The quantum Weyl group and the universal quantum \(R\)-matrix for affine Lie algebra \(A^{(1)}_ 1\).(English)Zbl 0776.17011
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 byV. G. Drinfel’d himself [Proc. Int. Congr. Math., Berkeley 1986, Vol. 1, 798-820 (1987;Zbl 0667.16003)] for \(s\ell(2)\), then byM. Rosso [Commun. Math. Phys. 124, 307-318 (1989;Zbl 0694.17006)] for \(s\ell(N)\) and finally byS. Levendorskij andYa. S. Soibel’man [J. Geom. Phys. 7, 241- 254 (1990;Zbl 0729.17009)], and independently byA. Kirillov andN. 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 byG. Lusztig [Adv. Math. 70, 237-249 (1988;Zbl 0651.17007)] (the quantum Weyl group was first considered in [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 byS. M. Khoroshkin andV. 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 byY.-Z. Zhan andM. Gould [Lett. Math. Phys. 29, 19-31 (1993)]).
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 byS. M. Khoroshkin andV. 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 byY.-Z. Zhan andM. Gould [Lett. Math. Phys. 29, 19-31 (1993)]).
Reviewer: N.Andruskiewitsch (Bonn)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
Keywords:
quantum groups;quantum Yang-Baxter equation;universal \(R\)-matrix;affine Kac-Moody algebra;braid group automorphismsCitations:
Zbl 0667.16003;Zbl 0694.17006;Zbl 0729.17009;Zbl 0723.17014;Zbl 0651.17007;Zbl 0708.46029;Zbl 0758.17011;Zbl 0744.17015Cite
Full Text:DOI
References:
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