# Infinite Dimensional Lie Algebras

@inproceedings{Kac1983InfiniteDL, title={Infinite Dimensional Lie Algebras}, author={Victor G. Kac}, year={1983} }

Introduction Notational conventions 1. Basic definitions 2. The invariant bilinear form and the generalized casimir operator 3. Integrable representations of Kac-Moody algebras and the weyl group 4. A classification of generalized cartan matrices 5. Real and imaginary roots 6. Affine algebras: the normalized cartan invariant form, the root system, and the weyl group 7. Affine algebras as central extensions of loop algebras 8. Twisted affine algebras and finite order automorphisms 9. Highest…

No Paper Link Available

## 624 Citations

### Double affine Lie algebras and finite groups

- Mathematics
- 2009

We introduce and begin to study Lie theoretical analogs of symplectic reflection algebras for a finite cyclic group, which we call "cyclic double affine Lie algebra". We focus on type A : in the…

### EXTENSIONS AND BLOCK DECOMPOSITIONS FOR FINITE-DIMENSIONAL REPRESENTATIONS OF EQUIVARIANT MAP ALGEBRAS

- Mathematics
- 2015

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. The irreducible…

### Conjugacy theorems for loop reductive group schemes and Lie algebras

- Mathematics
- 2011

The conjugacy of split Cartan subalgebras in the finite-dimensional simple case (Chevalley) and in the symmetrizable Kac–Moody case (Peterson–Kac) are fundamental results of the theory of Lie…

### Vertex Algebras and Coordinate Rings of Semi-infinite Flags

- MathematicsCommunications in Mathematical Physics
- 2019

The direct sum of irreducible level one integrable representations of affine Kac-Moody Lie algebra of (affine) type ADE carries a structure of P/Q-graded vertex operator algebra. There exists a…

### The Semi-Infinite Cohomology of Affine Lie Algebras

- Mathematics
- 1998

Abstract:We study the semi-infinite or BRST cohomology of affine Lie algebras in detail. This cohomology is relevant in the BRST approach to gauged WZNW models. Our main result is to prove necessary…

### Formal Hecke algebras and algebraic oriented cohomology theories

- MathematicsSelecta Mathematica
- 2013

In the present paper, we generalize the construction of the nil Hecke ring of Kostant–Kumar to the context of an arbitrary formal group law, in particular, to an arbitrary algebraic oriented…

### An overview of generalised Kac-Moody algebras on compact real manifolds

- MathematicsJournal of Geometry and Physics
- 2022

### Realizations of the Witt and Virasoro Algebras and Integrable Equations

- MathematicsJournal of Nonlinear Mathematical Physics
- 2019

In this paper we study realizations of infinite-dimensional Witt and Virasoro algebras. We obtain a complete description of realizations of the Witt algebra by Lie vector fields of first-order…

### Invariants of Automorphic Lie Algebras

- Mathematics
- 2015

Automorphic Lie Algebras arise in the context of reduction groups introduced in the late 1970s [35] in the field of integrable systems. They are subalgebras of Lie algebras over a ring of rational…