The Lie Theory

This chapter discusses the Lie theory. In the Lie theory, an essential role is played by the so-called three main Lie theorems, which in totality lead to the determination of a unique correspondence between the (local) Lie groups and Lie algebras. These theorems have been established for arbitrary Banach groups and Lie algebras, for which the parametric space can also be infinite dimensional. An important fact is that in every class of isomorphic groups, there is a canonical representative, that is, a group all of whose one-parameter subgroups are straight lines. This result is obtained in the third inverse Lie theorem using a differential equation whose solution is constructed from composite operators in the form of a known SCH-series. The domain of convergence of the SCH-series, for an arbitrary Banach space, is evaluated. Together with this, one more fundamental result of the Lie theory about the analyticity of the multiplication law is obtained. The main facts of the Lie algebra theory are expounded recapitulatively, and some important theorems, for example, the Levi theorem about separation of a radical, are presented without proof.