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S S symmetry Review Poisson–Lie Groups and Gauge Theory Catherine Meusburger Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen, Germany; [email protected] Abstract: We review Poisson–Lie groups and their applications in gauge theory and integrable sys- tems from a mathematical physics perspective. We also comment on recent results and developments and their applications. In particular, we discuss the role of quasitriangular Poisson–Lie groups and dynamical r-matrices in the description of moduli spaces of flat connections and the Chern–Simons gauge theory. Keywords: Poisson–Lie groups; Lie bialgebras and r-matrices; integrable models; Poisson homogeneous spaces 1. Poisson–Lie Groups This review article originated from a lecture series on Poisson–Lie groups given at the 12th International ICMAT Summer School on Geometry, Mechanics and Control in Santiago de Compostela. It is meant to provide an introduction that is accessible to physicists and mathematicians with no background on this topic, while at the same time covering current results. Citation: Meusburger, C. Poisson-Lie Due to the large body of work on Poisson–Lie groups and their numerous applications, Groups and Gauge Theory. Symmetry it is impossible to do justice to all the work on this topic. For this reason, we focus on 2021, 13, 1324. https://doi.org/ certain aspects and omit many others. For each result, we either cite the work where it 10.3390/sym13081324 was first discovered or, where appropriate and available, refer the reader to a textbook or review article and the references therein. Academic Editors: Ángel Ballesteros, Short and accessible introductions to the topic with emphasis on different aspects Giulia Gubitosi and Francisco J. are given in lecture notes by A. G. Reyman [1], by Y. Kosmann-Schwarzbach [2], by Herranz M. Audin [3], which also contain detailed lists of references. A good introduction to Poisson geometry and Poisson–Lie groups is the textbook [4]. Other textbooks that cover Received: 15 June 2021 Poisson–Lie groups are [5,6]. For a textbook presenting Poisson–Lie groups from the Accepted: 19 July 2021 point of view of integrable systems, see [7]. For an accessible introduction to dynamical Published: 22 July 2021 r-matrices, see the textbook [8]. Publisher’s Note: MDPI stays neutral Motivation with regard to jurisdictional claims in A Poisson–Lie group is a Lie group that is also a Poisson manifold in such a way that published maps and institutional affil- its multiplication is a Poisson map. Poisson–Lie groups arise in the description of integrable iations. systems and in the context of two-and three-dimensional gauge theories. They can be viewed as the classical counterparts of quantum group symmetries in certain quantum systems, such as quantum integrable systems and quantised Chern–Simons or BF-Theories. Both, quantum groups and Poisson–Lie groups, admit an infinitesimal description in terms Copyright: © 2021 by the authors. of Lie algebras with additional structure that is the infinitesimal counterpart of the Poisson Licensee MDPI, Basel, Switzerland. structure. Such a Lie algebra is called a Lie bialgebra. This article is an open access article One motivation for Poisson–Lie groups arises from Lie group actions on Poisson distributed under the terms and manifolds. Phase spaces of physical systems are usually Poisson manifolds, and Lie group conditions of the Creative Commons Attribution (CC BY) license (https:// actions on these manifolds arise from physical or gauge symmetries of these systems. This creativecommons.org/licenses/by/ includes integrable systems and constrained Hamiltonian systems. 4.0/). Symmetry 2021, 13, 1324. https://doi.org/10.3390/sym13081324 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 1324 2 of 39 The functions that are invariant under such Lie group actions play a distinguished role. They correspond to functions on their orbit space and Poisson commute with the Hamiltonians for the Lie group actions. In this sense, they are conserved quantities of the system. If the group action under consideration represents a gauge symmetry, the group action relates different points on an extended, non-physical phase space that correspond to the same physical state of the system. In this case, the invariant functions represent functions on the gauge invariant phase space, the physical observables of the theory. The functions that are invariant under a Lie group action form a subalgebra of the algebra of all functions on the Poisson manifold. The question is if they form a Poisson subalgebra, or, in other words, if the Poisson bracket of two invariant functions is again invariant. This must hold in particular for the conserved quantities of an integrable system and for the physical observables of a theory with gauge symmetries. Poisson actions of Poisson–Lie groups are a natural setting in which this is the case. Definition 1. 1.A Poisson manifold is a smooth manifold M together with a map f , g : C¥(M) × C¥(M) ! C¥(M), the Poisson bracket, that satisfies ¥ (a) antisymmetry: f f1, f2g = −f f2, f1g for all f1, f2 2 C (M). (b) Leibniz identity: f f1 · f2, f3g = f f1, f3g f2 + f f2, f3g f1. (c) Jacobi identity: ff f1, f2g, f3g + ff f2, f3g, f1g + ff f3, f1g, f2g = 0. 2. A Poisson map from a Poisson manifold (M, f , gM) to a Poisson manifold (N, f , gN) is a smooth map f : M ! N that satisfies ¥ f f1 ◦ f, f2 ◦ fgM = f f1, f2gN ◦ f 8 f1, f2 2 C (N). Example 1. 1. Every smooth manifold M is a Poisson manifold with the trivial Poisson structure f f1, f2g ¥ = 0 for all f1, f2 2 C (M). The identity map idM : M ! M is a Poisson map for any Poisson manifold M. 2. If M and N are Poisson manifolds, then M × N becomes a Poisson manifold with the product Poisson structure f f1, f2gM×N(m, n) = f f1(−, n), f2(−, n)gM(m) + f f1(m, −), f2(m, −)gN(n). (1) The projections pM : M × N ! M, (m, n) 7! m and pN : M × N ! N (m, n) 7! n are Poisson maps. For all Poisson maps f : M ! M0 and y : N ! N0, the map f × y : M × N ! M0 × N0, (m, n) 7! (f(m), y(n)) is a Poisson map. 3. A symplectic manifold (M, w) is a smooth manifold M together with a non-degenerate closed 2-form w on M. Every symplectic manifold (M, w) is a Poisson manifold with the Poisson bracket f g = = ( −) f1, f2 X f1 . f2 with the vector field X f defined by d f w X f , . (2) ¥ The non-degeneracy of w is required to define the vector field X f for any function f 2 C (M). The antisymmetry of w ensures the antisymmetry, and the closedness of w ensures the Jacobi identity for the Poisson bracket. 4. A Poisson manifold with a Poisson bracket of the form (2) is called a symplectic Poisson manifold. A Poisson manifold M is symplectic if and only if f f1, f2g(m) = 0 for all ¥ f2 2 C (M) implies that d f1(m) = 0. In this case, (2) defines a 2-form w on M, and the Jacobi identity ensures that w is closed. Symmetry 2021, 13, 1324 3 of 39 5. For every smooth manifold M, the cotangent bundle T∗ M has a canonical symplectic structure. If we interpret functions on M and vector fields on M as functions on T∗ M, their Poisson brackets are given by f f1, f2g = 0 fX1, f1g = X1. f1 fX1, X2g = [X1, X2], ¥ where f1, f2 2 C (M),X1, X2 2 Vec(M) and [ , ] denotes the Lie bracket of vector fields [X, Y]. f = X.(Y. f ) − Y.(X. f ) 8X, Y 2 Vec(M), f 2 C¥(M). (3) We now consider Lie group actions on Poisson manifolds. In the following, we assume that all Lie groups are finite-dimensional. Definition 2. 1. A Lie group G is a smooth manifold G with a group structure such that the multiplication map m : G × G ! G, (g, h) ! gh and the inversion i : G ! G, g 7! g−1 are smooth. 2. A Lie group action of G on a smooth manifold M is a smooth map B : G × M ! M, (g, m) 7! g B m with (gh) B m = g B (h B m) 1 B m = m 8g, h 2 G, m 2 M. • The set G B m = fg B m j g 2 Gg for m 2 M is called the orbit of m 2 M. • The set GnM = fG B m j m 2 Mg of orbits is called the orbit space. • The set of invariant functions on M is denoted ¥ G ¥ C (M) = f f 2 C (M) j f (g B m) = f (m) 8m 2 M, g 2 Gg. Example 2. 1. If G and H are Lie groups, then G × H is a Lie group with (g, h) · (g0, h0) = (gg0, hh0). It is called the direct product of G and H. 2. Rn and Cn with the usual addition are Lie groups. 3. Any closed subgroup of the group GL(n, C) of invertible n × n-matrices is a Lie group. A Lie group of this form is called a matrix Lie group. This includes the • special linear groups SL(n, C) = fM 2 Mat(n × n, C) j det M = 1g SL(n, R) = fM 2 Mat(n × n, R) j det M = 1g • unitary groups U(n) = fM 2 Mat(n × n, C) j M† M = 1g • special unitary groups SU(n) = U(n) \ SL(n, C) • orthogonal groups T −1p 0 O(p, q) = fM 2 Mat(n × n, R) j M Ip,q M = Ip,qg Ip,q = 0 1q O(n) = O(0, n) with p, q ≥ 0, p + q = n • special orthogonal groups SO(p, q) = O(p, q) \ SL(n, R) SO(n) = SO(0, n) Symmetry 2021, 13, 1324 4 of 39 • symplectic groups T 0 1n Sp(n, R) = fM 2 Mat(2n × 2n, R) j M In M = Ing In = −1n 0 4.

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