Consider $\mathfrak{g}$ a finitedimensional Lie algebra over the field $\textbf{k}$. If A is an associative algebra, we are searching for functions from $\textbf{C}\times \textbf{C}$ to $A\otimes A$ such that: $$r^{12}(u',v)r^{13}(u+u',v+v')r^{23}(u+u',v')r^{23}(u,v)+r^{13}(u,v+v')r^{23}(u',v')=0$$ For $u,u',v,v'\in\mathbb{C}$. This is known as the associative YangBaxter equation with spectral parameters. Has the set of the solutions been unravelled when $A=U(\mathfrak{g})$ is the universal envelopping algebra of $\mathfrak{g}$? In fact, I am searching for solutions which have the following unitarity condition: $$r^{12}(x,y)=r^{21}(x,y)$$

1$\begingroup$ What is the relation between $A$ and $g$? $\endgroup$ – Bruce Westbury Apr 4 '12 at 16:21

$\begingroup$ I reformulated a bit the question accordingly to your comment. I hope bob would agree with the changes I made. $\endgroup$ – DamienC Apr 5 '12 at 7:43

$\begingroup$ bob: If you register an account, it will be easier to edit your own question. $\endgroup$ – S. Carnahan♦ Apr 6 '12 at 2:28
I don't know about full classification results for $U(\mathfrak{g})$ in general (this is probably out of reach), but there are a bunch of very interesting constructions (together with partial classification resultas) when $A=M_n(\textbf{k})$ and/or when $\mathfrak{g}$ is a semisimple Lie algebra:
in the work of Travis Schedler: see e.g. http://arxiv.org/abs/math/0212258 (this is very much in the spirit of BelavinDrinfeld classification of solutions of the classical YangBaxter equation for a semi=simple $\mathfrak{g}$)
in the work of Igor Burban: see e.g. http://www.mi.unikoeln.de/~burban/AYBEnewsemistable.pdf (imo, this is beautiful geometric approach).
in the original work of Alexander Polishchuk: http://arxiv.org/abs/math/0008156

$\begingroup$ Ah, so it is ASSOCIATIVE YBE (not standard YBE) introduced by Polishchuk and Aguiar... Thanks for references... $\endgroup$ – Alexander Chervov Apr 5 '12 at 8:48