Drinfeld-Kohno Lie algebra

Let n>2n\gt 2. The Drinfel’d-Kohno Lie algebra is a Z\mathbf{Z}-algebra L nL_n defined by generators t ij=t jit_{ij} = t_{ji}, 1ijn1\leq i\neq j\leq n 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 nX_n of nn distincts points in the complex plane. Hence, it can be used to define a flat connection on X nX_n, which is universal among Knizhnik-Zamolodchikov equations.

Therefore, it induces a monodromy representation of π 1(X n)\pi_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\mathbf{C}-pro-unipotent completion of PB nPB_n is an isomorphism. Drinfeld showed using associators that the same holds true over Q\mathbf{Q}.

In particular, U(L nQ)U(L_n \otimes \mathbf{Q}) is isomorphic to the associated graded of Q[PB n]\mathbf{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)U(L_n \otimes \mathbf{Q}) may be identified with the algebra of horizontal chord diagrams.

The universal enveloping U(L n)QU(L_n)\otimes \mathbf{Q} is a Koszul algebra.

Cf. Drinfeld-Kohno theorem

Last revised on September 13, 2012 at 22:01:14. See the history of this page for a list of all contributions to it.