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 On generalized Witt algebras
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On generalized Witt algebras Ree, Rimhak
Abstract
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. 107121], 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 nonsimple 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.
Item Metadata
Title 
On generalized Witt algebras

Creator  
Publisher 
University of British Columbia

Date Issued 
1955

Description 
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. 107121], 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 nonsimple 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.

Genre  
Type  
Language 
eng

Date Available 
20120221

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080649

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.