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Let $\frak{g}$ be a simple Lie algebra of A,B,C,or D series type. Moreover, let $U_q(\frak{g})$ be its Drinfeld-Jimbo quantized enveloping algebra, and $G_q$ the quantized enveloping algebra. As is shown, for example, in Chapter 9 of Klimyk and Schmudgen, each $G_q$ can be realized as a quotient of an FRT algebra (which is to say, roughly, that they can be constructed from an R-matrix in the Yang--Baxter sense).

Does such a presentation exist for the exceptional quantum groups $E_q(6)$, $E_q(7), E_q(8), F_q(4)$, and $G_q(2)$?

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I think a good reference is this paper by Jin-Ma. R-matrices for E7 and F4 are computed while references are given for E6 and G2. I am not aware if E8 has been taken care of.

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