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On generalized Witt algebras Ree, Rimhak


Let ƣ be a commutative associative algebra over a field Φ of characteristic p > 0 , and ϑ(ƣ) the derivation algebra of ƣ. A subalgebra [symbol omitted] of ϑ(ƣ) is called regular if f D ɛ for any f ɛ ƣ and D ɛ [symbol omitted]. For a regular subalgebra [symbol omitted] of ϑ(ƣ), if there exist D₁,…, D[subscript]m ɛ [symbol omitted] such that every D ɛ [symbol omitted] is expressed uniquely as D = f₁D₁+ ... + f[subscript]mD[subscript]m, where e f[subscript]i ɛ ƣ, then [symbol omitted] is said to be defined by the system {D₁, …, D[subscript]m} and is denoted by the notation [symbol omitted] (ƣ ; D₁, ..., D[subscript]m). In this dissertation, the family of [symbol omitted] Lie algebras of the type [symbol omitted]( ƣ; D₁, …, D[subscript]m) is studied. It is shown that if ƣ is a field then all algebras in [symbol omitted] are simple except when p = 2, m = 1. It is also shown that if Φ is algebraically closed then every simple algebra in [symbol omitted] is a generalized Witt algebra of the type defined by I. Kaplansky [Bull. Amer. Math. Soc. vol. 10 (1943), pp. 107-121], and, conversely, that every generalized Witt algebra belongs to [symbol omitted]. A simpler form of the generalized Witt algebras is given. By using this form, the problem of whether every generalized Witt algebra can be defined over GF(p) is partly solved. It is shown also that a subfamily [symbol omitted] of [symbol omitted] , consisting for the most part of non-simple algebras, has a remarkable property: every algebra in [symbol omitted] has the same ideal theory as that of a commutative associative algebra. Jacobson's result on automorphisms of the derivation algebras of the group algebras of commutative groups of the type (p, ..., p) is extended to generalized Witt algebras, and, finally, it is shown that m is an invariant of the algebra [symbol omitted] = [symbol omitted] (ƣ; D₁, ..., D[subscript]m) if [symbol omitted] is normal simple.

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