Basic concepts of differential algebra by Wuerfl A.

By Wuerfl A.

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Let t be the radical of g, and k' an extension of k. Then: (i) The radical of 9 ® k' is t ® k'. (ii) g is semi-simple if and only if 9 ® k' is semi-simple. 2 (iii). The ideal t ® k' of 9 ® k' is solvable and g ® k'/t ® k' = (9/t) ® k' is semi-simple from (ii), whence (i). Let a be an ideal of 9, K the Killing form of g, and {1 the orthogonal subspace of a with respect to K. 7. LEMMA. J,§S UE ALGEBRAS a non-null soll•able ideal of g. Then {1 is an ideal of g which is complementary · to a, so that g = axb.

J is nilpotent. 16. LEMMA. Let V be a cector space, g = g{(V), and x E g. potent, then ad 0 x is nilpotent. If x is nil- Let n be an integer such that x" = 0, and f = ad 0x. If y E g, then f"'(y) is the sum of terms of the form ±x1yxl, where i + j = m. Hence f2n(y) = 0. 17. PROPOSITION. J in V such that el·ery element of g(g} is nilpotent. Then: (i) e is strictly triangularizable. (ii) The Lie algebra Q(g) is nilpotent. 16}, and hence is nilpotent. 1 5). 12. 18. LEMMA. Let V be a finite-dimensional vector space, let u and l' be endomorphisms of V, and let p be an integer such that (ad u)Pl' = 0.

Let a be orthogonal to 9 with respect to the Killing form, and let us assume that a =1= 0. 20). 24). Hence the radical of 9 is not null. 3. 2. is termed semisimple. 4. 2 (ii)), hence the adjoint representation of g is injective. 5. PROPOSITION. ln be Lie algebras. Jn is semisimple if and only if 9 1, ••• , On are semi-simple. 4. 6. PROPOSITION. Let t be the radical of g, and k' an extension of k. Then: (i) The radical of 9 ® k' is t ® k'. (ii) g is semi-simple if and only if 9 ® k' is semi-simple.

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