Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure

Neshveyev

2015

Transformation Groups

Self Cite

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.

Section: Degenerate Quantized Universal Enveloping Algebramentioning

confidence: 99%

Section: Degenerate Quantized Universal Enveloping Algebramentioning

confidence: 99%

Section: Theorem 28 There Is a Surjective Unital˚-homomorphismmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The algebras U q pg; Sq are particular examples of algebras studied in [2]. They can be obtained by an appropriate rescaling of the generators of U q pgq and sending the parameters to 0.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Quantum Flag Manifolds as Quotients of Degenerate Quantized Universal Enveloping Algebras

Neshveyev

2015

Transformation Groups

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Ribbon Braided Module Categories, Quantum Symmetric Pairs and Knizhnik–Zamolodchikov Equations

Neshveyev

Tuset

et al. 2019

Commun. Math. Phys.

Self Cite

Let u be a compact semisimple Lie algebra, and σ be a Lie algebra involution of u. Let Rep q (u) be the ribbon braided tensor C * -category of admissible Uq(u)-representations for 0 < q < 1. We introduce three module C * -categories over Rep q (u) starting from the input data (u, σ). The first construction is based on the theory of 2-cyclotomic KZ-equations. The second construction uses the notion of quantum symmetric pair as developed by G. Letzter. The third construction uses a variation of Drinfeld twisting. In all three cases the module C * -category is ribbon twist-braided in the sense of A. Brochier-this is essentially due to B. Enriquez in the first case, is proved by S. Kolb in the second case, and is closely related to work of J. Donin, P. Kulish, and A. Mudrov in the third case. We formulate a conjecture concerning equivalence of these ribbon twist-braided module C * -categories, and confirm it in the rank one case.

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

Advances in Mathematics

Matassa

2020