Equivariant vector bundles over quantum spheres

Abstract

We quantize homogeneous vector bundles over an even complex sphere S 2n as onesided projective modules over its quantized coordinate ring. We realize them in two different ways: as locally finite C-homs between pseudo-parabolic Verma modules and as induced modules of the quantum orthogonal group. Based on this alternative, we study representations of a quantum symmetric pair related to S 2n q and prove their complete reducibility.

Contravariant Form on Tensor Product of Highest Weight Modules

“…However, from the technical point of view, one of them may be easier to work with than the other. Such an example can be found in [12], where one of the spaces V + Z , Z + V is easy to identify while the other is unknown. The canonical form on V ⊗ Z translates the question of complete reducibility to non-degeneracy of a matrix, which is finite-dimensional when restricted to each weight subspace.…”

“…A quantim conjugacy class or a coadjoint orbit is realized by linear endomorphisms on such a module whose tensor product with finite-dimensional modules supports "representations" of a trivial bundle. Its non-trivial sub-bundles correspond to invariant idempotents that project the tensor product onto submodules [12,13].…”

Pseudo-parabolic category over quaternionic projective plane

“…In Section 3, we prove semi-simplicity of the category O t . It is an illustration of the complete reducibility criterion for the tensor products based on a contravariant form and Zhelobenko extremal cocycle [M4,M5,Zh]. We show that for every finite-dimensional quasi-classical U q (g)module V the tensor product V ⊗ M t is completly reducible and its simple submodules are in bijection with simple k-submodules in the classical g-module V .…”