We study the quantum finite W-algebras W(gl_N,f), associated to the Lie algebra gl_N, and its arbitrary nilpotent element f. We construct for such an algebra an r_1 x r_1 matrix L(z) of Yangian type, where r_1 is the number of maximal parts of the partition corresponding to f. The matrix L(z) is the quantum finite analogue of the operator of Adler type which we introduced in the classical affine setup. As in the latter case, the matrix L(z) is obtained as a generalized quasideterminant. It should encode the whole structure of W(gl_N,f), including explicit formulas for generators and the commutation relations among them. We describe in all detail the examples of principal, rectangular and minimal nilpotent elements.

Finite W-algebras for gl_N / DE SOLE, Alberto; Victor, G. Kac; Valeri, Daniele. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - ELETTRONICO. - 327:(2018), pp. 173-224. [10.1016/j.aim.2017.06.016]

Finite W-algebras for gl_N

DE SOLE, ALBERTO;Daniele, Valeri
2018

Abstract

We study the quantum finite W-algebras W(gl_N,f), associated to the Lie algebra gl_N, and its arbitrary nilpotent element f. We construct for such an algebra an r_1 x r_1 matrix L(z) of Yangian type, where r_1 is the number of maximal parts of the partition corresponding to f. The matrix L(z) is the quantum finite analogue of the operator of Adler type which we introduced in the classical affine setup. As in the latter case, the matrix L(z) is obtained as a generalized quasideterminant. It should encode the whole structure of W(gl_N,f), including explicit formulas for generators and the commutation relations among them. We describe in all detail the examples of principal, rectangular and minimal nilpotent elements.
2018
Finite w-algebra; quasideterminant; rees algebra; Yangian
01 Pubblicazione su rivista::01a Articolo in rivista
Finite W-algebras for gl_N / DE SOLE, Alberto; Victor, G. Kac; Valeri, Daniele. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - ELETTRONICO. - 327:(2018), pp. 173-224. [10.1016/j.aim.2017.06.016]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/972803
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