188
D. Valeri
Classical Finite W -Algebras
The classical finite W -algebra W fin (g, f ) is a PA, which can be viewed as the
algebra of functions on the so-called Slodowy slice S(g, f ), introduced by Slodowy
while studying the singularities associated to the coadjoint nilpotent orbits of g [49].
Finite W -Algebras
The first appearance of the finite W -algebras W fin (g, f ) was in a paper of Kostant
[39]. He constructed the finite W -algebra for principal nilpotent f ∈ g (in which
case it is commutative), and proved that it is isomorphic to the center of the universal
enveloping algebra U(g). The construction was then extended in [41] for even
nilpotent element f ∈ g. The general definition of finite W -algebras W fin (g, f ),
for an arbitrary nilpotent element f ∈ g, appeared later in a paper by Premet [46].
Finite W -algebras have deep connection with geometry and representation theory
of simple finite-dimensional Lie algebras, with the theory of primitive ideals, and
the Yangians, see [9, 43, 46, 47].
Classical Affine W -Algebras
The classical affine W -algebras W aff (g, f ) were introduced, for principal nilpotent
element f , in the seminal paper of Drinfeld and Sokolov [24]. They were introduced
as Poisson algebras of functions on an infinite dimensional Poisson manifold, and
they were used to study KdV-type integrable bi-Hamiltonian hierarchies of PDE’s,
nowadays known as Drinfeld–Sokolov hierarchies. Later, there have been several
papers aimed at the construction of generalized Drinfeld–Sokolov hierarchies [10,
12, 23, 26, 29, 30]. In [15], the classical W -algebras W aff (g, f ) were described as
PVA, and the theory of generalized Drinfeld–Sokolov hierarchies was formalized in
a more rigorous and complete way [16, 18, 20].
Quantum Affine W -Algebras
They have been extensively discussed in the Introduction. A review of the subject
up to the early 1990s may be found in the collection of a large number of reprints
on W -algebras [7]. Recently, it has been shown that they are at the base of an
unexpected connections of vertex algebras with the geometric invariants called the
Higgs branches in the four-dimensional N = 2 superconformal field theories [2, 5].
D. Valeri
Classical Finite W -Algebras
The classical finite W -algebra W fin (g, f ) is a PA, which can be viewed as the
algebra of functions on the so-called Slodowy slice S(g, f ), introduced by Slodowy
while studying the singularities associated to the coadjoint nilpotent orbits of g [49].
Finite W -Algebras
The first appearance of the finite W -algebras W fin (g, f ) was in a paper of Kostant
[39]. He constructed the finite W -algebra for principal nilpotent f ∈ g (in which
case it is commutative), and proved that it is isomorphic to the center of the universal
enveloping algebra U(g). The construction was then extended in [41] for even
nilpotent element f ∈ g. The general definition of finite W -algebras W fin (g, f ),
for an arbitrary nilpotent element f ∈ g, appeared later in a paper by Premet [46].
Finite W -algebras have deep connection with geometry and representation theory
of simple finite-dimensional Lie algebras, with the theory of primitive ideals, and
the Yangians, see [9, 43, 46, 47].
Classical Affine W -Algebras
The classical affine W -algebras W aff (g, f ) were introduced, for principal nilpotent
element f , in the seminal paper of Drinfeld and Sokolov [24]. They were introduced
as Poisson algebras of functions on an infinite dimensional Poisson manifold, and
they were used to study KdV-type integrable bi-Hamiltonian hierarchies of PDE’s,
nowadays known as Drinfeld–Sokolov hierarchies. Later, there have been several
papers aimed at the construction of generalized Drinfeld–Sokolov hierarchies [10,
12, 23, 26, 29, 30]. In [15], the classical W -algebras W aff (g, f ) were described as
PVA, and the theory of generalized Drinfeld–Sokolov hierarchies was formalized in
a more rigorous and complete way [16, 18, 20].
Quantum Affine W -Algebras
They have been extensively discussed in the Introduction. A review of the subject
up to the early 1990s may be found in the collection of a large number of reprints
on W -algebras [7]. Recently, it has been shown that they are at the base of an
unexpected connections of vertex algebras with the geometric invariants called the
Higgs branches in the four-dimensional N = 2 superconformal field theories [2, 5].