Metrics \(\mathcal H_s\) and behaviours as \(s\downarrow \frac {1} {2}\) on loop groups
Let \(G\) be a semi-simple and compact Lie group carrying an \(\text{Ad}_G\)-invariant metric \(\langle.,.\rangle\,.\) For the Laplace operator \(\Delta=-d^2/d\theta^2\) on the circle \(S^1\) and \(s>0\) define on the space of smooth functions from \(S^1\) into the Lie algebra \(\mathfrak{g}\) the following metric: \[ | h| ^2_s:=\int_{S^1} \langle (1+\Delta)^sh,h\rangle \frac{d\theta}{2\pi}\,, \] then \({\mathcal H}_s({\mathfrak g})=\{h\in L^2(S,{\mathcal g}), | h| _s<\infty\}.\) From the point of view of stochastic analysis \(s=1/2\) is a critical value. The behaviour of the Ricci tensors \(\text{ Ric}^s\) is studied as \(s\) converges to \(1/2\) from above.
scientific article; zbMATH DE number 2107434