# Commutators for quantum Lie algebras

Can the usual definition of a Lie algebra via commutators be simply adapted to quantum Lie algebras? Graphically you have the IHX scheme, with the X being a virtual crossing (so to say). Does it suffice to replace it with a real (knot-like) crossing and adapt the structure constants and coefficients before IHX a bit?
EDIT: $[x,y]_q=xy-q*yx$ and replacing all brackets by this q-bracket is a valid definition, but it does not answer my question, since in tensor form, yx is xy times a virtual crossing, so to say, and virtual and real crossings don't mix very well. I do have a conjecture for the formula, though. It is probably correct at q=1 (I guess it's eq. 4.36 of "Birdtracks", modulo a funny weight factor which is due to my choice of gauge, or that I use undirected graphs), and this graphic should make it clear (the blue J is the adjoint irrep, everything else can be any irrep):
http://imgur.com/sz8Nylh
The formula I'm seeking would then follow by applying it to a trivalent node made from two defining and one adjoint irrep. I will subsequently verify (or falsify :-) it by applying it to some simple examples like SU2(q) but that doesn't replace a proof...

• Based on the use of the terms "crossing" and "virtual crossing" this appears to be a question about knot invariants. Certainly, it makes no sense as a question about "representation theory" (of what?), or even about Lie algebras in the traditional algebraic realm. Nov 27 '16 at 16:54