W -Algebras via Lax Type Operators
Daniele Valeri
Abstract W -algebras are certain algebraic structures associated to a finitedimensional Lie algebra g 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 g 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.
Keywords W-algebras · Lax type operators · Generalized quasideterminants ·
Integrable Hamiltonian hierarchies · (Twisted) Yangians
1 Introduction
The first quantum affine W -algebra, the so-called Zamolodchikov W 3 -algebra
[50], appeared in the physics literature in the study of 2-dimensional Conformal
Field Theory. Further generalizations of this algebra were provided soon after
[25, 40]. Physicists thought of these algebras as “non-linear” infinite dimensional
Lie algebras extending the Virasoro Lie algebra. In [27] the affine W -algebras
W κ (g, f ) (κ is called the level), for a principal nilpotent element f ∈ g, were
described as vertex algebras obtained via a quantization of the Drinfeld–Sokolov
Hamiltonian reduction, which was used in [24] to construct classical affine W -
algebras. In particular, for sl 2 one gets the Virasoro vertex algebra, and for sl 3 the
Zamolodchikov’s W 3 algebra. The construction was finally generalized to arbitrary
nilpotent element f in [36–38]. In these papers, affine W -algebras were applied to
D. Valeri ()
School of Mathematics and Statistics, University of Glasgow, Glasgow, UK
e-mail: daniele.valeri@glasgow.ac.uk
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_17
181
Daniele Valeri
Abstract W -algebras are certain algebraic structures associated to a finitedimensional Lie algebra g 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 g 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.
Keywords W-algebras · Lax type operators · Generalized quasideterminants ·
Integrable Hamiltonian hierarchies · (Twisted) Yangians
1 Introduction
The first quantum affine W -algebra, the so-called Zamolodchikov W 3 -algebra
[50], appeared in the physics literature in the study of 2-dimensional Conformal
Field Theory. Further generalizations of this algebra were provided soon after
[25, 40]. Physicists thought of these algebras as “non-linear” infinite dimensional
Lie algebras extending the Virasoro Lie algebra. In [27] the affine W -algebras
W κ (g, f ) (κ is called the level), for a principal nilpotent element f ∈ g, were
described as vertex algebras obtained via a quantization of the Drinfeld–Sokolov
Hamiltonian reduction, which was used in [24] to construct classical affine W -
algebras. In particular, for sl 2 one gets the Virasoro vertex algebra, and for sl 3 the
Zamolodchikov’s W 3 algebra. The construction was finally generalized to arbitrary
nilpotent element f in [36–38]. In these papers, affine W -algebras were applied to
D. Valeri ()
School of Mathematics and Statistics, University of Glasgow, Glasgow, UK
e-mail: daniele.valeri@glasgow.ac.uk
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_17
181