras in type A

Nov 5, 2015

18 pages

Published in:

Adv.Math. 285 (2015) 1358-1375

Published: Nov 5, 2015

DOI:

10.1016/j.aim.2015.07.038

View in:

AMS MathSciNet

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Abstract: (Elsevier)

Given a simple Lie algebra g and an element μ∈g⁎ , the corresponding shift of argument subalgebra of S(g) is Poisson commutative. In the case where μ is regular, this subalgebra is known to admit a quantization, that is, it can be lifted to a commutative subalgebra of U(g) . We show that if g is of type A , then this property extends to arbitrary μ , thus proving a conjecture of Feigin, Frenkel and Toledano Laredo. The proof relies on an explicit construction of generators of the center of the affine vertex algebra at the critical level.

Poisson Lie bracket
Shift of argument subalgebra
Vinberg problem

References(21)

Figures(0)

[2]

Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution

A.V. Bolsinov

- Math.USSR Izv. 38 (1992) 69

[3]

The index of centralizers of elements of reductive Lie algebras

J.-Y. Charbonnel
,
A. Moreau

- Doc.Math. 15 (2010) 387

[4]

Algebraic properties of Manin matrices 1

A. Chervov
,
G. Falqui
,
V. Rubtsov

- Adv.Appl.Math. 43 (2009) 239

[5]

On higher order Sugawara operators

A.V. Chervov
,
A.I. Molev

[6]

Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence

A. Chervov
(
- Moscow, ITEP
)
,
D. Talalaev
(
- Moscow, ITEP
)

e-Print:
- hep-th/0604128

[7]

Affine Kac–Moody algebras at the critical level and Gelfand–Dikii algebras

B. Feigin
,
E. Frenkel

- Int.J.Mod.Phys.A 7 (1992) 197

[8]

Opers with irregular singularity and spectra of the shift of argument subalgebra

B. Feigin
,
E. Frenkel
,
L. Rybnikov

- Duke Math.J. 155 (2010) 337

[9]

Gaudin models with irregular singularities

- Adv.Math. 223 (2010) 873-948
•
e-Print:
- math/0612798
•
DOI:
- 10.1016/j.aim.2009.09.007

[10]

Langlands Correspondence for Loop Groups, Cambridge Studies in Advanced Mathematics, vol. 103

E. Frenkel

[11]

The quantum MacMahon Master Theorem

S. Garoufalidis
,
T.T.Q. Lê
,
D. Zeilberger

- Proc.Nat.Acad.Sci. 103 (2006) 13928

[12]

Euler equation on finite-dimensional Lie groups

A.S. Mishchenko
,
A.T. Fomenko

- Math.USSR Izv. 12 (1978) 371

[13]

Period Integrals of CY and General Type Complete Intersections

Bong H. Lian
(
- Brandeis U.
)
,
Shing-Tung Yau
(
- Harvard U., Dept. Math.
)

- Invent.Math. 191 (2013) 1
•
e-Print:
- 1105.4872
•
DOI:
- 10.1007/s00222-012-0391-6

[14]

The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for $\widehat{\mathfrak{gl}}_{m|n}$

- Moscow Math.J. 14 (2014) 1, 83-119

[15]

Bethe subalgebras in twisted Yangians

M. Nazarov
,
G. Olshanski

- Commun.Math.Phys. 178 (1996) 483

[16]

The argument shift method and maximal commutative subalgebras of Poisson algebras

D.I. Panyushev
,
O.S. Yakimova

- Math.Res.Lett. 15 (2008) 239

[17]

The shift of invariants method and the Gaudin model

L.G. Rybnikov

- Funct.Anal.Appl. 40 (2006) 188

[18]

The quantum Gaudin system

D.V. Talalaev

- Funct.Anal.Appl. 40 (2006) 73

[19]

On some commutative subalgebras in the universal enveloping algebra of the Lie algebra gl(n,C)

A.A. Tarasov

- Sb.Math. 191 (2000) 1375

[20]

The maximality of some commutative subalgebras in Poisson algebras of semisimple Lie algebras

A.A. Tarasov

- Russ.Math.Surveys 57 (2002) 1013

[21]

Some commutative subalgebras of a universal enveloping algebra

E.B. Vinberg

- Math.USSR Izv. 36 (1991) 1

[22]

The index of centralizers of elements in classical Lie algebras

O.S. Yakimova

- Funct.Anal.Appl. 40 (2006) 42

Quantization of the shift of argument subalgebras in type A

Citations per year

Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution

The index of centralizers of elements of reductive Lie algebras

Algebraic properties of Manin matrices 1

On higher order Sugawara operators

Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence

Affine Kac–Moody algebras at the critical level and Gelfand–Dikii algebras

Opers with irregular singularity and spectra of the shift of argument subalgebra

Gaudin models with irregular singularities

Langlands Correspondence for Loop Groups, Cambridge Studies in Advanced Mathematics, vol. 103

The quantum MacMahon Master Theorem

Euler equation on finite-dimensional Lie groups

Period Integrals of CY and General Type Complete Intersections

The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for glundefinedm∣n\widehat{\mathfrak{gl}}_{m|n}gl​m∣n​

Bethe subalgebras in twisted Yangians

The argument shift method and maximal commutative subalgebras of Poisson algebras

The shift of invariants method and the Gaudin model

The quantum Gaudin system

On some commutative subalgebras in the universal enveloping algebra of the Lie algebra gl(n,C)

The maximality of some commutative subalgebras in Poisson algebras of semisimple Lie algebras

Some commutative subalgebras of a universal enveloping algebra

The index of centralizers of elements in classical Lie algebras

The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for $\widehat{\mathfrak{gl}}_{m|n}$