Two Integral Equations.
نویسنده
چکیده
that g* has the same derived algebra as g itself and that every ideal in g is also an ideal in t*. Let g be any algebraic Lie algebra. Denote by b the radical of g (i.e., the largest solvable ideal in g) and by n the largest ideal of g composed only of nilpotent matrices. By Levi's theorem, g is the direct sum of t and of a semi-simple subalgebra J. It can be proved that f is the direct sum of n and of Abelian algebra a whose matrices are semi-simple and commute with those of J. Let g be any subalgebra of gl(n, K); then it can be shown that the derived algebra g' is algebraic. Moreover, g' can be "defined by its invariants," in the sense that any matrix which admits as its invariants all the common invariants of all matrices in g' lies itself in g'. Our result applies in particular to any semi-simple Lie algebra g of gt(n, K), which is identical with its derived algebra g'. Moreover, our method of proof shows more generally that any subalgebra g of gt(n, K) whose radical is composed only of nilpotent matrices is algebraic and is defined by its invariants. If A is any algebra (associative or not) over the field K, the derivations of A form a Lie algebra which is easily seen to be algebraic. Finally, let it be mentioned that the notion of algebraic Lie algebras can be used with advantage in the exposition of the theory of semi-simple Lie algebras, notably in establishing Cartan's criterion of semi-simplicity and Lie's theorem on solvable Lie algebras. Barring the recourse to the algebraic closure of the basic field in the proof of Theorem 3 of the paper quoted above,2 one obtains in this way a rational proof of Cartan's criterion and of Lie's theorem.
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ورودعنوان ژورنال:
- Proceedings of the National Academy of Sciences of the United States of America
دوره 31 7 شماره
صفحات -
تاریخ انتشار 1945