# Download An introduction to the theory of linear spaces by Shilov G. PDF

By Shilov G.

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**Additional info for An introduction to the theory of linear spaces**

**Example text**

Applying the induction hypothesis we see that (ha , hal is a linear combination of b's with h E H. ,[h1 ,ha]) is also a linear combination of h's with h e H. d. = 6. ) Let X be a set and let Fx be the free group on X. Let Fl be the descending central series of Fx, defined by F} = Fx and F (Fx,F;-l), for n > 1. The 888Oci~ted graded group is, as we know, a Lie algebra, given by x= grFx = EgrA Fx , 00 grft Fx = F • =1 In particular, grl Fx onX. 1. 2. The group' F~ / FR+l are free Z-module, and if X II finite witA CardX = d, thea rk(Fx/F + ) = lien).

3. As a compact Lie group over R giving , we can take a real torus, (R/Z)R. , which is a closed subgroup of the orthogonal group of linear transformations of • leaving fixed the Killing form of •. Since that form is definite, the latter group, and hence Aut a, is compact. The Lie algebra of Aut. is the algebra of derivations of a, which is isomorphic to. 3). 3. Suppose now c = 0 (hence 9 is semisimple) and let G be a connected Lie group with Lie algebra 9. We have a canonical homomorphism: Ad : G --+ Aut 9 .

Exercises 1. Let g be a Lie algebra, let ~ be its radical, and let i be the intersection of the kernels of the irreducible representation of g. a) Show that i = [9,~] = Dg n~. ) b) Show that x e ~ belongs to i if and only if e( x) is nilpotent for every representation U of 9. 2. Let 9 be a Lie algebra and let B(x,y) be a non-degenerate invariant symmetric bilinear form on g. a) Let X,II e ,. Show the equivalence of: (i) II e Imad(z). (ii) B(II,Z) = 0 for all z which commute with x. b) Assume 9 semisimple.