Published online by Cambridge University Press: 04 August 2010
Introduction
In this paper we give a short survey on infinite-dimensional Witt type Lie algebras and their simple subalgebras mainly over a field of characteristic zero. We give a list of papers related to this subject, but we did not intend to make a complete list.
We denote by k the ground field of any characteristic unless otherwise specified.
Some history
Let k be a field of characteristic p > 0, and W be a vector space over k with basis {Di | 0 ≤ i < p}. Define the multiplication by [Di, Dj] = (i – j)Di+j, and this makes W a Lie algebra. W is called a Witt algebra. We denote this algebra by Wz|pz, and the following result is well known [Se67].
TheoremIf p ≠ 2 then Wz/pz is a simple Lie algebra.
This result was the starting point for later research, and the finite cyclic group Z/pZ was replaced by several groups.
Kaplansky [K54] has generalized in the following form: Let V be a vector space over k, and G be an additive subgroup of the dual space V* of V. Let I be an index set of a basis of V, and we denote an element a of V* by a = (ai)i∈I, where ai ∈ k. Assume that G is a total additive group, that is, the only element α = (αi)i∈I, where αi = 0 except for finite i, such that ∑iaiαi = 0 for any a = (ai)i∈I ∈ G is the zeroelement.
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