# Lectures on Wakimoto modules, opers and the center at the critical level

@article{Frenkel2002LecturesOW,
title={Lectures on Wakimoto modules, opers and the center at the critical level},
author={Edward Frenkel},
journal={arXiv: Quantum Algebra},
year={2002}
}
• E. Frenkel
• Published 2 October 2002
• Mathematics
• arXiv: Quantum Algebra
165 Citations

### Geometric realizations of Wakimoto modules at the critical level

• Mathematics
• 2006
We study the Wakimoto modules over the affine Kac-Moody algebras at the critical level from the point of view of the equivalences of categories proposed in our previous works, relating categories of

### Self-extensions of Verma modules and differential forms on opers

• Mathematics
Compositio Mathematica
• 2006
We compute the algebras of self-extensions of the vacuum module and the Verma modules over an affine Kac–Moody algebra $\hat{\mathfrak g}$ in suitable categories of Harish-Chandra modules. We show

### On the Endomorphisms of Weyl Modules over Affine Kac–Moody Algebras at the Critical Level

• Mathematics
• 2009
We present an independent short proof of the recently established result that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of

### Fusion and convolution: applications to affine Kac-Moody algebras at the critical level

• Mathematics
• 2005
Let g be a semi-simple Lie algebra, and let g^ be the corresponding affine Kac-Moody algebra. Consider the category of g^-modules at the critical level, on which the action of the Iwahori subalgebra

### On Higher-Order Sugawara Operators

• Mathematics
• 2009
The higher Sugawara operators acting on the Verma modules over the affine Kac-Moody algebra at the critical level are related to the higher Hamiltonians of the Gaudin model due to the work of Feigin,

### GEOMETRIC REALIZATION OF THE SEGAL-SUGAWARA CONSTRUCTION

• Mathematics
• 2004
We apply the technique of localization for vertex algebras to the Segal- Sugawara construction of an "internal" action of the Virasoro algebra on affine Kac- Moody algebras. The result is a lifting

### Jantzen sum formula for restricted Verma modules over affine Kac-Moody algebras at the critical level

For a restricted Verma module of an affine Kac-Moody algebra at the critical level we describe the Jantzen filtration and give an alternating sum formula which corresponds to the Jantzen sum formula

### Affine opers and conformal affine Toda

• C. Young
• Mathematics
Journal of the London Mathematical Society
• 2021
For g a Kac–Moody algebra of affine type, we show that there is an AutO ‐equivariant identification between FunOpg(D) , the algebra of functions on the space of g ‐opers on the disc, and W⊂π0 , the

### The linkage principle for restricted critical level representations of affine Kac–Moody algebras

• Mathematics
Compositio Mathematica
• 2012
Abstract We study the restricted category 𝒪 for an affine Kac–Moody algebra at the critical level. In particular, we prove the first part of the Feigin–Frenkel conjecture: the linkage principle for

### Localization of g^-modules on the affine Grassmannian

• Mathematics
• 2005
We consider the category of modules over the affine Kac-Moody algebra g^ of critical level with regular central character. In our previous paper math.RT/0508382 we conjectured that this category is

## References

SHOWING 1-10 OF 42 REFERENCES

### AFFINE KAC-MOODY ALGEBRAS AT THE CRITICAL LEVEL AND GELFAND-DIKII ALGEBRAS

• Mathematics
• 1992
We prove Drinfeld's conjecture that the center of a certain completion of the universal enveloping algebra of an affine Kac-Moody algebra at the critical level is isomorphic to the Gelfand-Dikii

### Wakimoto modules for twisted affine lie algebras

We construct Wakimoto modules for twisted a!ne Lie algebras , and interpret this construction in terms of vertex algebras and their twisted modules. Using the Wakimoto construction, we prove the

### Affine Kac-Moody algebras and semi-infinite flag manifolds

• Mathematics
• 1990
We study representations of affine Kac-Moody algebras from a geometric point of view. It is shown that Wakimoto modules introduced in [18], which are important in conformal field theory, correspond

### Representations of affine Kac-Moody algebras, bosonization and resolutions

• Mathematics
• 1990
We study boson representations of the affine Kac-Moody algebras and give an explicit description of primary fields and intertwining operators, using vertex operators. We establish the resolution of

### Gaudin model, Bethe Ansatz and critical level

• Mathematics
• 1994
We propose a new method of diagonalization oif hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on the Wakimoto modules over affine algebras at the

### Integrable Hierarchies and Wakimoto Modules

• Mathematics
• 1999
In our earlier papers we proposed a new approach to integrable hierarchies of soliton equations and their quantum deformations. We have applied this approach to the Toda field theories and the

### Representations of Affine Kac-Moody Algebras

In the first chapter we explained how simple finite-dimensional Lie algebras can be completely characterized in terms of their Cartan matrices or Dynkin diagrams. The same holds for an arbitrary

### Semi-infinite Weil complex and the Virasoro algebra

• Mathematics
• 1991
We define a semi-infinite analogue of the Weil algebra associated an infinite-dimensional Lie algebra. It can be used for the definition of semi-infinite characteristic classes by analogy with the

### Fock representations of the affine Lie algebraA1(1)

The aim of this note is to show that the affine Lie algebraA1(1) has a natural family πμ, υ,v of Fock representations on the spaceC[xi,yj;i ∈ ℤ andj ∈ ℕ], parametrized by (μ,v) ∈C2. By corresponding

### Wakimoto Realizations of Current Algebras: An Explicit Construction

• Mathematics
• 1997
Abstract:A generalized Wakimoto realization of can be associated with each parabolic subalgebra of a simple Lie algebra according to an earlier proposal by Feigin and Frenkel. In this paper the