Witt Algebra Research Articles

Classification of Generalized Witt Algebras Over Algebraically Closed Fields

Classification of Generalized Witt Algebras Over Algebraically Closed Fields

On Lie algebras of rank one

where h is any additive mapping of G into F (see [1, p. 138]). Each of these algebras is a simple Lie algebra, for which uo spans a one-dimensional Cartan subalgebra, with one-dimensional root spaces spanned by the u,. We shall prove that under certain hypotheses, these are the only Lie algebras over F of rank one. Kaplansky in [5] has proved that any restricted simple Lie algebra over F of rank one is either 3-dimensional or the Witt algebra. He has also obtained a number of results on (not necessarily restricted) Lie algebras of rank one, and in particular has proved [5, Theorem 4] that if L is a Lie algebra of dimension > 3 over F, and if L has a one-dimensional Cartan subalgebra such that multiplication between La and L__ is nondegenerate for every nonzero root a (i.e., no nonzero element of a root space La annihilates all of La), then all root spaces are onedimensional and the roots form a group under addition. The result we shall obtain is the following:

Automorphisms of Generalized Witt Algebras Have Fixed Points

Automorphisms of Generalized Witt Algebras Have Fixed Points

Automorphisms of generalized Witt algebras have fixed points

and let 2 be the algebra of derivations of W, considered as a restricted Lie algebra. The generalized Witt algebras are precisely all algebras of this form; they were defined by Jacobson in [2], and the name results from the fact that the specialization n = 1 and 2j = 1 yields the usual Witt algebras. The purpose of this paper is to prove that for p > 5 all automorphisms of these algebras have nontrivial fixed points; actually, we show that the fixed point set of any such automorphism is a vector space of dimension > n. As pointed out by Jacobson in [2 ], it follows from the fact that 2 is simple (as an ordinary Lie algebra) that all automorphisms of the Lie structure of 5 preserve the restricted structure as well; thus the automorphisms of 2 as a Lie algebra and the automorphisms of 2 as a restricted Lie algebra are the same, and we need not distinguish between them. Let K be the algebraic closure of F and extend W to WK = W 0 FK and 2 to 2K = 20 FK in the usual way. We will identify 2K with the algebra of derivations of (K as indicated by Jacobson in [1, p. 213]. An automorphism oa of 2 extends to an automorphism (0K of 2K in the obvious way. The fixed points of oare the zeroes of the linear transformation I-o-; hence they form a subspace of 2 whose dimension is equal to the dimension of the set of zeroes of the extended linear transformation (I-_)KR=IRK, which in turn is the set of fixed points of 0K. Hence it will be sufficient to consider the dimension of the fixed point set of CK, i.e., we may restrict ourselves to the case in which the base field is algebraically closed. Thus from now on we will assume that F is algebraically closed; then we have 4/'CF for all i. Let yi=xi/V for each i; then the yi's also generate f, and yp =?O for all i. It follows from this and a dimension argument that W can

Note on Generalized Witt Algebras

Throughout this note K will denote a field of characteristic p > 0. Let I be the set {1,2, … , m}, and a finite additive group of functions on I with values in K. We assume that is total in the sense that, for any λ1, … , λm in K Σi=imλiσ(i) = 0 for all a in G implies all σi = 0. It is clear that is an elementary p-group. Let pn be the order of . A generalized Witt algebra is defined as an algebra over K with basis elements {e(σ, i)∣ σ ∈ , i ∈ I} and the multiplication table is a simple Lie algebra except when p = 2, m = 1.

On Generalized Witt Algebras

On Generalized Witt Algebras

On generalized Witt algebras

On generalized Witt algebras