Download PDF by Karl-Hermann Neeb: Invariant Subsemigroups of Lie Groups

By Karl-Hermann Neeb

ISBN-10: 0821825623

ISBN-13: 9780821825624

This paintings offers the 1st systematic remedy of invariant Lie semi teams. simply because those semi teams supply fascinating types for area instances generally relativity, this paintings should be invaluable to either mathematicians and physicists. it's going to additionally attract engineers attracted to bi-invariant regulate structures on Lie teams. Neeb investigates closed invariant subsemigroups of Lie teams that are generated by way of one-parameter semi teams and the units of infinitesimal turbines of such semi teams - invariant convex cones in Lie algebras.In addition, a characterization of these finite-dimensional genuine Lie algebras containing such cones is bought. the worldwide a part of the speculation bargains with globality difficulties (Lie's 3rd theorem for semi groups), controllability difficulties, and the facial constitution of Lie semi teams. Neeb additionally determines the constitution of the common compactification of an invariant Lie semigroup and exhibits that the lattice of idempotents is isomorphic to a lattice of faces of the cone twin to the cone of infinitesimal turbines.

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It is clear that Q(Ix,x) = q(Ix,y) = [Ix,y] and = [—x,x] — 0 and Q(x,[h,y]) + Q(Iy,[h,Ix}) = [Ix,[h,y]] - [y,[h,Ix]] = [h,[lx,y]]=0 because [Ix,y] G Z($)- Suppose that Q(M{,Mj) ^ {0}. Then there exist simple Qi -modules V{ C M; and Vj C Mj such that [V^Vj] / {0} . Let a G V such that a([Vi, Vj-J) ^ {0}. VixVj-* m,,(v,v') H-> a([v,v']) is a skew-symmetric gi -invariant bilinear form. Let x G Vj. Then Vj1- = [y G Vi : /3(y,Vj) = {0}} is a gi-submodule of Vi which has to be {0}. Therefore /? is non-degenerate.

Therefore ip' is equivariant with respect to the action of W on f} and the action of the Weyl group of £ on f). 151] to see that W is generated by the reflections on the hyperplanes kero; for w G 0^- since this holds for the compact Lie group lNN0 fy . 2. We call W = W(t),g) the Weyl group of g with respect to the compactly embedded Cart an algebra f). 3. For s G g w and h G f) we have that ((h)[x,Ix] = w(h)Q(x).

42. Let t = su(2) and V =

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Invariant Subsemigroups of Lie Groups by Karl-Hermann Neeb


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