# associative Yang-Baxter on U(g)

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: $$r^{12}(x,y)=-r^{21}(-x,-y)$$

• What is the relation between $A$ and $g$? – Bruce Westbury Apr 4 '12 at 16:21
• I reformulated a bit the question accordingly to your comment. I hope bob would agree with the changes I made. – DamienC Apr 5 '12 at 7:43
• bob: If you register an account, it will be easier to edit your own question. – 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: