Lie groups, convex cones, and semigroups.

*(English)*Zbl 0701.22001
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.

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 |