A Drinfeld type presentation of affine \(\imath\)quantum groups. I: Split ADE type. (English) Zbl 1514.17017
The affine quantum group admits two presentations: the Serre presentation introduced by Drinfeld-Jimbo, and the current presentation, also known as the Drinfeld presentation [V. G. Drinfel’d, Sov. Math., Dokl. 36, No. 2, 212–216 (1988; Zbl 0667.16004); translation from Dokl. Akad. Nauk SSSR 269, 13–17 (1987)]. The isomorphism between these two presentations, stated by V. Drinfeld, was proved by J. Beck [Commun. Math. Phys. 165, No. 3, 555–568 (1994; Zbl 0807.17013)] and I. Damiani [J. Algebra 161, No. 2, 291–310 (1993; Zbl 0803.17003); Publ. Res. Inst. Math. Sci. 48, No. 3, 661–733 (2012; Zbl 1297.17009)].
The authors of this article study the current presentation of the universal \(\imath\)quantum groups \(\widetilde{U}^{\imath}\) of split affine \(ADE\) type. By extending the results of P. Baseilhac and S. Kolb in [Transform. Groups 25, No. 2, 363–389 (2020; Zbl 1439.81058)], the authors established Drinfeld type new relations among the generators of the universal \(q\)-Onsager algebra \(\widetilde{U}^{\imath}(\widehat{sl}_2)\). In the higher rank case the authors used the braid group action on \(\widetilde{U}^{\imath}\), which is realized by reflection functors in \(\imath\)Hall algebras considered in the work [M. Lu and W. Wang, Commun. Math. Phys. 381, No. 3, 799–855 (2021; Zbl 1479.17029)].
The authors of this article study the current presentation of the universal \(\imath\)quantum groups \(\widetilde{U}^{\imath}\) of split affine \(ADE\) type. By extending the results of P. Baseilhac and S. Kolb in [Transform. Groups 25, No. 2, 363–389 (2020; Zbl 1439.81058)], the authors established Drinfeld type new relations among the generators of the universal \(q\)-Onsager algebra \(\widetilde{U}^{\imath}(\widehat{sl}_2)\). In the higher rank case the authors used the braid group action on \(\widetilde{U}^{\imath}\), which is realized by reflection functors in \(\imath\)Hall algebras considered in the work [M. Lu and W. Wang, Commun. Math. Phys. 381, No. 3, 799–855 (2021; Zbl 1479.17029)].
Reviewer: Marijana Butorac (Rijeka)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
Keywords:
affine quantum groups; Drinfeld presentation; \(\imath\)quantum groups; quantum symmetric pairs; \(q\)-Onsager algebraCitations:
Zbl 0667.16004; Zbl 0807.17013; Zbl 0803.17003; Zbl 1297.17009; Zbl 1439.81058; Zbl 1479.17029References:
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