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**Infinite dimensional Lie groups with applications to mathematical physics.**
*(English)*
Zbl 1063.22020

Recent developments and achievements in the theory of infinite-dimensional Lie groups and their physical applications (1978, 1984, 1987, 1990, 1991, 1994, Schmid; 1996, 2000, 2003, Eichhorn, Schmid) are reviewed.

The work is organized as follows: 1. Introduction. 2. Infinite-dimensional Lie groups. (Basic definitions and properties. Classical Lie groups. Classical results in finite dimensions which are not true in infinite dimensions. Infinite-dimensional examples). 3. Diffeomorphism groups. (Overview of \(\text{Diff}(M)\). The manifold structure on \(C(M,M)\). The diffeomorphism group \(\text{Diff}(M)\). ILH-Lie groups and ILH-Lie algebras. Gauge groups). 4. Subgroups of diffeomorphism groups and applications. (Volume preserving diffeomorphisms and fluid dynamics. Canonical transformations (symplectomorphisms) and plasma physics. Contact transformations on \(\dot T *M\). Globally Hamiltonian vector fields. The group of quantomorphisms. The group of gauge transformations and quantum field theory). 5. BRST symmetry. (Quantum chromodynamics and quantum electrodynamics. Symmetries. Quantization. BRST symmetries. Anomalies. The Wess-Zumino consistency condition). 6. Lie groups of pseudodifferential and Fourier integral operators. (Pseudodifferential operators \(\Psi DO\). Fourier integral operators FIO). 7. \(\text{Diff}(M)\) and FIO for non-compact manifolds. Application to fluid dynamics and quantization. (Bounded geometry. Bounded maps \(C^{\infty,r}(M, N)\). The bounded diffeomorphism group \(\text{Diff}^{p,r}(M)\). Volume preserving and symplectic diffeomorphisms. Contact transformations on \(\dot T* M\). Pseudodifferential operators and Fourier integral operators on open manifolds). 8. Applications to fluid dynamics and quantization. (The KdV equation and the group of Fourier integral operators. The KdV equation and the Lie group UFIO\(*\). Hydrodynamics and the diffeomorphism group \(\text{Diff}^{\infty, r}_\mu(M)\). \(\Psi DO\) and quantization).

The work is organized as follows: 1. Introduction. 2. Infinite-dimensional Lie groups. (Basic definitions and properties. Classical Lie groups. Classical results in finite dimensions which are not true in infinite dimensions. Infinite-dimensional examples). 3. Diffeomorphism groups. (Overview of \(\text{Diff}(M)\). The manifold structure on \(C(M,M)\). The diffeomorphism group \(\text{Diff}(M)\). ILH-Lie groups and ILH-Lie algebras. Gauge groups). 4. Subgroups of diffeomorphism groups and applications. (Volume preserving diffeomorphisms and fluid dynamics. Canonical transformations (symplectomorphisms) and plasma physics. Contact transformations on \(\dot T *M\). Globally Hamiltonian vector fields. The group of quantomorphisms. The group of gauge transformations and quantum field theory). 5. BRST symmetry. (Quantum chromodynamics and quantum electrodynamics. Symmetries. Quantization. BRST symmetries. Anomalies. The Wess-Zumino consistency condition). 6. Lie groups of pseudodifferential and Fourier integral operators. (Pseudodifferential operators \(\Psi DO\). Fourier integral operators FIO). 7. \(\text{Diff}(M)\) and FIO for non-compact manifolds. Application to fluid dynamics and quantization. (Bounded geometry. Bounded maps \(C^{\infty,r}(M, N)\). The bounded diffeomorphism group \(\text{Diff}^{p,r}(M)\). Volume preserving and symplectic diffeomorphisms. Contact transformations on \(\dot T* M\). Pseudodifferential operators and Fourier integral operators on open manifolds). 8. Applications to fluid dynamics and quantization. (The KdV equation and the group of Fourier integral operators. The KdV equation and the Lie group UFIO\(*\). Hydrodynamics and the diffeomorphism group \(\text{Diff}^{\infty, r}_\mu(M)\). \(\Psi DO\) and quantization).

Reviewer: A. A. Bogush (Minsk)

### MSC:

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

58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |

81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |

22E70 | Applications of Lie groups to the sciences; explicit representations |