Lie groups, convex cones, and semigroups. (English) Zbl 0701.22001

Hilgert, Joachim; Hofmann, Karl Heinrich; Lawson, Jimmie D.

Oxford Mathematical Monographs. Oxford etc.: Clarendon Press. xxxviii, 645 p. £55.00 (1989).

The book is the first attempt to expose systematically the Lie theory of differentiable semigroups and, mainly, of subsemigroups of Lie groups.
Chapter 1 contains preliminary facts about convex cones in a real vector space. A wedge in a vector space \(L\) is a closed convex cone \(W\subset L\) with vertex 0, its edge is the vector subspace \(H(W)=W\cap (-W)\). A wedge \(W\) is called generating if \(L=W-W.\)
Chapter 2 develops the theory of wedges in a real Lie algebra \(L\). A Lie wedge in \(L\) is a wedge \(W\) such that \(e^{\mathrm{ad}\;x} W=W\) for all \(x\in H(W)\). An invariant wedge is one satisfying \(e^{\mathrm{ad}\;x} W=W\) for all \(x\in L\). A Lie semialgebra is a wedge \(W\subset L\) which is locally closed under the Campbell-Hausdorff multiplication in \(L\). In particular, any wedge containing \([L,L]\) is invariant; such an invariant wedge is called trivial. The conditions forcing a Lie semialgebra to be invariant or trivial are discussed.
Chapter 3 deals with the theory of invariant generating wedges in a Lie algebra. In particular, the theorem of Kostant and Vinberg on the existence of invariant generating wedges in a semisimple linear Lie algebra is proved.
Chapter 4 is devoted to the local theory of subsemigroups. With any local subsemigroup \(S\) of a real Lie group \(G\) one associates its tangent wedge \(L(S)\) in the tangent Lie algebra \(L(G)\) of \(G\). Then \(L(S)\) is a Lie wedge and, conversely, any Lie wedge in \(L(G)\) is the tangent wedge of a local subsemigroup. The case of a local Banach-Lie group \(G\) is actually dealt with.
In Chapter 5 the global theory of subsemigroups is started. A subsemigroup \(S\) of a Lie group \(G\) is called preanalytic if the subgroup \(G(S)\) generated by \(S\) is arcwise connected (and hence has the structure of a Lie subgroup). With any preanalytic subsemigroup S the tangent Lie wedge \(L(S)\subset L(G)\) is associated. A preanalytic subsemigroup \(S\) is called infinitesimally generated if \(\exp L(S)\subset S\subset \overline{\langle\exp L(S)\rangle}\) (the closed subsemigroup of \(G(S)\) generated by \(\exp L(S)\)) and if \(G(S)\) is generated by \(\exp L(S)\).
In Chapter 6 the following problem is solved: when is a Lie wedge \(W\subset L(G)\) global, i.e. has the form \(L(S)\) for an infinitesimally generated subsemigroup \(S\) in \(G\)? One constructs the left-invariant wedge field \(W(g)\) on \(G\) such that \(W(e)=W\) and one proves that \(W\) is global if and only if there exists an exact 1-form \(\omega\) on \(G\) such that \(\omega_ g\geq 0\) on any \(W(g)\), \(g\in G\), and \(\omega_ e>0\) on \(W\setminus H(W).\)
Chapter 7 presents an abstract Lie theory of differentiable semigroups and of their embedding into Lie groups.

Reviewer: A.L.Onishchik

MSC:

22-02	Research exposition (monographs, survey articles) pertaining to topological groups
22Exx	Lie groups
17Bxx	Lie algebras and Lie superalgebras
52A20	Convex sets in \(n\) dimensions (including convex hypersurfaces)
17B05	Structure theory for Lie algebras and superalgebras
22A15	Structure of topological semigroups
22A22	Topological groupoids (including differentiable and Lie groupoids)
22E60	Lie algebras of Lie groups
22E05	Local Lie groups

Keywords:

differentiable semigroups; subsemigroups of Lie groups; convex cones; wedges; real Lie algebra; Lie wedge; Lie semialgebra; Campbell-Hausdorff multiplication; invariant wedge; invariant generating wedges; semisimple linear Lie algebra; local subsemigroup; local Banach-Lie group; preanalytic subsemigroup; infinitesimally generated subsemigroup