Selecta Mathematica

, Volume 24, Issue 5, pp 4617–4657 | Cite as

A Lax type operator for quantum finite W-algebras

  • Alberto De SoleEmail author
  • Victor G. Kac
  • Daniele Valeri


For a reductive Lie algebra \(\mathfrak {g}\), its nilpotent element f and its faithful finite dimensional representation, we construct a Lax operator L(z) with coefficients in the quantum finite W-algebra \(W(\mathfrak {g},f)\). We show that for the classical linear Lie algebras \(\mathfrak {gl}_N\), \(\mathfrak {sl}_N\), \(\mathfrak {so}_N\) and \(\mathfrak {sp}_N\), the operator L(z) satisfies a generalized Yangian identity. The operator L(z) is a quantum finite analogue of the operator of generalized Adler type which we recently introduced in the classical affine setup. As in the latter case, L(z) is obtained as a generalized quasideterminant.


Quantum finite \(\mathcal {W}\)-algebra Generalized quasideterminant Twisted Yangians Operators of twisted Yangian type Kazhdan filtration Rees algebra 

Mathematics Subject Classification

17B08 17B63 17B35 17B80 


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We are grateful to the referee for careful reading of the paper, for several corrections and various enlightening observations. The first author would like to acknowledge the hospitality of MIT, where he was hosted during the spring semester of 2017, when this project started. The second author would like to acknowledge the hospitality of the University of Rome La Sapienza during his visit in Rome in January 2017. The third author is grateful to the University of Rome La Sapienza for its hospitality during his several visits in 2016 and 2017. All three authors are extremely grateful to IHES for their kind hospitality during the summer of 2017, when the paper was completed. The first author is supported by National FIRB Grant RBFR12RA9W, National PRIN Grant 2015ZWST2C, and University Grant C26A158K8A, the second author was supported by an NSF grant.


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaSapienza Università di RomaRomeItaly
  2. 2.Department of MathematicsMITCambridgeUSA
  3. 3.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

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