I learned from a professor that $$ R=Id+(q-1)r+ o(q-1), $$ where $R$ is a quantum $R$-matrix and $r$ is the corresponding classical $r$-matrix. Here $o(q-1)$ denotes a term of the form $A(q-1)^2$, where $A$ doesn't depend on $q$ and $A$ is some matrix.
Are there some references about this fact? Thank you very much.
Are there some references about this fact?
Yes, the best references are:
V.G. Drinfeld, Quantum Groups, in Proc. of the Int. Conf. of Mathematicians (Berkeley, 1986)
V.G. Drinfeld, Sov. Math. Dokl.28 (1983) 667
M. Jimbo, Lett. Math. Phys. 10 (1985) 63, ibid. 11 (1986) 247
M. Jimbo, Commun. Math. Phys. 102 (1986) 537
E. K. Sklyanin, Funct. Anal. Appl. 16(1983) 263; ibid. 17 (1984) 273
L.D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtadzhyan, Quantization of Lie Groups and Lie Algebras, in Algebraic Analysis, vol. I, (Academic Press, 1988), p. 129