The author considers an elliptic analogue of the Knizhnik-Zamolodchikov equations – the consistent system of linear differential equations arising from the elliptic solution of the classical Yang-Baxter equation for the Lie algebra sl N. The solutions of this system are interpreted as traces of products of intertwining operators between certain representations of the affine Lie algebra sl N. A ...

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of the book by N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry , Birkhäuser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a quantum group or the ...

After introducing double derivations of $n$-Lie algebra $L$ we‎ ‎describe the relationship between the algebra $mathcal D(L)$ of double derivations and the usual‎ ‎derivation Lie algebra $mathcal Der(L)$‎. ‎In particular‎, ‎we prove that the inner derivation algebra $ad(L)$‎ ‎is an ideal of the double derivation algebra $mathcal D(L)$; we also show that if $L$ is a perfect $n$-Lie algebra‎ ‎wit...

We discuss the mutually opposite procedures of deformations and contractions of Lie algebras. Our main purpose is to illustrate the fact that, with appropriate combinations of both procedures, we obtain new Lie algebras. Firstly, we discuss low-dimensional Lie algebras, and these simple examples illustrate that, whereas for every contraction there exists a reverse deformation, the converse is n...

I gave two talks on some of my results on toroidal Lie algebras in the conference Functional Analysis VIII held in Dubrovnik, Croatia in June 2003. But these results have been submitted to research journals and will appear soon. So I decided to write an expository article on Toroidal Lie Algebras covering my results for the proceedings of Functional Analysis VIII. I will introduce the definitio...

In this contributed presentation, we discuss and compare the mutually opposite procedures of deformations and contractions of Lie algebras. We suggest that with appropriate combinations of both procedures one may construct new Lie algebras. We first discuss low-dimensional Lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation, the converse is ...

Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to whom they are related via Schur-Weyl duality. In this review paper, we give a glimpse on some aspects of their very rich repre...

• ei, fi. e1 = 1⊗ E1, e2 = 1⊗ E2; f1 = 1⊗ F1, f2 = 1⊗ F2. • e0, f0, h0. e0 = t⊗ [F1, F2] = t⊗ E31. f0 = t −1 ⊗ [E1, E2] = t−1 ⊗ E13. h0 = [e0, f0] = −1⊗ (H1 +H2) + c = −1⊗ (E11 + E33) + c • H. H = 1⊗H ⊕ Cc⊕ Cd. Note that c is just the central element c = h0 + h1 + h2. • Π. α1 = 1 = 2, α1(c) = α1(d) = 0 α2 = 1 = 2, α1(c) = α1(d) = 0 θ = α1 + α2 = 1 − 3, θ(c) = θ(d) = 0 δ : δ(1⊗H) = δ(c) = 0, δ(d...

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of the book N. Chriss, V. Ginzburg ‘Representation Theory and Complex Geometry’ , Birkhäuser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a Quantum group or the Iwaho...

Braided-Lie bialgebras have been introduced by Majid, as the Lie versions of Hopf algebras in braided categories. In this paper we extend previous work of Majid and of ours to show that there is a braided-Lie bialgebra associated to each inclusion of Kac–Moody bialgebras. Doing so, we obtain many new examples of infinite-dimensional braided-Lie bialgebras. We analyze further the case of untwist...