Editing Quantum group (section)

====Case 1: ''q'' is not a root of unity====
Strictly, the quantum group ''U''<sub>''q''</sub>(''G'') is not quasitriangular, but it can be thought of as being "nearly quasitriangular" in that there exists an infinite formal sum which plays the role of an [[R-matrix|''R''-matrix]].  This infinite formal sum is expressible in terms of generators ''e<sub>i</sub>'' and ''f<sub>i</sub>'', and Cartan generators ''t''<sub>''λ''</sub>, where ''k<sub>λ</sub>'' is formally identified with ''q''<sup>''t''<sub>''λ''</sub></sup>.  The infinite formal sum is the product of two factors,{{citation needed|reason=I could not find this in references or anywhere else. Chari-Pressley has a different formula.|date=July 2016}}
:<math>q^{\eta \sum_j t_{\lambda_j} \otimes t_{\mu_j}}</math>
and an infinite formal sum, where ''λ''<sub>''j''</sub> is a basis for the dual space to the Cartan subalgebra, and ''μ''<sub>''j''</sub> is the dual basis, and ''η'' = ±1.

The formal infinite sum which plays the part of the [[R-matrix|''R''-matrix]] has a well-defined action on the tensor product of two irreducible highest weight modules, and also on the tensor product of two lowest weight modules.  Specifically, if ''v'' has weight ''α'' and ''w'' has weight ''β'', then 
:<math>q^{\eta \sum_j t_{\lambda_j} \otimes t_{\mu_j}}\cdot(v \otimes w) = q^{\eta (\alpha,\beta)} v \otimes w,</math>
and the fact that the modules are both highest weight modules or both lowest weight modules reduces the action of the other factor on ''v'' ⊗ ''W'' to a finite sum.

Specifically, if ''V'' is a highest weight module, then the formal infinite sum, ''R'', has a well-defined, and [[invertible]], action on ''V'' ⊗ ''V'', and this value of ''R'' (as an element of End(''V'' ⊗ ''V'')) satisfies the [[Yang–Baxter equation]], and therefore allows us to determine a representation of the [[braid group]], and to define quasi-invariants for [[knot (mathematics)|knots]], [[link (knot theory)|links]] and [[braid theory|braids]].