I've come across a reference in a paper to the 



> 
> Hecke relation for the universal R-matrix of a quasi-triangular Hopf algebra.
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> 
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I've looked around, standard references, online etc, but can't seem to find what this Hecke relation is. Can anyone point me in the right direction, or even better, tell me what it is?



Reference for the Hecke relation for the universal R-matrix
For the Drinfeld-Jimbo quantum universal enveloping algebras, see Proposition 24 of Chapter 8 in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. This relation is just in the type A situation, for $\mathfrak{gl}\_n$ or $\mathfrak{sl}\_n$. The relation they get is
$$
 (\hat{R} - q)(\hat{R} + q^{-1}) = 0,
$$
where $R$ is the R-matrix for the vector representation of $U\_q(\mathfrak{g})$, $\hat{R} = \tau \circ R$, where $\tau$ is the tensor flip, and all the conventions are those of Klimyk and Schmudgen.


In other situations, the map $\hat{R}$ has more than just two eigenvalues, so the spectral decomposition is more complicated, and the characteristic polynomial (i.e. the Hecke relation) is more complicated. 


