Bojko Bakalov

Cohomology of Conformal Algebras

Abstract:The notion of a conformal algebra encodes an axiomatic description of the operator product expansion of chiral fields in conformal field theory. On the other hand, it is an adequate tool for the study of infinite-dimensional Lie algebras satisfying the locality property. The main examples of such Lie algebras are those “based” on the punctured complex plane, such as the Virasoro algebra and loop Lie algebras. In the present paper we develop a cohomology theory of conformal algebras with coefficients in an arbitrary module. It possesses standards properties of cohomology theories; for example, it describes extensions and deformations. We offer explicit computations for the most important examples. To Bertram Kostant on his seventieth birthday

General methods for constructing bispectral operators

We present methods for obtaining new solutions to the bispectral problem. We achieve this by giving its abstract algebraic version suitable for generalizations. All methods are illustrated by new classes of bispectral operators.

Bispectral Algebras of Commuting Ordinary Differential Operators

Abstract:We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank N. It combines and unifies the ideas of Duistermaat–Grünbaum and Wilson. Our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators. The method also provides new broad families of bispectral algebras which may help to penetrate deeper into the problem.

TWISTED MODULES OVER LATTICE VERTEX ALGEBRAS

For any integral lattice

On the Lego-Teichmüller game

Q

Jacobi identity for vertex algebras in higher dimensions

, one can construct a vertex algebra

Highest weight modules over the

V_Q

W_{1+\infty}

called a lattice vertex algebra. If

algebra and the bispectral problem

\sigma

Infinite dimensional Lie algebras in 4D conformal quantum field theory

is an automorphism of