# Quantum Orbit Method in the Presence of Symmetries

## Abstract

**:**

## 1. Introduction

## 2. Quantization via Symplectic Groupoid

- pre-quantization;
- choice of Lagrangian polarization;
- Bohr–Sommerfeld conditions;

**Definition**

**1**

- 1.
- the curvature of ∇ equals the symplectic form ω;
- 2.
- σ is a norm one, groupoid cocycle, valued in the pull-back line bundle ${\partial}^{\ast}{E}^{\ast}$;
- 3.
- σ is covariantly constant, i.e., $d\sigma +{\partial}^{\ast}\theta \sigma =0$, where θ is a local primitive of ω.

**Definition**

**2**

**Definition**

**3.**

**Definition**

**4.**

**Proposition**

**1**

## 3. Review of Some Easy Cases

#### 3.1. Trivial Poisson

#### 3.2. Standard symplectic Euclidean space

#### 3.3. Linear Poisson

#### 3.4. Symplectic torus

## 4. Poisson Structures and Symmetry

**Definition**

**5.**

**Definition**

**6.**

- δ is a one-cocycle with respect to the adjoint representation, i.e.,:$$\delta \left([X,Y]\right)={\mathrm{ad}}_{X}^{2}\delta \left(Y\right)-{\mathrm{ad}}_{Y}^{2}\delta \left(X\right)$$where ${\mathrm{ad}}^{2}=\mathrm{ad}\otimes 1+1\otimes \mathrm{ad}$ is the extension of the adjoint action of $\mathfrak{g}$ on ${\wedge}^{2}\mathfrak{g}$.
- ${\delta}^{\ast}:{\wedge}^{2}{\mathfrak{g}}^{\ast}\to {\mathfrak{g}}^{\ast}$ is a Lie bracket.

**Theorem**

**1.**

**Definition**

**7.**

**Proposition**

**2.**

- Let H be a coisotropic subgroup of G. There exists a unique Poisson structure $\widehat{\pi}$ on $G/H$ such that the natural projection $G\to G/H$ is a Poisson map. With respect to the quotient action:$$g\xb7\left({g}_{1}H\right)=\left(g{g}_{1}\right)H$$the manifold $G/H$ is a Poisson homogeneous space, having at least one zero-dimensional leaf ${p}_{0}=H/H$.
- Let $(M,{\pi}_{M})$ be a Poisson homogeneous space having at least one zero-dimensional leaf ${p}_{0}$. Its stabilizer ${H}_{{p}_{0}}=\{g\in G\phantom{\rule{0.166667em}{0ex}}:g\xb7{p}_{0}={p}_{0}\}$ is then a coisotropic subgroup of G such that the canonical G-equivariant diffeomorphism between M and $G/{H}_{{p}_{0}}$ is a Poisson diffeomorphism.

#### 4.1. Abelian Poisson Structures

## 5. Quantum Heisenberg Group

## 6. Compact Quantum Homogeneous Spaces

**Theorem**

**2**

- To what extent are these results valid without the assumption if G is not a standard compact Poisson–Lie group?
- Does the quantum orbit method still hold for coisotropic quotients?

- It is known that there are obstructions to the existence of a real Lagrangian groupoid polarization on standard $SU\left(2\right)$: it is reasonable to expect such an obstruction to hold for more general standard compact Poisson–Lie groups G (though this is still to be proven; see [7], proposition 7.5).
- In some cases, covariant Poisson structures on homogeneous spaces come in pencils [13,42], and as such, it is reasonable to expect that a multiplicative integrable system can be constructed, giving rise to a mildly singular Lagrangian polarization. In fact, the above program was carried out successfully for $\mathbb{C}{\mathbb{P}}^{N}$ in [13], allowing an explicit description for the groupoid ${C}^{\ast}$-algebra, equivalent to the one given in terms of generators and relations.
- The groupoid ${C}^{\ast}$-algebra, in the case of Poisson quotients, was used to prove the quantum orbit correspondence for the Bruhat–Poisson structure on $\mathbb{C}{\mathbb{P}}^{n}$ in [5]. Work in progress is to prove that this holds also for the non-standard case.
- It could be very interesting to work out in some detail the case of the Poisson structure on the space of Stokes matrices introduced in [43]. This Poisson structure admits a natural interpretation in terms of the coisotropic quotient of the standard dual $SU{\left(n\right)}^{\ast}$ and can be quantized in terms of generators and relations (see [19] and the references therein). When $n=3$, its symplectic groupoid is explicitly known [44]. The underlying symplectic foliation is determined by level sets of the Markoff polynomial $xyz-({x}^{2}+{y}^{2}+{z}^{2})$ and is of particular interest since, different from the standard compact case, there appears both simply connected and non-simply connected leaves. As explained in the next section, this should reflect in a more subtle behavior of the orbit correspondence where irreducible representations of the homotopy group of leaves (rigged leaves) should be taken into account.

## 7. Quantizing Locally Abelian Poisson Manifolds

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kirillov, A. Lectures on the Orbit Method, Graduate Studies in Mathematics; American Mathematical Society: Providence, RI, USA, 2004. [Google Scholar]
- Vogan, D. Review of “Lectures on the orbit method” by A. A. Kirillov. Bull. Am Math. Soc.
**2005**, 42, 535–544. [Google Scholar] [CrossRef] [Green Version] - Neshveyev, S.; Tuset, L. Quantized algebras of functions with Poisson stablizers. Commun. Math. Phys.
**2012**, 312, 223–225. [Google Scholar] [CrossRef] [Green Version] - Stokman, J.V.; Dijkhuizen, M. Quantized flag manifolds and ∗-irreducible representations. Commun. Math. Phys.
**1999**, 203, 297–324. [Google Scholar] [CrossRef] [Green Version] - Ciccoli, N. A new approach to quantum orbit method for standard quantum CP
^{N}. Rend. Sem. Mat. Pol. Torino**2016**, 74, 45–53. [Google Scholar] - Weinstein, A. Noncommutative geometry and geometric quantization. In Symposium on Symplectic Geometry and Mathematical Physics in Honor of Jean-Marie Souriau; Birkhäuser: Boston, MA, USA, 1991. [Google Scholar]
- Hawkins, E. A groupoid approach to quantization. J. Symp. Geom.
**2008**, 6, 61–125. [Google Scholar] [CrossRef] [Green Version] - Crainic, M.; Loja-Fernandes, R. Integrability of Lie brackets. Ann. Math.
**2003**, 157, 575–620. [Google Scholar] [CrossRef] [Green Version] - Mackenzie, K. Lie Groupoids and Lie Algebroids in Differential Geometry; London Mathematical Society Lecture Series; Cambridge University Press: Cambridge, UK, 2005; Volume 213. [Google Scholar]
- Weinstein, A.; Xu, P. Extensions of symplectic groupoids and quantization. J. Reine Angew. Math.
**1991**, 417, 159–189. [Google Scholar] - Renault, J. Groupoid C
^{∗}Algebras. In Lecture Notes in Mathematics; Springer: New York, NY, USA, 1985; Volume 793. [Google Scholar] - Jotz-Lean, M. Foliated groupoids and infinitesimal ideal systems. Indag. Math.
**2014**, 25, 1019–1053. [Google Scholar] [CrossRef] - Bonechi, F.; Ciccoli, N.; Qiu, J.; Tarlini, M. Quantization of the Poisson manifold from integrability of the modular class. Commun. Math. Phys.
**2014**, 331, 851–885. [Google Scholar] [CrossRef] [Green Version] - Williams, D. Groupoid C*-Algebras, a Tool-Kit; American Mathematical Society: Providence, RI, USA, 2019. [Google Scholar]
- Brenken, B. Representations and automorphisms of the irrational rotation algebra. Pac. J. Math.
**1984**, 111, 257–282. [Google Scholar] [CrossRef] [Green Version] - Drinfeld, V. Quantum groups. Zap. Nauchnykh Semin. POMI
**1986**, 155, 18–49. [Google Scholar] - Korogodskij, L.I.; Soibelman, Y. Algebra of Functions on Quantum Groups I, Math. Surveys and Monographs; American Mathematical Society: Providence, RI, USA, 1998; Volume 56. [Google Scholar]
- Gomez, X. Classification of three-dimensional Lie bialgebras. J. Math. Phys.
**2000**, 41, 4349–4359. [Google Scholar] [CrossRef] - Ciccoli, N.; Gavarini, F. A global quantum duality principle for coisotropic subgroups and homogeneous spaces. Doc. Math.
**2014**, 19, 333–380. [Google Scholar] - Lu, J.H. Momentum mappings and reductions of Poisson Lie group actions. In Seminaire Sud-Rhodanien de Geometrie Berkeley; Springer: New York, NY, USA, 1991; pp. 209–226. [Google Scholar]
- Zakrzewski, S. Poisson homogeneous spaces. In Quantum Groups (Karpacz, 1994); Lukierski, J., Popowicz, Z., Sobczyk, J., Eds.; PWN: Warszaw, Poland, 1995; pp. 629–639. [Google Scholar]
- Ciccoli, N.; Sheu, A.J.-L. Covariant Poisson structures on complex Grassmannians. Commun. Anal. Geom.
**2006**, 14, 443–474. [Google Scholar] [CrossRef] [Green Version] - Bonechi, F.; Ciccoli, N.; Staffolani, N.; Tarlini, M. On the integration of Poisson homogeneous spaces. J. Geom. Phys.
**2008**, 58, 1519–1529. [Google Scholar] [CrossRef] - Bursztyn, H.; Iglesias-Ponte, D.; Lu, J.H. Dirac geometry and integration of Poisson homogeneous spaces. arXiv
**2019**, arXiv:1905.11453. [Google Scholar] - Xu, P. Poisson manifolds associated with group actions and classical triangular r-matrices. J. Funct. Anal.
**1993**, 112, 218–240. [Google Scholar] [CrossRef] [Green Version] - Khang, B.J. Non compact quantum groups arising from Heisenberg type Lie bialgebra. J. Oper. Theory
**2000**, 44, 303–334. [Google Scholar] - Khang, B.J. ∗-representations of a q-Heisenberg group algebra. Houst. J. Math.
**2002**, 28, 529–552. [Google Scholar] - Khang, B.J. Dressing orbits and a quantum Heisenberg group algebra. Ill. J. Math.
**2004**, 48, 609–634. [Google Scholar] [CrossRef] - Cahen, M.; Ohn, C. Bialgebra structures on the Heisenber algebra. Bull. Acad. Roy Belgique
**1989**, 75, 315–321. [Google Scholar] - Szymczak, S.; Zakrzewski, S. Quantum deformation of the Heisenberg group obtained by geometric quantization. J. Geom. Phys.
**1990**, 7, 553–567. [Google Scholar] [CrossRef] - Celeghini, E.; Giachetti, R.; Sorace, N.; Tarlini, M. The quantum Heisenberg group H(1)
_{q}. J. Math. Phys.**1991**, 32, 1155–1158. [Google Scholar] [CrossRef] - Lu, J.H.; Weinstein, A. Poisson–Lie groups, dressing transformation and Bruhat decomposition. J. Diff. Geom.
**1990**, 31, 501–526. [Google Scholar] [CrossRef] - Karolinsky, E. A classification of Poisson homogeneous spaces of complex reductive Poisson–Lie groups. Banach Cent. Publ.
**2000**, 51, 103–108. [Google Scholar] - Khoroshkin, S.; Radul, A.; Rubtsov, V. A family of Poisson structures on Hermitian symmetric spaces. Commun. Math. Phys.
**1992**, 152, 299–315. [Google Scholar] [CrossRef] - Soibelman, Y. Orbit method for the algebra of functions on quantum groups and coherent states I. Int. Math. Res. Not.
**1993**, 6, 151–163. [Google Scholar] [CrossRef] - Levendorskij, S.L.; Soibelman, Y. Algebras of functions on compact quantum groups, Schubert cells and quantum tori. Commun. Math. Phys.
**1991**, 139, 141–170. [Google Scholar] [CrossRef] - Stokman, J. The quantum orbit method for generalized flag manifolds. Math. Res. Lett.
**2003**, 10, 469–481. [Google Scholar] [CrossRef] [Green Version] - Ballesteros, A.; Gubitosi, G.; Gutierrez-Sagredo, I.; Herranz, F.J. The κ-Newtonian and κ-Carrollian algebras and their noncommutative spacetimes. Phys. Lett. B
**2020**, 805, 135461. [Google Scholar] [CrossRef] - Ballesteros, A.; Herranz, F.J.; Meusberger, C.; Naranjo, P. Twisted (2+ 1) κ-AdS algebra, Drinfel’d doubles and non-commutative spacetimes. Symmetry Integr. Geom. Methods Appl.
**2014**, 10, 52. [Google Scholar] [CrossRef] [Green Version] - Ballesteros, A.; Herranz, F.J.; Meusberger, C. Three-dimensional gravity and Drinfel’d doubles: Spacetimes and symmetries from quantum deformations. Phys. Lett. B
**2010**, 687, 375–381. [Google Scholar] [CrossRef] [Green Version] - Lu, J.H.; Weinstein, A. Grupoides symplectiques doubles de groupes de Lie-Poisson. Compt. Rend. Sci. Ser. I Math.
**1989**, 309, 951–954. [Google Scholar] - Bonechi, F.; Qiu, J.; Tarlini, M. Complete integrability from Poisson-Nijenhuis structures on compact Hermitian symmetric spaces. J. Symp. Geom.
**2018**, 16, 1167–1208. [Google Scholar] [CrossRef] [Green Version] - Ugaglia, M. On a Poisson structure on the spaces of Stokes matrices. Int. Math. Res. Not.
**1999**, 9, 473–493. [Google Scholar] [CrossRef] - Bondal, A. A symplectic groupoid of triangular bilinear forms and the braid group. Izv. Math.
**2004**, 68, 659. [Google Scholar] [CrossRef] - Ciccoli, N. Quantum orbit method for the CLM 3-sphere. 2021; submitted. [Google Scholar]
- Auslander, L.; Kostant, B. Polarization and unitary representations of solvable Lie groups. Invent. Math.
**1971**, 14, 255–354. [Google Scholar] [CrossRef] - Ciccoli, N. Quantum orbit method for locally Abelian Poisson manifolds. 2021; in preparation. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ciccoli, N.
Quantum Orbit Method in the Presence of Symmetries. *Symmetry* **2021**, *13*, 724.
https://doi.org/10.3390/sym13040724

**AMA Style**

Ciccoli N.
Quantum Orbit Method in the Presence of Symmetries. *Symmetry*. 2021; 13(4):724.
https://doi.org/10.3390/sym13040724

**Chicago/Turabian Style**

Ciccoli, Nicola.
2021. "Quantum Orbit Method in the Presence of Symmetries" *Symmetry* 13, no. 4: 724.
https://doi.org/10.3390/sym13040724