We describe various combinatorial aspects of the Birkhoff–Connes–Kreimer factorization in perturbative renormalisation. The analog of Bogoliubov’s preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal characters.

We rewrite the Bagger-Lambert action for any Lie 3-algebra as a standard ChernSimons action coupled to matter. We use this action to compute self-energies and vertex corrections at one-loop order. Non-renormalization of the coupling constant comes out as a direct consequence of the Lie 3-algebra structure underlying the Lie algebra. [email protected]

We associate to any germ of an analytic variety a Lie algebra of tangent vector fields, the tangent algebra. Conversely, to any Lie algebra of vector fields an analytic germ can be associated, the integral variety. The paper investigates properties of this correspondence: The set of all tangent algebras is characterized in purely Lie algebra theoretic terms. And it is shown that the tangent alg...

Every affine structure on Lie algebra g defines a representation of g in aff(Rn). If g is a nilpotent Lie algebra provided with a complete affine structure then the corresponding representation is nilpotent. We describe noncomplete affine structures on the filiform Lie algebra Ln. As a consequence we give a nonnilpotent faithful linear representation of the 3-dimensional Heisenberg algebra. 200...

Aslaksen, H., Determining summands in tensor products of Lie algebra representations, Journal of Pure and Applied Algebra 93 (1994) 135-146. We give some results that enable us to find certain summands in tensor products of Lie algebra representations. We concentrate on the splitting of tensor squares into their symmetric and antisymmetric parts. Our results are valid for any Lie algebra of arb...

We introduce and study a Hopf algebra containing the descent algebra as a sub-Hopf-algebra. It has the main algebraic properties of the descent algebra, and more: it is a sub-Hopf-algebra of the direct sum of the symmetric group algebras; it is closed under the corresponding inner product; it is cocommutative, so it is an enveloping algebra; it contains all Lie idempotents of the symmetric grou...

We solve the Jacobi identity of metric 3-Lie algebras for which an associated Lie algebra either does not admit bi-invariant 4-forms or it is the sum of up to three abelian directions and a semi-simple Lie algebra. In all cases, we find that the structure constants of the metric 3-Lie algebras are sums of orthogonal simple 4-forms verifying a conjecture in math/0211170. In particular, there is ...

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...

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 ...

I give a short proof of the following algebraic statement: if V is a simple vertex algebra, then the underlying Lie conformal algebra is either abelian, or it is an irreducible central extension of a simple Lie conformal algebra. This provides many examples of non-finite simple Lie conformal algebras, and should prove useful for classification purposes.