Quantum Flag Manifolds as Quotients of Degenerate Quantized Universal Enveloping Algebras

Abstract

Let g be a semi-simple Lie algebra with fixed root system, and U q pgq the quantization of its universal enveloping algebra. Let S be a subset of the simple roots of g. We show that the defining relations for U q pgq can be slightly modified in such a way that the resulting algebra U q pg; Sq allows a homomorphism onto (an extension of) the algebra PolpG q {K S,q q of functions on the quantum flag manifold G q {K S,q corresponding to S. Moreover, this homomorphism is equivariant with respect to a natural adjoint action of U q pgq on U q pg; Sq and the standard action of U q pgq on PolpG q {K S,q q.

I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type

“…Proof. To see that we get an algebra homomorphism from U q (b), we can apply the same argument as in [5,Lemma 2.11]. It then follows immediately that we have also an algebra homomorphism from U q (b − ) by applying the * -operation.…”

“…The key observation, allowing to connect the algebraic with the analytic framework, will be a variation on the fact that the Heisenberg algebra of the nilpotent part of the quantized Borel algebra can be realized as the function algebra on the (big open) quantum Schubert cell associated to g, see [10,Section 10] and the more recent works [30,11,5].…”

Quantum flag manifolds, quantum symmetric spaces and their associated universal K-matrices

“…These quantum flag manifolds are studied intensively both from the algebraic and the operator algebraic viewpoint, see e.g. [88,87,24,25,21,26,23,37,54,13,77,16,81].…”

Locally compact quantum groups

“…For example this method gives a way of defining SU q (1,1). In [12] series of quantum groups have been found as well.…”