Consider $\mathfrak{g}$ a finite-dimensional 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 Yang-Baxter 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: $$r^{12}(x,y)=-r^{21}(-x,-y)$$

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    $\begingroup$ 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 semi-simple Lie algebra:

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  • $\begingroup$ 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

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