The heat semigroup and integrability of Lie algebras
Let \({\mathfrak g}\) be a finite-dimensional Lie algebra. A family \(V=\{V(x)\); \(x\in {\mathfrak g}\}\) of closed operators on a Banach space B is a representation of \({\mathfrak g}\) if the common domain \(B_{\infty}(V)\) of arbitrary finite products of elements from V is norm dense and if V respects the Lie brackets. Such a representation is denoted by \((B_{\infty},{\mathfrak g},V)\). If \(x_ 1,...,x_ n\) is a basis for \({\mathfrak g}\) then the Laplacian of the representation \((B_{\infty},{\mathfrak g},V)\) is defined by \(\Delta =- \sum^{n}_{1}V(x_ j)^ 2\). The authors consider various conditions under which the Laplacian is closable and generates a continuous one parameter semigroup S. They derive growth conditions of the type \(\| V(x_{i_ 1})\cdot...\cdot V(x_{i_ k})S_ ta\| \leq c_ k\| a\| /t^{k/2}\) from \((*)\quad \| V(x_ i)S_ ta\| \leq c\| a\| /t^{1/2}.\) These estimates are used to show that under certain circumstances (*) implies that a representation of \({\mathfrak g}\) can be integrated to a continuous representation of the simply connected Lie group G with Lie algebra \({\mathfrak g}\). Conversely it is shown that the differential representation \(d\pi\) of \({\mathfrak g}\) associated to a continuous representation \(\pi\) of G satisfies (*). The proof uses heat equation methods and the observation that the estimates essentially only have to be proved for the regular representation on \(L^ 1(G)\). The paper contains an example section which illustrates some aspects of the main results for sl(2,\({\mathbb{R}})\) and the Heisenberg algebra. Finally the \(L^ 1\)-estimates for the heat kernel derived earlier in the paper are extended to weighted \(L^ 1\)-spaces.
scientific article; zbMATH DE number 4081860
scientific article