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Simple subalgebras of generalized Witt algebras of characteristic zero

Published online by Cambridge University Press:  04 August 2010

Naoki Kawamoto
Affiliation:
Maritime Safety Academy, 5-1, Wakaba, Kure 737, Japan
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
N. Ruskuc
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
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Summary

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] = (ij)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)iI, where aik. Assume that G is a total additive group, that is, the only element α = (αi)iI, where αi = 0 except for finite i, such that ∑iaiαi = 0 for any a = (ai)iIG is the zeroelement.

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Publisher: Cambridge University Press
Print publication year: 1999

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