2020/06/03 by Andrew Jaramillo, Jaramillo, Andrew, Garrett Johnson +1
Mathematics · Physics and Astronomy · #16T15 #16T20 (Secondary) #17B37 (Primary) #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2006.02462
openalex publication_date 2020/06/03 · openalex created_date 2020/06/12 · openalex updated_date 2026/07/28
Let PJ be the standard parabolic subgroup of SLn obtained by deleting a subset J of negative simple roots, and let PJ = LJUJ be the standard Levi decomposition. Following work of the first author, we study the quantum analogue θ: \mathcal Oq(PJ) →\mathcal Oq(LJ) ⊗ \mathcal Oq(PJ) of an induced coaction and the corresponding subalgebra \mathcal Oq(PJ)co θ ⊆ \mathcal Oq(PJ) of coinvariants. It was shown that the smash product algebra \mathcal Oq(LJ)# \mathcal Oq(PJ)co θ is isomorphic to \mathcal Oq(PJ). In view of this, \mathcal Oq(PJ)co θ -- while it is not a Hopf algebra -- can be viewed as a quantum analogue of the coordinate ring \mathcal O(UJ). In this paper we prove that when q∈ \mathbbK is nonzero and not a root of unity, \mathcal Oq(PJ)co θ is isomorphic to a quantum Schubert cell algebra \mathcal Uq+[w] associated to a parabolic element w in the Weyl group of \mathfraksl(n). An explicit presentation in terms of generators and relations is found for these quantum Schubert cells.