Note on outer derivations of Lie algebras

Abstract

Let O be the set of Lie algebras L over a field Φ satisfying the conditions that LφL and Z(L)Φ(0\ where Z(L) denotes the center of L. Clearly every non-trivial nilpotent Lie algebra belongs to O. It is known ([4], [6], [8], [13]) that every Le Ό has an outer derivation. In [13] we have introduced the notion of Lie algebras of type (Γ) and shown that every Lie algebra L of type (T) such that LΦL admits an outer derivation belonging to 3ΐ, the radical of the derivation algebra ®(Z). It has been also shown that if L e Ό is not of type (Γ) there exists an abelίan ideal of ©(£) containing an outer derivation. From these observations it seems to be interesting to study the case where L is of type (T) such that L = L. The main purpose of this note is to give a detailed consideration to the case just mentioned. Some additional remarks will be also given. In Section 2 we shall show that a Lie algebra L of type (Γ) such that dim Z(L)Φ1 or Φ is of characteristic 2 admits an outer derivation in 3ΐ and that a Lie algebra L of type (T) such that L = L\ dim Z{L) — \ and Φ is of characteristic Φ 2 admits an outer derivation in 9ΐ if and only if L does (Theorem 2.2). In Section 35 we shall show that a Lie algebra L over a field of characteristic 0 admits a semisimple outer derivation in 9ΐ if the radical of L does (Proposition 3.1), and based on this result, for a Lie algebra L of type (Γ) such that L = L and dim Z(£) = l, we shall give several properties of the radical of L\ each of which ensures the existence of a semisimple outer derivation in 31 (Theorem 3.6). In [12] we have studied the existence of the automorphisms of L, when Φ is of characteristic 0, outside the connected algebraic group such that the corresponding Lie algebra is the algebraic hull 3KL)* of $(L), the ideal of ®(£) consisting of all inner derivations of Z. The final section 4 will be devoted to the discussions about the existence of the derivations of L e O which are contained in 9ΐ but not in