Generalizing Feingold-Frenkel’s construction we use Weyl bosonic fields to construct toroidal Lie algebras of types An, Bn, Cn and Dn of level −1,−2,−1/2 and −2 respectively. In particular, our construction also gives new bosonic construction for the orthogonal Lie algebras in the cases of affine Lie algebras.

By exploring the description of chiral blocks in terms of co-invariants, a proof of the Verlinde formula for WZW models is obtained which is entirely based on the representation theory of affine Lie algebras. In contrast to other proofs of the Verlinde formula, this approach works for all untwisted affine Lie algebras. As a by-product we obtain a homological interpretation of the Verlinde multi...

This paper provides a survey for the latest developments in the theory of affine q-Schur algebras and Schur–Weyl duality between affine quantum gln and affine type A Hecke algebras. More precisely, we will establish, on the one side, an isomorphism between the double Ringel–Hall algebra D△(n) of a cyclic quiver △(n) and the quantum loop algebra of gln, and establish, on the other side, explicit...

We study and give a complete classification of good Zgradings of all simple finite-dimensional Lie algebras. This problem arose in the quantum Hamiltonian reduction for affine Lie algebras.

We extend Yangian double to super (or graded) case and give its Drinfeld generators realization by Gauss decomposition. Quantum algebras(in general meaning, it includes Yangians and quantum affine algebras etc.) are new algebraic structures discovered about ten years ago [1-4], they play important roles in the study of soluble statistical models and quantum fields theory. Quantum universal enve...

We construct vertex representations of quantum affine algebras of ADE type, which were first introduced in greater generality by Drinfeld and Jimbo. The limiting special case of our construction is the untwisted vertex representation of affine Lie algebras of Frenkel-Kac and Segal. Our representation is given by means of a new type of vertex operator corresponding to the simple roots and satisf...

This paper is about Toroidal Lie algebras which generalize the notion of an Affine Lie algebra. We study Verma type modules for these Toroidal algebras and prove an irreducibility criterion when the number of variables is two. We use the fact that the universal enveloping algebra is an Ore domain to obtain facts about the Verma type modules. Moreover, we are able to characterize the Affine Kac-...

We give a example of non nilpotent faithful representation of a filiform Lie algebra. This gives one counter-example of the conjecture saying that every affine connection on a filiform Lie group is complete. 1. Affine connection on a nilpotent Lie algebra 1.1. Affine connection on nilpotent Lie algebras. Definition 1. Let g be a n-dimensional Lie algebra over R. It is called affine if there is ...