Abstract
We construct equivariant vector bundles over quantum projective spaces making use of parabolic Verma modules over the quantum general linear group. Using an alternative realization of the quantized coordinate ring of projective space as a subalgebra in the algebra of functions on the quantum group, we reformulate quantum vector bundles in terms of quantum symmetric pairs. In this way, we prove complete reducibility of modules over the corresponding coideal stabilizer subalgebras, via the quantum Frobenius reciprocity.
Highlights
We quantize equivariant vector bundles over complex projective spaces regarded as one-sided projective modules over coordinate rings
The main idea underlying quantization is to realize the module of global sections of an associated vector bundle by a vector space of linear maps between certain highest weight modules over the classical/quantum total group
From this point of view the problem was addressed within the theory of dynamical twist in [DM1], for homogeneous spaces with Levi isotropy subgroups
Summary
We quantize equivariant vector bundles over complex projective spaces regarded as one-sided projective modules over coordinate rings. The main idea underlying quantization is to realize the module of global sections of an associated vector bundle by a vector space of linear maps between certain highest weight modules over the classical/quantum total group. From this point of view the problem was addressed within the theory of dynamical twist in [DM1], for homogeneous spaces with Levi isotropy subgroups. G., algebras of functions) admit a similar approach with regard to semi-simple conjugacy classes (consisting of semi-simple elements) with non-Levi isotropy subgroups, [M1, AM] Those findings suggest a uniform quantization scheme for associated vector bundles over all semi-simple conjugacy classes, making use of highest weight modules. We switch to quantum symmetric pairs and give a realization of projective (left) Cq[Pn]-modules through B-invariants, to the classical induced representations
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