Drinfeld-Kohno Lie algebra

Let n>2. The Drinfel’d-Kohno Lie algebra is a Z-algebra L n defined by generators t ij=t ji, 1ijn subject to the relations

[t ij,t kl]=0,[t ij,t ik+t jk]=0,1ijkln[t_{ij}, t_{kl}] = 0, \,\,\,[t_{ij}, t_{ik}+t_{jk}] = 0, \,\,\,\,\,1\leq i\neq j\neq k\neq l\leq n

It is the holonomy Lie algebra? of the configuration space X n of n distincts points in the complex plane. Hence, it can be used to define a flat connection on X n, which is universal among Knizhnik-Zamolodchikov equations.

Therefore, it induces a monodromy representation of π 1(X n) which is isomorphic to the pure braid group:

PB nexp(L nC)PB_n \longrightarrow \exp(L_n \otimes \mathbf{C})

It was shown by Kohno that the extension of this map to the C-pro-unipotent completion of PB n is an isomorphism. Drinfeld showed using associators that the same holds true over Q.

In particular, U(L nQ) is isomorphic to the associated graded of Q[PB n] with respect to the filtration induced by powers of the augmentation ideal. Since it is known that this filtration coincides with the one induced by the Vassiliev skein relation, U(L nQ) may be identified with the algebra of horizontal chord diagrams?.

The universal enveloping U(L n)Q is a Koszul algebra.

Cf. Drinfeld-Kohno theorem

Revised on September 13, 2012 22:01:14 by Adrien B? (