Hilbert–Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

Abstract

We describe the exponential map from an infinite-dimensional Lie algebra to an infinitedimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is −∞.

The early proofs of the theorem of Campbell, Baker, Hausdorff, and Dynkin

“…For example, in this context, the so-called Baker-Campbell-Hausdorff groups are particularly significant: for related topics see Birkhoff (1936Birkhoff ( , 1938, Boseck et al (1981), Czyż (1989Czyż ( , 1994, Dynkin (1950), Glöckner (2002a,b,c), Glöckner and Neeb (2003), Gordina (2005), Hilgert and Hofmann (1986), Hofmann (1972Hofmann ( , 1975, Hofmann and Morris (2007), Hofmann and Neeb (2009), Neeb (2006), Omori (1997), Robart (1997Robart ( , 2004, Schmid (2010), Van Est and Korthagen (1964), Vasilescu (1972), and Wojtyński (1998). Finally, the interest in providing new and simpler proofs of the CBHD Theorem seems to have been renewed throughout the century: see e.g., Yosida (1937), Cartier (1956), Eichler (1968), Djoković (1975), Veldkamp (1980), and Tu (2004).…”

Möbius transformations and the configuration space of a Hilbert snake

“…Hilbert-Schmidt group of orthochronous Lorentz transformations. Given a Hilbert space H, we first recall results of [7], about some particular Lie sub-algebras of L(H) of the Lie algebra L(H) of bounded operators on H.…”

A Brownian Motion on the Diffeomorphism Group of the Circle

“…1.1 by using the theory of stochastic differential equations in Hilbert spaces as developed by G. DaPrato and J. Zabczyk in [2]. Using this method, M. Gordina [4][5][6] and M. Wu [8] constructed a Brownian motion in several Hilbert-Schmidt groups. The construction relied on the fact that these Hilbert-Schmidt groups are Hilbert Lie groups.…”