New PDF release: Algebra lineal y algunas de sus aplicaciones

By L.I. GOLOVINA

ISBN-10: 8440109717

ISBN-13: 9788440109712

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Let Xl : H --+ C· be the corresponding character. By the main lemma again, the set of all v E V such that U(h)v = Xl(h)v for all h E H is invariant under u( G) and hence is all of V. Hence, in particular g(x)v E Cv for each v E V. Since x was arbitrary in G, we conclude that dim V = 1. 1* are proved. Remark. In fact, Lie's theorem and its group theoretic analog imply each other directly. Granting the group statement, we get the Lie algebra statement in case k = C, by considering the connected Lie group attached to a given Lie algebra.

From this decomposition it is clear first that ~ acts diagonally, and commutes with the action of g. Thus, ~ = c is the center of g. To get 8, we can either quote Levi's theorem, or argue directly using the adjoint representation. Chapter VI. Semisimple Lie Algebras 51 Conversely, suppose conditions a) and b) are satisfied. By b) we have a decomposition of the vector space V of the form (*), where the Xi'S are characters (linear forms) on c. Since c is in the center of 8 = , x _, the eigenspaces VXi are stable under 8.

Let fJ be a subspace of 9 of codimension 1, containing Dg. Then IJI Dg is an ideal in slDg because the latter is abelian, and consequently fJ is an ideal in 9. By induction there is v E V, v 1= 0, and X : f) -+ k such that hv = X(h)v for all h E I). Let W = {W E V hw = X(h)w for all h E IJ }. By construction, W is a non-zero linear subspace of V, and using the main lemma we can show that W is stable under g. If w E W, x E g, then for h E I), hxw = xhw - [x, h]w = X(h)xw - X([x, h])w , I and since the last term is zero, it follows that xw E W.

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Algebra lineal y algunas de sus aplicaciones by L.I. GOLOVINA


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