This is a written version of the talk delivered on Feb 6, 1996 at the SMC conference , Amsterdam. Last Summer, P. Vogel noted that the dimensions of the exceptional Lie algebras are given by ?2 (+ 5)(? 6) (? 1) for an appropriate value of , as well as other uniform behavior of the exceptional Lie algebras with regard to the parameter. Inspired by this, P. Deligne conjectured that there might be...

Hom-Lie algebras are non-associative, non-commutative algebras generalizing Lie algebras by twisting the Jacobi identity by a homomorphism. The main examples are algebras of twisted derivations (i.e., linear maps with a generalized Leibniz rule). Such generalized derivations appear in all parts of number theory, so hom-Lie algebras appear to have a natural role to play in many number-theoretica...

We consider Yang-Mills theories with general gauge groups G and twists on the four torus. We find consistent boundary conditions for gauge fields in all instanton sectors. An extended Abelian projection with respect to the Polyakov loop operator is presented, where A 0 is independent of time and in the Cartan subalgebra. Fundamental domains for the gauge fixed A 0 are constructed for arbitrary ...

In the first section we recall some basic notions on Lie algebras. In a second time w e study the algebraic variety of complex n-dimensional Lie algebras. We present different notions of deformations : Gerstenhaber deformations, pertubations, valued deformations and we use these tools to study some properties of this variety. Finaly we introduce the concept of rigidity and we present some resul...

We describe an interesting relation between Lie 2-algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2-group is a categorified version of a Lie group. If G is a simply-connected compact simple Lie group...

Let g be a simple, finite-dimensional Lie (super)algebra, equipped with an embedding of sl2 inducing the minimal gradation on g. The corresponding minimal Walgebra W(g, e−θ) introduced by Kac and Wakimoto has strong generators in weights 1, 2, 3 2 , and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra A(g) where g ⊂ g denote...

Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer-Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of Kosmann-Schwarzbach. This approach provides a definition of an L∞-(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L∞-algebra structure as...

Space-time N = 2 and N = 4 superconformal algebras can be built using world-sheet free fields or, equivalently, free field representations of ˆ osp(1|2) and ŝl(2|1), respectively. A prescription for the calculation of space-time correlators is given. As applications we compute one and two-point correlation functions of Virasoro (superVirasoro) generators. We also present a possible scenario for...

Covariant tensor representations of glm|n occur as irreducible components of tensor powers of the natural (m + n)-dimensional representation. We construct a basis of each covariant representation and give explicit formulas for the action of the generators of glm|n in this basis. The basis has the property that the natural Lie subalgebras glm and gln act by the classical Gelfand–Tsetlin formulas...

We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor h...