Book contents
- Frontmatter
- Contents
- Introduction
- Galois groups through invariant relations
- Construction of Co3. An example of the use of an integrated system for computational group theory
- Embedding some recursively presented groups
- The Dedekind-Frobenius group determinant: new life in an old problem
- Group characters and π-sharpness
- Permutation group algorithms via black box recognition algorithms
- Nonabelian tensor products of groups: the commutator connection
- Simple subalgebras of generalized Witt algebras of characteristic zero
- Applications of the Baker-Hausdorff formula in the theory of finite p-groups
- Generalizations of the restricted Burnside problem for groups with automorphisms
- The ∑m-conjecture for a class of metabelian groups
- Rings with periodic groups of units II
- Some free-by-cyclic groups
- The residually weakly primitive geometries of the Suzuki simple group Sz(8)
- Semigroup identities and Engel groups
- Groups whose elements have given orders
- The Burnside groups and small cancellation theory
- Solvable Engel groups with nilpotent normal closures
- Nilpotent injectors in finite groups
- Some groups with right Engel elements
- The growth of finite subgroups in p-groups
- Symplectic amalgams and extremal subgroups
- Primitive prime divisor elements in finite classical groups
- On the classification of generalized Hamiltonian groups
- Permutability properties of subgroups
- When Schreier transversals grow wild
- Probabilistic group theory
- Combinatorial methods: from groups to polynomial algebras
- Formal languages and the word problem for groups
- Periodic cohomology and free and proper actions on ℝn × Sm
- On modules over group rings of soluble groups of finite rank
- On some series of normal subgroups of the Gupta-Sidki 3-group
Simple subalgebras of generalized Witt algebras of characteristic zero
Published online by Cambridge University Press: 04 August 2010
- Frontmatter
- Contents
- Introduction
- Galois groups through invariant relations
- Construction of Co3. An example of the use of an integrated system for computational group theory
- Embedding some recursively presented groups
- The Dedekind-Frobenius group determinant: new life in an old problem
- Group characters and π-sharpness
- Permutation group algorithms via black box recognition algorithms
- Nonabelian tensor products of groups: the commutator connection
- Simple subalgebras of generalized Witt algebras of characteristic zero
- Applications of the Baker-Hausdorff formula in the theory of finite p-groups
- Generalizations of the restricted Burnside problem for groups with automorphisms
- The ∑m-conjecture for a class of metabelian groups
- Rings with periodic groups of units II
- Some free-by-cyclic groups
- The residually weakly primitive geometries of the Suzuki simple group Sz(8)
- Semigroup identities and Engel groups
- Groups whose elements have given orders
- The Burnside groups and small cancellation theory
- Solvable Engel groups with nilpotent normal closures
- Nilpotent injectors in finite groups
- Some groups with right Engel elements
- The growth of finite subgroups in p-groups
- Symplectic amalgams and extremal subgroups
- Primitive prime divisor elements in finite classical groups
- On the classification of generalized Hamiltonian groups
- Permutability properties of subgroups
- When Schreier transversals grow wild
- Probabilistic group theory
- Combinatorial methods: from groups to polynomial algebras
- Formal languages and the word problem for groups
- Periodic cohomology and free and proper actions on ℝn × Sm
- On modules over group rings of soluble groups of finite rank
- On some series of normal subgroups of the Gupta-Sidki 3-group
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] = (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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- Groups St Andrews 1997 in Bath , pp. 455 - 459Publisher: Cambridge University PressPrint publication year: 1999