The geometry of infinite-dimensional groups.

*(English)*Zbl 1153.22001
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 51. Berlin: Springer (ISBN 978-3-540-77262-0/hbk). xii, 304 p. (2009).

The aim of this monograph is an overview of various classes of infinite-dimensional Lie groups and applications, mainly to Hamiltonian mechanics, fluid dynamics, integrable systems and complex geometry. The authors present the unifying ideas of the theory by concentrating on specific types and examples of infinite-dimensional Lie groups.

The introductory Chapter I collects some key notions and facts from the theory of Lie groups and Hamiltonian systems. Beginning with the definition of Lie group and the main related concepts of its Lie algebra, the adjoint and coadjoint representations, central extensions of Lie groups and algebras are introduced, some notions from symplectic geometry including Arnold’s formulation of the Euler equations on a Lie group (i.e. equations for the geodesic flow with respect to a one-sided invariant metric on the group) are presented and many finite- and infinite-dimensional dynamical systems are described, including the Korteweg-de Vries (KdV) equation and the equations of magnetohydrodynamics. The Marsden-Weinstein Hamiltonian reduction as a method often used to describe complicated Hamiltonian systems is presented.

Chapter II is central in the monograph. In Section 1 the loop group for a compact Lie group is introduced as one of the most studied types of infinite-dimensional groups. Here its universal central extension and the relevant Lie algebra (affine Kac-Moody algebra) are constructed and corresponding coadjoint orbits are classified. Section 2 deals with the Lie group \(\text{Diff}(S^1)\) of orientation-preserving diffeomorphisms of the circle and its Lie algebra \(\text{Vect}(S^1)\) of smooth vector fields on the circle and also with their central extensions. It is shown that the Lie algebra of vector fields on the circle admits a unique nontrivial central extension – the Virasoro algebra, which gives rise to a central extension of the Lie group of circle diffeomorphisms – the Virasoro-Bott group. It turns out that the coadjoint orbit of the Virasoro-Bott group can be classified in a manner similar to that for the orbits of the loop groups. The Euler equations corresponding to right-invariant metrics on the Virasoro-Bott group, i.e. the KdV equation governing waves in shallow water and the related PDE among them, are described. The Euler nature of the KdV shows its complete integrability. Section 3 is devoted to various diffeomorphism groups, among them the groups of volume-preserving diffeomorphisms, their geometry and coadjoint orbits. The Euler geodesic equations for such groups and their generalizations deliver the Euler equations for ideal incompressible fluids on manifolds and also the equations of compressible fluids and magnetohydrodynamics. Here it is also discussed how invariants of the coadjoint action of certain diffeomorphism groups are related to knot theory and to the symplectic structure on the space of immersed curves in \(\mathbb R^3\). Section 4 studies the group of pseudodifferential symbols on the circle, which contains the algebra of differential operators as a subalgebra. This group can be endowed with the structure of a Poisson Lie group, where the relevant Poisson structure is given by the Adler-Gelfand-Dickey brackets. Naturally corresponding to this group are the dynamical systems such as the Kadomtsev-Petviashvili hierarchy, the higher \(n\)-KdV equations and the nonlinear Schrödinger equation. Section 5 deals with generalizations of the loop groups, elliptic Lie groups and corresponding Lie algebras. It turns out that many constructions for the affine algebras have their analogues for the elliptic case. In particular, the coadjoint orbits of the corresponding elliptic group are classified by equivalence classes of holomorphic G-bundles over the elliptic curve. Here it is shown that the Calogero-Moser dynamical systems provide a bridge that unites the three classes of Lie algebras: there is a universal construction of a Hamiltonian reduction on the dual of a Lie algebra, which for the finite-dimensional simple Lie algebras, affine Lie algebras, and elliptic Lie algebras leads to the Calogero-Moser integrable systems with, respectively, rational, trigonometric and elliptic potentials.

Chapter III discusses applications of the parallelism between the affine and elliptic Lie algebras, which resembles the “real-complex” correspondence. The infinite-dimensional Lie groups are groups of gauge transformations of principal bundles over real and complex surfaces. It is shown how the classification of coadjoint orbits of loop groups (resp. double loop groups) can be used to study the Poisson structure on the moduli space of flat connections (resp. semistable holomorphic bundles) on a Riemann surface (resp. a complex surface). This correspondence between real and complex cases leads to analogies between notions of differential topology (orientation, boundary, Stokes theorem) and those in algebraic geometry (a meromorphic differential form, its divisor of poles and Cauchy-Stokes formula). In the notion of polar homology these analogies are formalized.

Ten appendices contain several topics serving as explanation of some facts used in the basic text: root systems; compact Lie groups; Krichever-Novikov algebras; Kähler structures on the Virasoro and loop group coadjoint orbits; diffeomorphism groups and optimal mass transport; metrics and diameters of the group of Hamiltonian diffeomorphisms; semidirect extensions of the diffeomorphism; the Drinfeld-Sokolov reduction; the algebra \(\mathfrak{gl}_\infty\); torus actions on the moduli space of flat connections.

The introductory Chapter I collects some key notions and facts from the theory of Lie groups and Hamiltonian systems. Beginning with the definition of Lie group and the main related concepts of its Lie algebra, the adjoint and coadjoint representations, central extensions of Lie groups and algebras are introduced, some notions from symplectic geometry including Arnold’s formulation of the Euler equations on a Lie group (i.e. equations for the geodesic flow with respect to a one-sided invariant metric on the group) are presented and many finite- and infinite-dimensional dynamical systems are described, including the Korteweg-de Vries (KdV) equation and the equations of magnetohydrodynamics. The Marsden-Weinstein Hamiltonian reduction as a method often used to describe complicated Hamiltonian systems is presented.

Chapter II is central in the monograph. In Section 1 the loop group for a compact Lie group is introduced as one of the most studied types of infinite-dimensional groups. Here its universal central extension and the relevant Lie algebra (affine Kac-Moody algebra) are constructed and corresponding coadjoint orbits are classified. Section 2 deals with the Lie group \(\text{Diff}(S^1)\) of orientation-preserving diffeomorphisms of the circle and its Lie algebra \(\text{Vect}(S^1)\) of smooth vector fields on the circle and also with their central extensions. It is shown that the Lie algebra of vector fields on the circle admits a unique nontrivial central extension – the Virasoro algebra, which gives rise to a central extension of the Lie group of circle diffeomorphisms – the Virasoro-Bott group. It turns out that the coadjoint orbit of the Virasoro-Bott group can be classified in a manner similar to that for the orbits of the loop groups. The Euler equations corresponding to right-invariant metrics on the Virasoro-Bott group, i.e. the KdV equation governing waves in shallow water and the related PDE among them, are described. The Euler nature of the KdV shows its complete integrability. Section 3 is devoted to various diffeomorphism groups, among them the groups of volume-preserving diffeomorphisms, their geometry and coadjoint orbits. The Euler geodesic equations for such groups and their generalizations deliver the Euler equations for ideal incompressible fluids on manifolds and also the equations of compressible fluids and magnetohydrodynamics. Here it is also discussed how invariants of the coadjoint action of certain diffeomorphism groups are related to knot theory and to the symplectic structure on the space of immersed curves in \(\mathbb R^3\). Section 4 studies the group of pseudodifferential symbols on the circle, which contains the algebra of differential operators as a subalgebra. This group can be endowed with the structure of a Poisson Lie group, where the relevant Poisson structure is given by the Adler-Gelfand-Dickey brackets. Naturally corresponding to this group are the dynamical systems such as the Kadomtsev-Petviashvili hierarchy, the higher \(n\)-KdV equations and the nonlinear Schrödinger equation. Section 5 deals with generalizations of the loop groups, elliptic Lie groups and corresponding Lie algebras. It turns out that many constructions for the affine algebras have their analogues for the elliptic case. In particular, the coadjoint orbits of the corresponding elliptic group are classified by equivalence classes of holomorphic G-bundles over the elliptic curve. Here it is shown that the Calogero-Moser dynamical systems provide a bridge that unites the three classes of Lie algebras: there is a universal construction of a Hamiltonian reduction on the dual of a Lie algebra, which for the finite-dimensional simple Lie algebras, affine Lie algebras, and elliptic Lie algebras leads to the Calogero-Moser integrable systems with, respectively, rational, trigonometric and elliptic potentials.

Chapter III discusses applications of the parallelism between the affine and elliptic Lie algebras, which resembles the “real-complex” correspondence. The infinite-dimensional Lie groups are groups of gauge transformations of principal bundles over real and complex surfaces. It is shown how the classification of coadjoint orbits of loop groups (resp. double loop groups) can be used to study the Poisson structure on the moduli space of flat connections (resp. semistable holomorphic bundles) on a Riemann surface (resp. a complex surface). This correspondence between real and complex cases leads to analogies between notions of differential topology (orientation, boundary, Stokes theorem) and those in algebraic geometry (a meromorphic differential form, its divisor of poles and Cauchy-Stokes formula). In the notion of polar homology these analogies are formalized.

Ten appendices contain several topics serving as explanation of some facts used in the basic text: root systems; compact Lie groups; Krichever-Novikov algebras; Kähler structures on the Virasoro and loop group coadjoint orbits; diffeomorphism groups and optimal mass transport; metrics and diameters of the group of Hamiltonian diffeomorphisms; semidirect extensions of the diffeomorphism; the Drinfeld-Sokolov reduction; the algebra \(\mathfrak{gl}_\infty\); torus actions on the moduli space of flat connections.

Reviewer: Boris V. Loginov (Ul’yanovsk)

##### MSC:

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |

37K05 | Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) |

58B25 | Group structures and generalizations on infinite-dimensional manifolds |

53D30 | Symplectic structures of moduli spaces |

17B66 | Lie algebras of vector fields and related (super) algebras |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

17B68 | Virasoro and related algebras |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |