Hilbert--Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry (Q2573421)

The author studies infinite-dimensional Lie algebras \(\mathfrak g\subset B(H)\) of bounded operators on a Hilbert space \(H\), equipped with certain inner products and the corresponding norms \(|\cdot|\), such that their completion \({\mathfrak g}_\infty\) is again a subspace of \(B(H)\). In some cases \(\mathfrak g\) is assumed to be the direct limit of an increasing sequence \({\mathfrak g}_n\) of finite-dimensional Lie algebras. It is shown that in the case when \({\mathfrak g}_\infty\) is a Banach-Lie algebra, the exponential map is a diffeomorphism of a neighbourhood of 0 in \({\mathfrak g}_\infty\) onto a neighbourhood of 1 in the so-called Cameron-Martin group \(G_{CM}({\mathfrak g}_\infty)\subset B(H)\) associated with \({\mathfrak g}_\infty\). Moreover, if \({\mathfrak g}=\bigcup_n{\mathfrak g}_n\), the group \(\bigcup_nG_n\) is dense in \(G_{CM}({\mathfrak g}_\infty)\). For some Lie algebras \(\mathfrak g\) consisting of Hilbert-Schmidt operators, the Ricci curvature of the Levi-Civita connection associated with the inner product on \(\mathfrak g\) is also studied. There are examples where the Ricci curvature is \(-\infty\).