# nLab Drinfeld-Kohno Lie algebra

Let $n>2$. The Drinfel’d-Kohno Lie algebra is a $Z$-algebra ${L}_{n}$ defined by generators ${t}_{\mathrm{ij}}={t}_{\mathrm{ji}}$, $1\le i\ne j\le n$ subject to the relations

$\left[{t}_{\mathrm{ij}},{t}_{\mathrm{kl}}\right]=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left[{t}_{\mathrm{ij}},{t}_{\mathrm{ik}}+{t}_{\mathrm{jk}}\right]=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}1\le i\ne j\ne k\ne l\le n$[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 ${\pi }_{1}\left({X}_{n}\right)$ which is isomorphic to the pure braid group:

${\mathrm{PB}}_{n}⟶\mathrm{exp}\left({L}_{n}\otimes C\right)$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 ${\mathrm{PB}}_{n}$ is an isomorphism. Drinfeld showed using associators that the same holds true over $Q$.

In particular, $U\left({L}_{n}\otimes Q\right)$ is isomorphic to the associated graded of $Q\left[{\mathrm{PB}}_{n}\right]$ 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, $U\left({L}_{n}\otimes Q\right)$ may be identified with the algebra of horizontal chord diagrams?.

The universal enveloping $U\left({L}_{n}\right)\otimes Q$ is a Koszul algebra.

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