associative Yang-Baxter on U(g)

Question

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

Answer 1

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:

• in the work of Travis Schedler: see e.g. http://arxiv.org/abs/math/0212258 (this is very much in the spirit of Belavin-Drinfeld classification of solutions of the classical Yang-Baxter equation for a semi=simple $\mathfrak{g}$)

• in the work of Igor Burban: see e.g. http://www.mi.uni-koeln.de/~burban/AYBEnewsemistable.pdf (imo, this is beautiful geometric approach).

• in the original work of Alexander Polishchuk: http://arxiv.org/abs/math/0008156