Victor G. Kac

Infinite-Dimensional Lie Algebras

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-weight modules over Kac-Moody algebras 10. Integrable highest-weight modules: the character formula 11. Integrable highest-weight modules: the weight system and the unitarizability 12. Integrable highest-weight modules over affine algebras 13. Affine algebras, theta functions, and modular forms 14. The principal and homogeneous vertex operator constructions of the basic representation Index of notations and definitions References Conference proceedings and collections of paper.

A sketch of Lie superalgebra theory

This article deals with the structure and representations of Lie superalgebras (ℤ2-graded Lie algebras). The central result is a classification of simple Lie superalgebras over ℝ and ℂ.

Quasifinite highest weight modules over the Lie algebra of differential operators on the circle

AbstractWe classify positive energy representations with finite degeneracies of the Lie algebraW1+∞ and construct them in terms of representation theory of the Lie algebra

Characters and fusion rules for

\hat gl(\infty ,R_m )

-algebras via quantized Drinfel'd-Sokolov reduction

of infinites matrices with finite number of non-zero diagonals over the algebraRm=ℂ[t]/(tm+1). The unitary ones are classified as well. Similar results are obtained for the sin-algebras.

The n-Component KP Hierarchy and Representation Theory

The problem of representing an integer as a sum of squares of integers has had a long history. One of the first after antiquity was A. Girard who in 1632 conjectured that an odd prime p can be represented as a sum of two squares iff p ≡ 1 mod 4, and P. Fermat in 1641 gave an “irrefutable proof” of this conjecture. The subsequent work on this problem culminated in papers by A.M. Legendre (1798) and C.F. Gauss (1801) who found explicit formulas for the number of representations of an integer as a sum of two squares. C.G. Bachet in 1621 conjectured that any positive integer can be represented as a sum of four squares of integers, and it took efforts of many mathematicians for about 150 years before J.-L. Lagrange gave a proof of this conjecture in 1770.

Geometric interpretation of the partition function of 2D gravity

Using the cohomological approach toW-algebras, we calculate characters and fusion coefficients for their representations obtained from modular invariant representations of affine algebras by the quantized Drinfeld-Sokolov reduction.

Integrable Highest Weight Modules over Affine Superalgebras and Appell's Function

It is the aim of the present article to give all formulations of the n-component KP hierarchy and clarify connections between them. The generalization to the n-component KP hierarchy is important because it contains many of the most popular systems of soliton equations, like the Davey–Stewartson system (for n=2), the two-dimensional Toda lattice (for n=2), the n-wave system (for n⩾3) and the Darboux–Egoroff system. It also allows us to construct natural generalizations to the Davey–Stewartson and Toda lattice systems. Of course, the inclusion of all these systems in the n-component KP hierarchy allows us to construct their solutions by making use of vertex operators.