nLab Poisson Lie group

Idea

A Poisson Lie group (often written Poisson-Lie group) or Lie Poisson group $G$ is a Lie group and a Poisson manifolds, the two structures being compatible such that the group product is a smooth map of Poisson manifolds where $G\times G$ has the product Poisson structure.

Warning: the inverse map is NOT a Poisson map unless G has the trivial Poisson structure, in fact it is an anti-Poisson map.

Deformation quantizations of Poisson Lie groups are Hopf algebras. The usual quantum groups have smaller number of quantum subgroups (i.e. Hopf quotient algebras) than the corresponding Lie group has, namely only only those whose classical limits are not only Lie subgroups but Poisson Lie subgroups.

Properties

Relation to quantum groups

One can regard Poisson groups as the classical limit of quantum groups: a theorem by Drinfeld established a bijection between connected, simply connected Poisson Lie groups and Lie bialgebras.

Examples

Additive Poisson Lie groups

A Lie-Poisson structure is an additive Poisson Lie group (e.g. Kosmann-Schwarzbach 04, p. 46).

If $H$ is any (finite dimensional) Lie group then the dual $(T_e H)^*$ of its tangent Lie algebra has a canonical bracket introduced by Kirillov which makes it into a Poisson Lie group. To this aim one identifies $(T_e H)^*$ with its own tangent space $T_u (T_e H)^*$ and interprets the differential $df$ of a function $f:(T_e H)^*\to \mathbb{R}$ as a function $(T_e H)^*\to ((T_e H)^*)^*\cong T_e H$ where the finite dimensionality is used. Then Kirillov defines

$$\lbrace f_1, f_2\rbrace (u) = \langle [(df_1)_u, (df_2)_u, u\rangle$$

Given two Lie groups $H,K$, the Lie algebra homomorphisms $T_e H \to T_e K$ are in 1-1 correspondence with the Poisson Lie maps $(T_e K)^* \to (T_e H)^*$.

Related concepts

• dressing action

• Poisson Lie 2-group

• Unrelated is the concept of the Lie group that integrates the Lie algebra inside a Poisson algebra. This is instead called a quantomorphism group.

References

• V. Chari, A. Pressley, A guide to quantum groups, Camb. Univ. Press 1994

• S. Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.

• Gloria Marí Beffa, A transverse structure for the Lie-Poisson bracket on the dual of the Virasoro algebra, Pacific J. Math. 163 (1994), no. 1, 43–72, euclid

• M. A. Semenov-Tian-Shansky, Группы Пуассона–Ли. Квантовый принцип двойственности и скрученный квантовый дубль, Teoret. Mat. Fiz., 1992, 93:2, 302–329 (in Russian) pdf; in English: Poisson–Lie groups. The quantum duality principle and the twisted quantum double, Theoret. and Math. Physics 1992, 93:2, 1292–1307 doi; Poisson groups and dressing transformations, Zap. Nauchn. Sem. LOMI, 150 (1986), 119–142 mathnet.ru; Poisson Lie groups, quantum duality principle, and the quantum double, in: Math. aspects of conformal and topological field theories and quantum groups (South Hadley, MA, 1992), 219–248, Contemp. Math. 175, AMS 1994.

• M. A. Semenov-Tian-Shansky, Classical $r$-matrices, Lax equations, Poisson Lie groups and dressing transformations, in: Field theory, quantum gravity and strings, II (Meudon/Paris, 1985/1986), 174–214, Lec. Notes in Phys. 280, Springer 1987, MR89g:58098

• A. P. Fordy, A. G. Reyman, M. A. Semenov-Tian-Shansky, Classical $r$-matrices and compatible Poisson brackets for coupled KdV systems, Lett. Math. Phys. 17 (1989), no. 1, 25–29, doi, MR90e:58061

• A. Yu. Alekseev, A. Z. Malkin, Symplectic structures associated to Lie-Poisson groups, Comm. Math. Phys. 162 (1994), no. 1, 147–173.

• T. Tao’s blog: The Euler-Arnold equation

• Peter Olver, Applications of Lie groups to differential equations, Springer

• A. Cannas da Silva, Alan Weinstein, Geometric models for noncommutative algebras, Berkeley Math. Lec. Notes Series, AMS 1999, pdf

• Yvette Kosmann-Schwarzbach, Groupes de Lie-Poisson quasitriangulaires, in: Géométrie symplectique et mécanique (La Grande Motte, 1988), 161–177, Springer LNM 1416, 1990.

• Yvette Kosmann-Schwarzbach, Lie bialgebras, Poisson Lie groups and dressing transformations, in Integrability of Nonlinear Systems, Second edition, Lecture Notes in Physics 638, Springer-Verlag, 2004, pp. 107-173. (pdf)

• A. G. Reyman, Poisson structures related to quantum groups, in: Quantum groups and their applications in physics (Varenna, 1994), 407–443, Proc. Internat. School Phys. Enrico Fermi, 127, IOS, Amsterdam, 1996, MR97j:58052

• Nicola Ciccoli, Quantization of co-isotropic subgroups, Lett. Math. Phys. 42:2 (1997) 123–138, doi, MR98k:58252

• Renaud Brahami, Cluster X-varieties for dual Poisson-Lie groups I, II, arxiv/1005.5289, arxiv/1006.4583

• László Fehér, Ctirad Klimčík, Poisson-Lie generalization of the Kazhdan-Kostant-Sternberg reduction, Lett. Math. Phys. 87 (2009), no. 1-2, 125–138, doi, MR2010c:53122)

• V. Fock, A. B. Goncharov, Cluster X-varieties, amalgamation and Poisson-Lie groups, math.RT/0508408.

• Ivan Calvo, Fernando Falceto, David Garcia-Alvarez, Topological Poisson sigma models on Poisson-Lie groups, JHEP 0310 (2003) 033, arXiv.

• Philip Boalch, Stokes matrices and Poisson Lie groups, Invent. math. 146, 479-506 (2001), math.DG/0011062.

• Ctirad Klimčík, Pavol Ševera, T-duality and the moment map, IHES/P/96/70, hep-th/9610198; Poisson-Lie T-duality: open strings and D-branes, CERN-TH/95-339. Phys.Lett. B376 (1996) 82-89, hep-th/9512124

• Anton Alekseev, Ctirad Klimčík, Arkady Tseytlin, Quantum Poisson-Lie T-duality and WZNW model, Nucl. Phys. B458:430-444 (1996) hep-th/9509123

• David Li-Bland, Pavol Ševera, On deformation quantization of Poisson-Lie groups and moduli spaces of flat connections, arXiv/1307.2047