Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups

作者：Xu, Yongjun^*; Wang, Dingguo; Chen, Jialei

来源：Journal of Algebra and Its Applications, 2016, 15(10): 1650179.

DOI：10.1142/S0219498816501796

• Quantum group
• PBW-deformation
• coideal subalgebra
• braid group action
• root vector

We focus on a class of filtered quantum algebras B-q(g) which are both coideal sub-algebras of quantum groups and Poincare-Birkhoff-Witt (PBW)-deformations of their negative parts. In [Y. Xu and S. Yang, PBW-deformations of quantum groups, J. Algebra 408 (2014) 222-249], Xu and Yang proved that braid group actions on Bq(g) introduced by Kolb and Pellegrini can be used to define root vectors and construct PBW bases for B-q(g). In this present paper, for each element w in the Weyl group of g we first introduce a subspace B-w and a subalgebra B(w) of B-q(g), where B(w) can be considered as an analogue of quantum Schubert cell algebra. Then a sufficient and necessary condition on w is given for B-w = B(w). Moreover, we prove that B(w(1)) = B(w(2)) if and only if w(1) and w(2) can be generated by the same simple reflections. Finally, we characterize the algebra B(w) which can be obtained via an iterated Ore extension. Our results show that quantum groups and their PBW-deformations really have some different properties.

• 出版日期2016-12
• 单位曲阜师范大学; 北京航空航天大学