Quantum current operators. I: Zeros and poles of quantum current operators and the condition of quantum integrability
For the current realization of affine quantum groups \textit{V. G. Drinfeld} gave a simple comultiplication in a closed form. Since all integrable highest weight \(U_q(\widehat{\mathfrak{sl}}(2))\)-modules can be derived from tensors of fundamental representations, the authors study zeros and poles of quantum current operators by using this comultiplication and the construction [\textit{I. B. Frenkel} and \textit{N. Jing}, Proc. Natl. Acad. Sci. USA 85, 9373-9377 (1988; Zbl 0662.17006)]. The main result is that on any level \(k\) integrable \(U_q(\widehat{\mathfrak{sl}}(2))\)-module the matrix coefficients of \(x^\pm (z_1)x^\pm (z_2)\cdots x^\pm (z_{k+1})\) have zeros at \(z_2/z_1=z_3/z_2=\dots =z_{k+1}/z_{k}=q^{\pm 2}\), where \(x^+(z)\) and \(x^-(z)\) are the quantized current operators of \(U_q(\widehat{\mathfrak{sl}}(2))\) corresponding to the current operators \(e(z)\) and \(f(z)\) of \(\widehat{\mathfrak{sl}}(2)\). This condition of integrability for the quantum currents is a deformation of the corresponding classical condition \(e(z)^{k+1}=f(z)^{k+1}=0\) [cf. \textit{J. Lepowsky} and \textit{M. Primc}, Structure of the standard modules for the affine Lie algebra \(A_1^{(1)}\), Contemp. Math. 46, Am. Math. Soc, Providence (1985; Zbl 0569.17007)]. The authors also present the results about the zeros and poles of the quantum current operators of \(U_q(\widehat{\mathfrak{sl}}(n))\). For Part II see the review Zbl 0889.17008 below.
scientific article; zbMATH DE number 1051628