Quantum Orbit Method in the Presence of Symmetries (original) (raw)

Quantization of orbit bundles in

Let G be the complex general linear group and g its Lie algebra equipped with a factorizable Lie bialgebra structure; let U (g) be the corresponding quantum group. We construct explicit U (g)-equivariant quantization of Poisson orbit bundles O λ −→ O µ in g * .

On Quantum Orbit Method

1997

A version of quantum orbit method is presented for real forms of equal rank of quantum complex simple groups. A quantum moment map is constructed, based on the canonical isomorphism between a quantum Heisenberg algebra and an algebra of functions on a family of quantum G-spaces. For the series A, we construct some irreducible * -representations of Uqg which correspond to the semi-simple dressing orbits of minimal dimension in the dual Poisson Lie group. It is shown that some complimentary series representations correspond to some quantum 'tunnel' G-spaces which do not have a quasi-classical analog.

Invariant quantization in one and two parameters on semisimple coadjoint orbits of simple Lie groups

1998

We study one and two parameter quantizations of the function algebra on a semisimple orbit in the coadjoint representation of a simple Lie group subject to the condition that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group. We prove that the corresponding Poisson bracket must be the sum of the so-called R-matrix bracket and an invariant bracket. We classify such brackets for all semisimple orbits and show that they form a family of dimension equal to the rank equal to the second cohomology group of the orbit and then we quantize these brackets. A two parameter (or double) quantization corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on the orbit. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases.

Double Quantization on Some Orbits in the Coadjoint Representations of Simple Lie Groups

Communications in Mathematical Physics, 1999

Let A be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, G, with the Lie algebra g. We study one and two parameter quantizations A h and A t,h of A such that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group, U h (g).

Quantization of Poisson groups

Pacific Journal of Mathematics, 1998

Let G τ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfel'd structure of Poisson group; let H τ be its dual Poisson group. By means of quantum double construction and dualization via formal Hopf algebras, we construct new quantum groups U M q,φ (h)-dual of U M ′ q,φ (g)-which yield infinitesimal quantization of H τ and G τ ; we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for H τ what was known for G τ , completing the quantization of the pair (G τ , H τ).

Quantization of Poisson algebraic groups and Poisson homogeneous spaces

1995

This paper consists of two parts. In the first part we show that any Poisson algebraic group over a field of characteristic zero and any Poisson Lie group admits a local quantization. This answers positively a question of Drinfeld and generalizes the results of [BFGP] and [BP]. In the second part we apply our techniques of quantization to obtain some nontrivial examples of quantization of Poisson homogeneous spaces.

Quantization of Poisson groups -- II

1996

Let $ G^\tau $ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfeld structure of Poisson group; let $ H^\tau $ be its dual Poisson group. By means of Drinfeld's double construction and dualization via formal Hopf algebras, we construct new quantum groups $ U_{q,\phi}^M ({\frak h}) $ --- dual of $ U_{q,\phi}^{M'} ({\frak g}) $ --- which yield infinitesimal quantization of $ H^\tau $ and $ G^\tau ;westudytheirspecializationsatrootsof1(inparticular,theirclassicallimits),thusdiscoveringnewquantumFrobeniusmorphisms.Thewholedescriptiondualizefor; we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for ;westudytheirspecializationsatrootsof1(inparticular,theirclassicallimits),thusdiscoveringnewquantumFrobeniusmorphisms.Thewholedescriptiondualizefor H^\tau $ what was known for $ G^\tau ,completingthequantizationofthepair, completing the quantization of the pair ,completingthequantizationofthepair (G^\tau,H^\tau) $.