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W-Algebras via Lax Type Operators

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Quantum Theory and Symmetries

Part of the book series: CRM Series in Mathematical Physics ((CRM))

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Abstract

W-algebras are certain algebraic structures associated to a finite-dimensional Lie algebra and a nilpotent element f via Hamiltonian reduction. In this note we give a review of a recent approach to the study of (classical affine and quantum finite) W-algebras based on the notion of Lax type operators.

For a finite-dimensional representation of a Lax type operator for W-algebras is constructed using the theory of generalized quasideterminants. This operator carries several pieces of information about the structure and properties of the W-algebras and shows the deep connection of the theory of W-algebras with Yangians and integrable Hamiltonian hierarchies of Lax type equations.

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Acknowledgements

This review is based on the talk I gave at the XIth International Symposium Quantum Theory and Symmetry in Montréal. I wish to thank the organizers for the invitation and the hospitality.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Glasgow, Glasgow, UK

    Daniele Valeri

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  1. Daniele Valeri
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Correspondence to Daniele Valeri .

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Editors and Affiliations

  1. Département de physique, Université de Montréal, Montréal, QC, Canada

    M. B. Paranjape

  2. Département de physique, Université de Montréal, Montréal, QC, Canada

    Richard MacKenzie

  3. Department of Mathematics and Physics, SUNY Polytechnic Institute, Utica, NY, USA

    Zora Thomova

  4. Department of Mathematics and Statistics, Université de Montréal, Montréal, QC, Canada

    Pavel Winternitz

  5. Département de physique, Université de Montréal, Montréal, QC, Canada

    William Witczak-Krempa

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Valeri, D. (2021). W-Algebras via Lax Type Operators. In: Paranjape, M.B., MacKenzie, R., Thomova, Z., Winternitz, P., Witczak-Krempa, W. (eds) Quantum Theory and Symmetries. CRM Series in Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-55777-5_17

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