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LIEAUTOMORPHISMSOFINCIDENCEALGEBRAS
É,MYKOLAKHRYPCHENKO,.
descriptionoftheLieautomorphismsoftheincidencealgebraI(X,K).In
particular,weshowthattheyareingeneralnotproper.
Introduction
Liealgebraunderthecommutator[a,b]=ab−baofa,b∈-
phismofAonemeansanautomorphismoftheLiealgebra(A,[,]).Clearly,an
automorphismϕofAandthenegative−ψofananti-automorphismψofAareLie
,ifνisanR-linearcentral-valuedmaponAsuch
thatν([A,A])={0}andφiseitherϕor−ψasabove,thenφ+νisaLieendo-
morphismofA,which,undercertainassumptionsonν,
[9]provedthateachLieautomorphism
ofthefullmatrixringMn(R),nψ>2,overadivisionringR,char(R)6∈{2,3},
-
tiverings[13],simplerings[15]and,yetmoregenerally,betweenprimerings[14].
Ðoković[7]describedthegroupofLieautomorphismsoftheuppertriangularma-
trixalgebraTn(R),whereRisacommutativeunitalringwithtrivialidempotents.
ItfollowsfromhisdescriptionthateachLieautomorphismsofTn(R)isproper
().Cao[5]generalizedtheresultbyÐokovićtothecaseofcom-
beenindependentlyprovedbyMarcouxandSourour[12].Cecil[6]showedthatLie
isomorphismsbetweenblock-triangularmatrixalgebrasoveraUFDareproper.
TheincidencealgebraI(X,R)ofalocallyfiniteposetXoveracommutative
unitalringRisanaturalgeneralizationofTn(R).JordanandLiemapsonI(X,R)
(andevenonmoregeneralalgebras)havebeenactivelystudiedduringthelast5
arXiv:[]9Aug2021years(see[1,2,3,11,17,18,19]).Usually,allLie-typemapsonI(X,R)areproper.
ThisisnolongerthecaseforLieautomorphismsofI(X,K),whereKisafieldand
Xisfiniteandconnected,
severalfactsonthestructureof(I(X,K),[,])whichwillbeusedthroughoutthe
(X,K)to
thedescriptionofthosewhichwecallelementary().Section5is
(X,K)
intermsoftriples(θ,σ,c),whereθisabijectionofB={exy:x<ψy},monotone
onmaximalchainsinXandsatisfyingacertaincombinatorialconditionwhich
:16S50,17B60,17B40;secondary:16W10.
,incidencealgebra,automorphism,anti-
automorphism.
1:.
2É,MYKOLAKHRYPCHENKO,.
involvescyclesinX,σisa“1-cocycle-looking”maprelatedtoPθandcisasequence
of|X|elementsofKsuchthatci6=0().
(X,≤)beapartiallyorderedset(poset,forshort)andletx,y∈
⌊x,y⌋,thatis,⌊x,y⌋={z∈X:
x≤z≤y}.IfalltheintervalsofXarefinite,thenXissaidtobelocallyflnite.
AchaininXisalinearlyordered(under≤)
chainC⊆Xis|C|−,thelengthofafinitenon-emptysubset
Y⊆X,denotedbyl(Y),isthemaximumlengthofchainsC⊆,
l(Y)≤|Y|−,x1,...,xmofelementsofX,such
thatxiiscomparablewithxi+1and,moreover,l(⌊xi,xi+1⌋)=1(ifxi≤xi+1)or
l(⌊xi+1,xi⌋)=1(ifxi+1≤xi)foralli=0,...,m−,x1,...,xmis
saidtobeclosedifx0==xjfori6=
aclosedwalkx0,x1,...,xm=x0suchthatm≥4andxi=xj⇒{i,j}={0,m}
fori6=,y∈Xthereisa
pathx=x0,...,xm=.
incidencealgebraI(X,K)ofXoverKisthesetI(X,K)={f:X×X→K:
f(x,y)=0ifxy}endowedwiththeusualstructureofavectorspaceoverK
andtheproductdefinedby
X
(fg)(x,y)=f(x,t)g(t,y),
x≤t≤y
foranyf,g∈I(X,K).ThenI(X,K)isaK-algebrawithidentityδgivenby
(
1,ψψx=y,
δ(x,y)=
0,ψψx6=y.
Moreover,ifXisfinite,whichisthecasewedealwithinthispaper,I(X,K)admits
thestandardbasis{exy:x≤y},where
(
1,(u,v)=(x,y),
exy(u,v)=
0,(u,v)6=(x,y).
X
Indeed,f=f(x,y)exyforallf∈I(X,K).Wewilldenoteex=exx.
x≤y
LetB={exy:x<y}.Itisknown(see[16,])thattheJacobson
radicalofI(X,K)is
J(I(X,K))={f∈I(X,K):f(x,x)=0forallx∈X}=SpanKB.
Anelementf∈I(X,K)issaidtobediagonal,iff(x,y)=0forx6=
elementsformacommutativesubalgebraofI(X,K)spannedby{ex:x∈X},
whichwedenotebyD(X,K).Clearly,eachf∈I(X,K)canbeuniquelywritten
asf=fD+fJwithfD∈D(X,K)andfJ∈J(I(X,K)).
WedenotebyAut(I(X,K))andLAut(I(X,K))thegroupsofusualandLie
automorphismsofI(X,K),-
jectiveK-linearmapsofI(X,K),denotedbyGL(I(X,K)),andwiththegroup
Inn1(I(X,K))oftheinnerautomorphismsofI(X,K)consistingoftheconjuga-
tionsbyβ∈I(X,K)withβD=:.
LIEAUTOMORPHISMSOFINCIDENCEALGEBRAS3
(I(X,K))
FromnowonKwillbeafieldandXafiniteconnectedposetofcardinalityn.
∈I(X,K)commuteswitheuvifandonlyiff(x,u)=0
forx<u,f(v,y)=0fory>vandf(u,u)=f(v,v).
XX
feuv=euvf⇔f(x,y)exyeuv=f(x,y)euvexy
x≤yx≤y
XX
⇔f(x,u)exv=f(v,y)euy,
x≤u≤vu≤v≤y
whencethedesiredpropertyofffollows.
WerecallthatthederivedidealofaK-LiealgebraLis
[L,L]=SpanK{[a,b]:a,b∈L}.
ThesequenceofLieideals
L⊇[L,L]⊇[L,[L,L]]⊇···
invariantunderanyautomorphismofL.
LetJ1bethederivedidealofI(X,K)and
Jm=[J1,Jm−1]=SpanK{[f,g]:f∈J1,g∈Jm−1},ψm≥2.
Wethushavethefollowing.
∈LAut(I(X,K)),thenϕ(Jm)=Jmforallm≥1.
(I(X,K)).
,g∈I(X,K).Forallx∈X,wehave
(fg−gf)(x,x)=f(x,x)g(x,x)−g(x,x)f(x,x)=0.
Thus[f,g]∈J(I(X,K)).
Ontheotherhand,ifx<yinX,thenexy=exexy−exyex=[ex,exy]∈J1.
J=Span{e:l(⌊x,y⌋)≥m}=J(I(X,K))m.
mKxy
Inparticular,Jn=J(I(X,K))n={0}.ItfollowsthatJmisanideal(and,there-
fore,aLieideal)ofI(X,K).
=>1.
Leteu1v1,...,eumvm∈J(I(X,K)),i=1,...,···eumvm6=0,then
vi=ui+1,i=1,...,m−,u1<ψu2<···<ψum<ψvmand
eu1v1···eumvm=eu1vmwithl(⌊u1,vm⌋)≥,
J(I(X,K))m⊆Span{e:l(⌊x,y⌋)≥m}.
Kxy
AssumethatSpanK{exy:l(⌊x,y⌋)≥k}⊆Jkforsomepositiveintegerkand
letx<ysuchthatl(⌊x,y⌋)≥k+∈Jk+=t1<
···<tk+2=ybeachainin⌊x,y⌋oflengthk+
exy=extk+1etk+1y−etk+1yextk+1=[extk+1,etk+1y].:.
4É,MYKOLAKHRYPCHENKO,.
Observethatetk+1y∈J(I(X,K))andl(⌊x,tk+1⌋)≥-
sis,extk+1∈Jkand,therefore,exy∈Jk+1,,
SpanK{exy:l(⌊x,y⌋)≥m}⊆Jm.
Finally,letJk⊆J(I(X,K))∈J(I(X,K))and
g∈Jk,then[f,g]∈J(I(X,K))k+,Jm⊆J(I(X,K))m.
(J(I(X,K)),[,])is
J(I(X,K))⊇J(I(X,K))2⊇···⊇J(I(X,K))n−1⊇J(I(X,K))n={0}.
(I(X,K))is
Z=SpanK{exy:x<y,xisminimalandyismaximal}.
,v,x,y∈Xsuchthatuψ<ψv,xψ<ψy,xisminimalandyismaximal.
Then[exy,euv]=δyuexv−=u,thenyψ<ψv,whichcontradictsthe
=x,thenuψ<ψx,whichcontradictstheminimalityofx.
Thus,[exy,euv]=0and,therefore,Pexy∈Z.
Conversely,letg=x<yg(x,y)exy∈
thereexistsuψ<=euxgweconcludethatg(x,y)=0forallyψ>ψx
,ifyisnotmaximal,g(x,y)=0forallx<y.
(X,K)
WebeginthissectionbypresentingsomepropertiesofassociativeandLieideals
ofI(X,K)containedinJ(I(X,K)).
(X,K)suchthatI⊆J(I(X,K)).Iff∈I
andf(x,y)6=0,thenexy∈I.
∈Isuchthatf(x,y)6=<ψysinceI⊆J(I(X,K)).
Therefore,f(x,y)exy=[[ex,f],ey]∈I,whenceexy∈I.
(X,K)suchthatI⊆J(I(X,K)).Then
I=Span{exy:exy∈I}.1
K
GivenasubsetSofI(X,K),wewilldenotebyhSi()theideal(resp.
Lieideal)ofI(X,K)generatedbyS.
≤yonehashexyi=SpanK{euv:u≤x≤y≤v}.
≤x≤y≤v,theneuv=,theproducteabexyecd
isnonzeroifandonlyifb=xandy=c,inwhichcaseitequalseadwitha≤x≤
y≤d.
∈hSi,whereS⊆J(I(X,K)).Thentherearef∈Sand
x≤u<v≤ysuchthatf(u,v)6=0.
P
=igifihiwithgi,hiP∈IP(X,K)andfi∈(u,v)=0for
allx≤u<v≤y,then1=exy(x,y)=ix≤u<v≤ygi(x,u)fi(u,v)hi(v,y)=0,
acontradiction.
⊆J(I(X,K)),thenhSi=,everyLieidealof
I(X,K)containedinJ(I(X,K))isanideal.
1Foranassociativeidealthisiswell-known(see[8,]).:.
LIEAUTOMORPHISMSOFINCIDENCEALGEBRAS5
,hSiL⊆⊆hSiL,weonlyneedtoshow
∈hSiLwheneverexy∈∈
∈Sandx≤uψ<ψv≤ysuchthatf(u,v)6=
exy=f(u,v)−1[[exu,f],evy]∈hSiL.
I
⊆JmbeanidealofI(X,K)suchthatdim=1.
I∩Jm+1
Thenthereexistsauniqueexy∈Jm−Jm+1suchthatanyf∈I−Jm+1isofthe
formf=f(x,y)exy+jxy,wherejxy∈Jm+1.
∈I−Jm+∈Jm−Jm+1suchthatf(x,y)6=0.
I
,exy∈=1,exyistheonlyeuv∈Jm−Jm+1
I∩Jm+1
,f−f(x,y)exy∈Jm+1.
≤yandu≤≤u,thenhexyiheuvi={0}.
∈hexyiandg∈≤bsuchthat
X
06=(fg)(a,b)=f(a,c)g(c,b),
a≤c≤b
thenf(a,c)6=0andg(c,b)6=0forsomea≤c≤
a≤x≤y≤candc≤u≤v≤b,whencey≤u,acontradiction.
(I(X,K))intoasemidirectproduct
LetLi=SpanK{exy:exy∈Ji−Ji+1}=SpanLK{exy:l(⌊x,y⌋)=i},forany
integeri≥0,whereJ0:=I(X,K).ThenJi=k≥iLkasK-vectorspaces,i≥0.
∈LAut(I(X,K)),setϕe:I(X,K)→I(X,K)tobethe
K-linearmapdefinedasfollows:foranyexy,ifexy∈Li,then
ϕ(exy)=ϕe(exy)+jxy,
whereϕe(exy)∈Liandjxy∈Ji+1.
Clearly,ϕeiswell-.
→ϕeisagrouphomomorphismfromLAut(I(X,K))
toGL(I(X,K)).
∈GL(I(X,K)).Sinceϕeislinear,itsufficestocheck
≤yandtakeg∈I(X,K)suchthatϕ(g)=∈Li,
then,,g∈Ji,sog=f+hwithf∈Liandh∈Ji+
followsthatexy=ϕ(g)=ϕ(f)+ϕ(h)=ϕe(f)+j+ϕ(h),wherej∈Ji+,
exy−ϕe(f)=j+ϕ(h)inwhichexy−ϕe(f)∈Liandj+ϕ(h)∈Ji+,
ϕe(f)=exy(andj+ϕ(h)=0),whichimpliesthatϕeisonto.
Letϕ,ψ∈LAut(I(X,K)).Weneedtoprovethatϕ]◦ψ=ϕe◦≤y
withexy∈(exy)=ψe(exy)+jxy,wherejxy∈Ji+1,soϕ(ψ(exy))=
ϕ(ψe(exy))+ϕ(jxy)=ϕe(ψe(exy))+j′+ϕ(jxy)forsomej′∈Ji+(ψe(exy))∈
xyxy
Liandj′+ϕ(jxy)∈Ji+1,then(ϕ]◦ψ)(exy)=ϕe(ψe(exy)).
xy
LetusintroduceBi=B∩JiandX2={(x,y)∈X2:x<y}.
<:.
6É,MYKOLAKHRYPCHENKO,.
∈LAut(I(X,K)).Thereexistabijectionθ=θϕ:B→B
withθ(Bi)⊆Biandamapσ=σϕ:X2→K∗suchthat,foranyexy∈B,
<
ϕe(exy)=σ(x,y)θ(exy).(1)
∈Bi−Bi+=hexyi/(hexyi∩Ji+1).
Observethatdim(Ixy)=1sinceanyeuv∈hexyiwitheuv6=exybelongstoJi+1by
.
Notethatϕ(hexyi)⊆(hexyi)isanidealofI(X,K)
,ϕ(hexyi)/(ϕ(hexyi)∩Ji+1)andIxyareisomorphicas
(ϕ(hexyi)/(ϕ(hexyi)∩Ji+1))=dim(Ixy)=-
∈B−B,k∈K∗andg∈Jsuchthatϕ(e)=ke+g.
uvii+1i+1xyuv
Hence,ϕe(e)=:B→Bbyθ(e)=eandσ:X2→K∗
xyuvxyuv<
byσ(x,y)=,θ(Bi)⊆Bi.
,itsufficestoestablish
(exy)=σ(x,y)euvandϕe(eab)=σ(a,b)euvfor
someexy,eab∈Bi−Bi+(exy)=σ(x,y)euv+gandϕ(eab)=σ(a,b)euv+h
forsomeg,h∈Ji+,ϕ(σ(a,b)exy−σ(x,y)eab)=σ(a,b)g−σ(x,y)h∈
Ji+(a,b)exy−σ(x,y)eab∈Ji+
σ(x,y),σ(a,b)∈K∗,thenexy=eab(andσ(x,y)=σ(a,b)).
∈LAut(I(X,K))andexy∈Li,wherei>(exy)−
ϕe(exy)∈hθϕ(exy)i∩Ji+1.
(exy)=ϕe(exy