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 ...

We present an algebraic approach to string theory. An embedding of sl(2|1) in a super Lie algebra together with a grading on the Lie algebra determines a nilpotent subalgebra of the super Lie algebra. Chirally gauging this subalgebra in the corresponding Wess-Zumino-Witten model, breaks the affine symmetry of the Wess-Zumino-Witten model to some extension of the N = 2 superconformal algebra. Th...

The representation theory of Kac–Moody algebras, and in particular that of affine Lie algebras, has been extensively studied over the past twenty years. The representations that have had the most applications are the integrable ones, so called because they lift to the corresponding group. The affine Lie algebra associated to a finite-dimensional complex simple Lie algebra g is the universal (on...

In this paper, using generating functions we study two categories E and C of modules for twisted affine Lie algebras ĝ[σ], which were firstly introduced and studied in [Li] for untwisted affine Lie algebras. We classify integrable irreducible ĝ[σ]-modules in categories E and C, where E is proved to contain the well known evaluation modules and C to unify highest weight modules, evaluation modul...

These are the notes of my talk at the conference “Double affine Hecke algebras and algebraic geometry” (MIT, May 18, 2010). The goal of this talk is to discuss some results and conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products categorifies certain structures in the representation theory for affine Lie algebras. These conjectures arose fr...

We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro and for Heisenberg algebras.

If A is a unital associative algebra and u is invertible in A, one can define an algebra A, called the u-isotope of A, which is equal to A as a vector space but has a new product x ·u y = xuy. This isotope A (u) is again unital and associative but with a shifted identity element u. More generally there are definitions of isotope for several other classes of unital nonassociative algebras, notab...

The Lascoux-Leclerc-Thibon conjecture, reformulated and solved by S. Ariki, asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra g where g is gl ∞ or the affine Lie algebra A (1) l , and the irreducible representations correspond to the upper g...