The Universal Lie algebra

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 is the module generated by uni-trivalent diagrams whith n uni-valent vertices, and divided by the AS and IHX-relations. Actually this isomorphism induces an isomorphism from the direct sum of modules B n, n > 0 to P , where B ′ n is the submodule of Bn generated by connected diagrams. Therefore to understand Ac(S ), it’s enough to describe the modules B n. These modules are part of a more complicated object: the category of diagrams D. This category is a monoidal linear category. Every Lie algebra L equipped with a non singular symmetric invariant bilinear form induces a functor from D to the category of L-modules and, roughly speaking, these functors are the only one known. The purpose of this paper is to construct a monoidal category which looks like the category of module over a Lie algebra and which is universal in some sense. A lot of properties of this category is shown and many conjectures are given. In some sense this category is the universal Lie algebra, and every simple Lie (super)algebra is obtained by changing the coefficient ring.