The quantum Weyl group and the universal quantum \(R\)-matrix for affine Lie algebra \(A^{(1)}_ 1\)
As is well-known, one of the main achievements of the theory of quantum groups is to provide large families of solutions of the quantum Yang- Baxter equation, by means of the so-called universal \(R\)-matrix; recall that \(R\) is an element of \(U_ h({\mathfrak g})\hat\otimes U_ h({\mathfrak g})\), \(\mathfrak g\) a finite-dimensional complex Lie algebra. Whereas the existence of \(R\) was from the beginning theoretically established by Drinfel'd, its explicit formula was given first by \textit{V. G. Drinfel'd} himself [Proc. Int. Congr. Math., Berkeley 1986, Vol. 1, 798-820 (1987; Zbl 0667.16003)] for \(s\ell(2)\), then by \textit{M. Rosso} [Commun. Math. Phys. 124, 307-318 (1989; Zbl 0694.17006)] for \(s\ell(N)\) and finally by \textit{S. Levendorskij} and \textit{Ya. S. Soibel'man} [J. Geom. Phys. 7, 241- 254 (1990; Zbl 0729.17009)], and independently by \textit{A. Kirillov} and \textit{N. Yu. Reshetikhin} [Commun. Math. Phys. 134, 421-431 (1991; Zbl 0723.17014)], for the general case. The main tool of the proof in the last two papers is the ``quantum Weyl group'', a Hopf algebra built from both \(U_ h({\mathfrak g})\) and the group algebra of the braid group; the last acts on the first by (a version of) certain algebra automorphisms introduced by \textit{G. Lusztig} [Adv. Math. 70, 237-249 (1988; Zbl 0651.17007)] (the quantum Weyl group was first considered in [\textit{Ya. S. Soibel'man}, Algebra Anal. 2, 190-212 (1990; Zbl 0708.46029)]. However, a proof of the explicit formula for the universal \(R\)-matrix using instead of the quantum Weyl group, a ``quantum'' variant of the Cartan-Weyl basis was offered by \textit{S. M. Khoroshkin} and \textit{V. N. Tolstoj} [Funkts. Anal. Prilozh. 26, 85-88 (1992; Zbl 0758.17011); see also Commun. Math. Phys. 141, 599-617 (1991; Zbl 0744.17015)] who showed also that their approach is also available for \(U_ h({\mathfrak g})\), if \(\mathfrak g\) is an affine non-twisted Kac-Moody algebra. In the paper under review, the explicit formula for the universal \(R\)- matrix of the affine Kac-Moody algebra of type \(A_ 1^{(1)}\) is proved using braid group automorphisms; the authors remark however that it is not yet known how to prove the explicit formula in the general case by this method. (An even more explicit formula for the universal \(R\)-matrix of affine Kac-Moody algebras of rank 3 was recently given by \textit{Y.-Z. Zhan} and \textit{M. Gould} [Lett. Math. Phys. 29, 19-31 (1993)]).
scientific article; zbMATH DE number 202408