We continue a previous study on Γ-vertex algebras and their quasimodules. In this paper we refine certain known results and we prove that for any Z-graded vertex algebra V and a positive integer N , the category of V -modules is naturally isomorphic to the category of quasimodules of a certain type for V ⊗N . We also study certain generalizations of twisted affine Lie algebras and we relate suc...

We consider some unitary representations of infinite dimensional Lie algebras motivated by string theory on AdS3. These include examples of two kinds: the A,D,E type affine Lie algebras and the N = 4 superconformal algebra. The first presents a new construction for free field representations of affine Lie algebras. The second is of a particular physical interest because it provides some hints t...

We extend classical results of Kostant and al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogan’s conjecture on infinitesimal characters of Harish–Chandra modules in terms of Dirac cohomology. For our calculations we use ...

We study Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov structure must be solvable. Conversely we present an example of a nilpotent 2-step solvable Lie algebra without any Novikov structure. We construct Novikov structures on certain Lie algebras via classical r-matrices and via extensions. In the latter case we lift No...

The purpose of this thesis is to give a new construction for central extensions of certain classes of infinite dimensional Lie algebras which include multiloop Lie algebras as motivating examples. The key idea of this construction is to view multiloop Lie algebras as twisted forms. This perspective provides a beautiful bridge between infinite dimensional Lie theory and descent theory and is cru...

In this paper, we extend the Drinfeld current realization of quantum affine algebras Uq(ĝ) and of the Yangians in several directions: we construct current operators for non-simple roots of g, define a new braid group action in terms of the current operators and describe the universal R-matrix for the corresponding “Drinfeld” comultiplication in the form of infinite product and in the form of ce...

Energy-momentum tensor for the toroidal Lie algebras. Abstract. We construct vertex operator representations for the full (N + 1)-toroidal Lie algebra g. We associate with g a toroidal vertex operator algebra, which is a tensor product of an affine VOA, a sub-VOA of a hyperbolic lattice VOA, affine sl N VOA and a twisted Heisenberg-Virasoro VOA. The modules for the toroidal VOA are also modules...

We construct and study various dual pairs between finite dimensional classical Lie groups and infinite dimensional Lie algebras in some Fock representations. The infinite dimensional Lie algebras here can be either a completed infinite rank affine Lie algebra, the W1+∞ algebra or its certain Lie subalgebras. We give a formulation in the framework of vertex algebras. We also formulate several co...

is called a truncated current Lie algebra, or sometimes a generalised Takiff algebra. We shall describe a highest-weight theory for ĝ, and the reducibility criterion for the universal objects of this theory, the Verma modules. The principal motivation, beside the aesthetic, is that certain representations of affine Lie algebras are essentially representations of a truncated current Lie algebra....