Quantum coadjoint orbits in gl( n )*

Letters in Mathematical Physics, 2013

Let O be a closed Poisson conjugacy class of the complex algebraic Poisson group GL(n) relative to the Drinfeld-Jimbo factorizable classical r-matrix. Denote by T the maximal torus of diagonal matrices in GL(n). With every a ∈ O ∩ T we associate a highest weight module M a over the quantum group U q gl(n) and an equivariant quantization C ,a [O] of the polynomial ring C[O] realized by operators on M a . All quantizations C ,a [O] are isomorphic and can be regarded as different exact representations of the same algebra, C [O]. Similar results are obtained for semisimple adjoint orbits in gl(n) equipped with the canonical GL(n)-invariant Poisson structure. Mathematics Subject Classifications: 81R50, 81R60, 17B37.

Pacific Journal of Mathematics, 2001

In this paper we consider the problem of deformation quantization of the algebra of polynomial functions on coadjoint orbits of semisimple Lie groups. The deformation of an orbit is realized by taking the quotient of the universal enveloping algebra of the Lie algebra of the given Lie group, by a suitable ideal. A comparison with geometric quantization in the case of SU(2) is done, where both methods agree.

Israel Journal of Mathematics, 2003

We prove that the reflection equation (RE) algebra L R associated with a finite dimensional representation of a quasitriangular Hopf algebra H is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show that L R is a module algebra over the twisted tensor square H R ⊗ H and the double D(H). We define FRT-and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces. *

2013

In this paper we describe a multiparameter deformation of the function algebra of a semisimple coadjoint orbit. In the first section we use the representation of the Lie algebra on a generalized Verma module to quantize the Kirillov bracket on the family of semisimple coadjoint orbits of a given orbit type. In the second section we extend this construction to define a deformation in the category of representations of the quantized enveloping algebra. In an earlier paper we used cohomological methods to prove the existence of a two parameter family quantizing a compatible pair of Poisson brackets on any symmetric coadjoint orbit. This paper gives a more explicit algebraic construction which includes more general orbit types and which we prove to be flat in all parameters. 1 Quantizing the Kirillov bracket Assume we have the following data: a simple Lie algebra, G, over the complex field, C, with corresponding Lie group, G, and a Cartan subalgebra, H, together with a system of simple positive roots. Let G = N − ⊕ H ⊕ N + , be the corresponding Cartan decomposition, where N + , N − are the nilpotent subalgebras which are, respectively, the sum of positive root spaces and the sum of the negative root spaces. Denote by G * and H * the C linear duals of G and H. Given λ ∈ G * , let O λ be the orbit of λ under the coadjoint action of G on G *. Finally, let S(G) be the symmetric algebra of G considered as polynomial functions on G * and I λ the ideal of polynomials vanishing on the orbit O λ. Then F λ = S(G)/I λ is the algebra of functions on O λ , with Poisson structure given by the standard Kostant-Kirillov-Souriau (KKS) bracket {f, g}(λ) = λ, [df λ , dg λ ]. Consider the G orbits of semisimple elements, that is, linear functions conjugate under the coadjoint action to ones which vanishes on N + ⊕ N −. In any such orbit we can pick as origin an element, λ, which is the trivial extension to G of a functional on H. The stabilizer of such an element will be a subalgebra generated by H and a subset of the simple roots. Such a subalgebra is called a Levi subalgebra. The set of all λ with stabilizer equal to a fixed Levi algebra L is parametrized by a subspace of H * minus its intersection with a family of coordinate hyperplanes.

Symmetry, 2021

We review some of the main achievements of the orbit method, when applied to Poisson– Lie groups and Poisson homogeneous spaces or spaces with an invariant Poisson structure. We consider C∗-algebra quantization obtained through groupoid techniques, and we try to put the results obtained in algebraic or representation theoretical contexts in relation with groupoid quantization.

Journal of Algebra, 2019

We give simple formulas for the elements c k appearing in a quantum Cayley-Hamilton formula for the reflection equation algebra (REA) associated to the quantum group Uq(gl N), answering a question of Kolb and Stokman. The c k 's are certain canonical generators of the center of the REA, and hence of Uq(gl N) itself; they have been described by Reshetikhin using graphical calculus, by Nazarov-Tarasov using quantum Yangians, and by Gurevich, Pyatov and Saponov using quantum Schur functions; however no explicit formulas for these elements were previously known. As byproducts, we prove a quantum Girard-Newton identity relating the c k 's to the so-called quantum power traces, and we give a new presentation for the quantum group Uq(gl N), as a localization of the REA along certain principal minors.

Journal of Geometry and Physics, 1988

The functional integral for the quantization of the coadjoint orbits of the unitary and orthogonal groups is given by means of an explicit construction of the corresponding~Darbouiø> variables.

Eprint Arxiv Math 0111132, 2001

In this paper we make a review of the results obtained in previous works by the authors on deformation quantization of coadjoint orbits of semisimple Lie groups. We motivate the problem with a new point of view of the well known Moyal-Weyl deformation quantization. We consider only semisimple orbits. Algebraic and differential deformations are compared.

Regular and Chaotic Dynamics, 2008

We introduce a family of compatible Poisson brackets on the space of 2 × 2 polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the XXX Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top.

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.