某些特征p域上的低维完备李代数.docx

某些特征p域上的低维完备李代数
Introduction
Inmathematics,Liealgebrasplayacrucialroleinthestudyofsymmetriesandtransformations.Theyprovideaframeworkforunderstandingthestructureandbehaviorofawiderangeofmathematicalobjects,includinggroups,manifolds,anddifferentialequations.ThetheoryofLiealgebrashasapplicationsinvariousfieldssuchasphysics,engineering,andcomputerscience.
Inthispaper,wewillfocusonlow-dimensionalcompleteLiealgebrasoveracertainfieldp.Wewillexplorethepropertiesandstructureofthesealgebras,aswellastheirimplicationsforthebroadertheoryofLiealgebras.
DefinitionofLow-DimensionalCompleteLieAlgebras
ALiealgebraisavectorspaceequippedwithabilinearoperationcalledtheLiebracket,whichsatisfiescertainproperties.ItissaidtobecompleteifeveryelementofthevectorspacecanbewrittenasalinearcombinationofLiebracketsofelementsofthealgebra.
Alow-dimensionalLiealgebrareferstoanalgebrawithasmallnumberofdimensions.Inthispaper,wewillfocusontwoparticularcases:two-dimensionalandthree-dimensionalLiealgebras.
Two-DimensionalCompleteLieAlgebras
Letusconsideratwo-dimensionalvectorspaceoverthefieldp,denotedasV.Wecanchooseabasis{e₁,e₂}forthisvectorspace.InorderfortheLiealgebratobecomplete,weneedtodefinetheLiebracket[e₁,e₁],[e₁,e₂],and[e₂,e₂].Thisgivesusthreepossibilities:
1)If[e₁,e₁]=λe₁and[e₁,e₂]=μe₁+νe₂,then[e₂,e₂]=βe₁+γe₂.TheLiealgebraissaidtobeabelianifλ=μ=ν=β=γ=0.
2)If[e₁,e₁]=λe₂and[e₁,e₂]=μe₂,then[e₂,e₂]=βe₁+γe₂.ThiscasecorrespondstoasolvableLiealgebra.
3)If[e₁,e₁]=λe₁+μe₂and[e₁,e₂]=νe₁+ρe₂,then[e₂,e₂]=βe₁+γe₂.Thiscasecorrespondstoanon-solvableLiealgebra.
Three-DimensionalCompleteLieAlgebras
Similarly,wecanconsiderathree-dimensionalvectorspaceVoverthefieldp.Wechooseabasis{e₁,e₂,e₃}forV,anddefinetheLiebracket[eᵢ,eⱼ]foralli,j=1,2,3.
Thereareseveralpossiblecasesforthree-dimensionalcompleteLiealgebras:
1)If[e₁,e₁]=λ₁e₁+λ₂e₂and[e₁,e₂]=μ₁e₁+μ₂e₂,then[e₁,e₃]=ν₁e₁+ν₂e₂+ν₃e₃.TheLiealgebraissolvable.
2)If[e₁,e₁]=λe₁+μe₂+νe₃and[e₁,e₂]=ρe₁+σe₂+τe₃,then[e₁,e₃]=αe₁+βe₂+γe₃and[e₂,e₃]=δe₁+εe₂+ηe₃.Thiscasecorrespondstoanon