The Kontsevich integral of a knot K lies in an algebra of diagrams Ac(S ). This algebra is (up to completion) a symmetric algebra of a graded module P , where P is the set of primitive elements of Ac(S ). The elements of P are represented by S-diagrams K such that the complement of the circle in K is connected and non empty. On the other hand there is an isomorphism from ⊕Bn to Ac(S ), where Bn...

It is well known that the Lyndon words of length n can be used to construct a basis for the n-th homogeneous component of the free Lie algebra. We develop an algorithm that uses a dynamic programming table to efficiently generate the standard bracketing for all Lyndon words of length n, thus constructing a basis for the n-th homogeneous component of the free Lie algebra. The algorithm runs in l...

The equations of motion of quantum Yang Mills theory (in the planar ‘large N’ limit), when formulated in Loop-space are shown to have an anomalous term, which makes them analogous to the equations of motion of WZW models. The anomaly is the Jacobian of the change of variables from the usual ones i.e. the connection one form A, to the holonomy U . An infinite dimensional Lie algebra related to t...

In this paper we study the Green function for a second order noncoercive differential operator L on a connected, simply connected homogeneous manifold of negative curvature. Such a manifold is a solvable Lie group S = NA, a semi-direct product of a nilpotent Lie group N and an abelian group A = R. Moreover, for an H belonging to the Lie algebra a of A, the real parts of the eigenvalues of Adexp...

Abstract The aim of this paper is to generalise the construction n -ary Hom-Lie bracket by means an $$(n-2)$$ ( n - 2 ) -cochain given algebra super case inducing -Hom-Lie superalgebras. We study notion generalized deri...

This is a review article in which we will introduce, in a unifying fashion and with more intermediate steps in some difficult calculations, two infinitedimensional Lie algebras of quantum matrix models, one for the open string sector and one for the closed string sector. Physical observables of quantum matrix models in the large-N limit can be expressed as elements of these Lie algebras. We wil...

We construct the general supersymmetry algebra via the adjoint action on a semi-Hopf algebra which has a more general structure than a Hopf algebra. As a result we have an extended supersymmetry theory with quantum gauge group, i.e., quantised enveloping algebra of a simple Lie algebra. For the example, we construct the N =1 and generalized N =2 supersymmetry theory which leads to the Seiberg-W...