Drinfeld-Kohno Lie algebra

Let $n\gt 2$. The Drinfel’d-Kohno Lie algebra is a $\mathbf{Z}$-algebra $L_n$ defined by generators $t_{ij} = t_{ji}$, $1\leq i\neq j\leq n$ subject to the relations

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

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 $\pi_1(X_n)$ which is isomorphic to the pure braid group:

$PB_n \longrightarrow \exp(L_n \otimes \mathbf{C})$

It was shown by Kohno that the extension of this map to the $\mathbf{C}$-pro-unipotent completion of $PB_n$ is an isomorphism. Drinfeld showed using associators that the same holds true over $\mathbf{Q}$.

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

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

Revised on September 13, 2012 22:01:14
by Adrien B?
(129.215.104.113)