Already Delius gave a construction leading from any simple Lie algebra to a quantum Lie algebra $L$. The extension to semisimple seems simple (sorry :-) to me (see below). My question: Can this work for non-semisimple Lie algebras? My guess is "yes, for some subset", for the following reason.
If $L_1$ has the S matrix $S_1$ for irrep $R_1$, likewise $L_2$ $S_2$ $R_2$, then $L_1\bigoplus L_2$ has $S_1\bigotimes S_2$ for irrep $R_1\bigotimes R_2$ (just my hypothesis! - I hope I got the tensor operators right this time - example: choose two copies of $L=A_1$ with the spin $1/2$ irrep of dimension $2$, the S matrix has dimension $2^2$, the tensor product $4^2$, and $4$ is the lowest irrep dimension of $A_1\bigoplus A_1$, and it clebsches as $4\bigotimes 4=9+(3+3)_{ad}+1$.)
But now you can also "add" any two S matrices as $S_1 "\bigoplus" S_2$ (this is not exactly right, you must afterwards fill up all "cross" entries $12|21$ with an "1" to get the Yang-Baxter equation working - I am too lazy to actually prove it works -, but in any case the dimensions of the S matrices just add). Again, "add" the S matrices of two copies of $2$ of $A_1$, the S matrix has dimension $2^2$, the tensor product $4^2$, but it now clebsches as $4\bigotimes 4=4+4+(3+3)_{ad?}+1+1$. Already the two $1$ smell fishy, I guess the reduction is incomplete, in any case the result is something completely different than that of $A_1\bigoplus A_1$.
I've been told on SO that $\bigoplus$ is the only operation you can apply to any two Lie algebras. So two main possibilities exist - is one of them correct?
- Just because $S$ is a valid S matrix, it doesn't mean that it is the S matrix of some $L$. Question: Do any valid $S$ corresponds to some $L$? Do any $L$ (not only semisimple) have a valid $S$? (I guess this corresponds to my first question.)
- Some other operation $\bigodot$ can be applied to any two members of some subset of Lie algebras, the set generated under $\bigodot,\bigoplus$ is closed, and $S_1"\bigoplus"S_2$ indeed is the S matrix of $L_1\bigodot L_2$.