The variety of $2$-dimensional algebras over an algebraically closed field

0.21 MB·

by  Ivan Kaygorodov

Thevarietyof2-dimensionalalgebras overan algebraically closedfield 1
IvanKaygorodova,YuryVolkovb
aUniversidadeFederaldoABC,CMCC,SantoAndre´,Brazil.
bSaintPetersburgstateuniversity,SaintPetersburg,Russia.
7 E-mailaddresses:
1 IvanKaygorodov([email protected]),
YuryVolkov([email protected]).
0
2
Abstract.Theworkisdevotedtothevarietyof2-dimensionalalgebrasoveranalgebraicallyclosedfield. Firstly,weclassify

n such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the
a
varietyunderconsideration. Finally, we applyourresultsto obtainanalogousdescriptionsforthe subvarietiesofflexible, and
J
bicommutativealgebras.Inparticular,wedescriberigidalgebrasandirreduciblecomponentsforthesesubvarieties.
8
2 Keywords:2-dimensionalalgebras,orbitclosure,degeneration,rigidalgebra

]
A
1. INTRODUCTION
R
. Inthispaper,analgebraissimplyavectorspaceoverafieldwithbilinearbinaryoperationthatdoesn’thavetobeassociative.
h
Algebras of a fixed dimension form a variety with a natural action of a general linear group. The algebraic classification of
t
a algebras is the classification of isomorphism classes of algebras. In other words it is the classification of orbits under the
m
action of the generallinear group. There are many classifications for varietiesof algebrasof some fixed dimensionsatisfying
[ somepolynomialidentities. Forexample,thereexistalgebraicclassificationsof2-dimensionalpre-Liealgebras,3-dimensional

1 Novikovalgebras,4-dimensionalJordanalgebras,4-dimensionalLeibnizalgebras,and6-dimensionalLiealgebras.
v In this paper we classify all 2-dimensional algebras over an algebraically closed field. It is not the first work devoted to
3 this problem, classifications of differenttypeswere made in [1,10,24], but some of them contain inaccuracies, some of them
3
are notwellorganizedandallof themarenotconvenientforourmaingoal, thegeometricdescriptionof thealgebraicvariety
2
of 2-dimensionalalgebras. By this reason, we give a classification that is valid over an algebraically closed field of arbitrary
8
0 characteristic. Inthesamepartofthepaper,wealsodescribeautomorphismgroupsforallalgebrasunderconsideration. Note
. thatthereareotherinterestingstructuresonthesetof2-dimensionalalgebras.Oneofthem,thestructureofconservativealgebra
1
definedbyI.Kantor,wasdevelopedin[17,18].
0
7 In the main part of our paper we developthe geometryof the variety of 2-dimensionalalgebras. Namely, we describe the
1 closures of orbits of some sets with respect to Zariski topology. Firstly, we describe all possible degenerations, i.e. closures
: of orbits of one point sets. Degenerations is an interesting subject, which was studied in various papers (see, for example,
v
i [2–9,11–13,15,16,21–23,25]). Oneoftheproblemsinthisdirectionistodescribealldegenerationsinavarietyofalgebrasof
X
somefixeddimensionsatisfyingsomesetofidentities. Forexample,thisproblemwassolvedfor2-dimensionalpre-Liealgebras
r in[2],for3-dimensionalNovikovalgebrasin[3],for4-dimensionalLiealgebrasin[6],for4-dimensionalZinbielandnilpotent
a
Leibniz algebras in [16], for nilpotent 5- and 6-dimensional Lie algebras in [11,25], and for nilpotent 5- and 6-dimensional
Malcevalgebrasin[15]. Asanapplicationofourresults,onecaneasilyrecovertheresultsof[2].
Anotherinteresting notionconcerningdegenerationsis the so-called levelof an algebra. The algebrasof the first leveland
theassociative,LieandJordanalgebrasofthesecondlevelareclassifiedin[19,20]. Inthepapers[7–9],theauthordefinedthe
notionofaninfinitelevelanddescribedallanticommutativealgebrasthathaveaninfinitelevelnotgreaterthan3. Thisnotion
ismucheasier inthesense thattheinfinitelevelofan algebracanbe easilyexpressedintermsoftheusuallevel. Algebrasof
low dimensionplay a specialrole in problemsof such type, because theyhave small levels. The descriptionof degenerations
obtainedinthisworkallowstocomputethelevelforall2-dimensionalalgebras.
Thenextresultofthispaperisthedescriptionofclosuresoforbitsofsomeprincipalseriesthatappearduringthealgebraic
classification.Anotherimportantcharacteristicofavarietyisitspartitiontoirreduciblecomponents.Thenotionofarigidalgebra
is closely related to this characteristic, because orbit closures of such algebras form irreducible components. For example,
1
I.KaygorodovwassupportedbyFAPESP14/24519-8.
1
2
irreducible componentsand rigid algebras were classified for low dimensional associative (see [22,23]) and Jordan algebras.
Since the varietyof2-dimensionalalgebrasis simplyk8, it isclear thatthereis onlyone irreduciblecomponentandthere are
norigidalgebras. Thus,thisproblemisnotrelevantforthevarietyofall2-dimensionalalgebras. Nevertheless,itisrelevantfor
subvarieties. Inthelastpartweapplyourresultsaboutthevarietyofall2-dimensionalalgebrastoitssubvarietiesconsistingof
flexible and bicommutativealgebras. We describe all degenerationsand closuresof orbits in these varieties. In particular, we
classifytheirreduciblecomponentsandrigidalgebras.Ourresultsallowtogetsuchdescriptionsandclassificationsforvarieties
of2-dimensionalalgebrasdefinedbyanyidentitieswithoutanyproblems.
2. DEFINITIONS AND NOTATION
Duringthepaperwefixanalgebraicallyclosedfieldk,a2-dimensionalk-linearvectorspaceV andabasis{e ,e }ofV. All
1 2
spacesinthispaperareconsideredoverk,andwewritesimplydim,Homand⊗insteadofdimk,Homk and⊗k. Analgebra
AisasetwithastructureofavectorspaceandabinaryoperationthatinducesabilinearmapfromA×AtoA.
Sincethispaperisdevotedto2-dimensionalalgebras,wegivealldefinitionsandnotationonlyforthiscase,thougheverything
inthissectioncanberewrittenforanydimension.
ThesetA := Hom(V ⊗V,V) ∼= V∗ ⊗V∗⊗V isavectorspaceofdimension8. Thisspacehasastructureoftheaffine
2
varietyk8.Indeed,anyµ∈A isdeterminedby8structureconstantsck ∈k(i,j,k =1,2)suchthatµ(e ⊗e )=c1e +c2e .
2 ij i j ij 1 ij 2
AsubsetofA isZariski-closedifitcanbedefinedbyasetofpolynomialequationsinthevariablesck.
2 ij
ThegenerallineargroupGL(V)actsonA byconjugations:
2
(g∗µ)(x⊗y)=gµ(g−1x⊗g−1y)
forx,y ∈V,µ ∈ A andg ∈GL(V). Thus,A isdecomposedintoGL(V)-orbitsthatcorrespondtotheisomorphismclasses
2 2
of2-dimensionalalgebras.Thus,analgebraicclassificationof2-dimensionalalgebrasistheclassificationofGL(V)-orbits.
LetO(µ)denotetheorbitofµ ∈ A undertheactionofGL(V)andO(µ)denotetheZariskiclosureofO(µ). LetAandB
2
betwo2-dimensionalalgebrasandµ,λ∈A representAandBrespectively.WesaythatAdegeneratestoBandwriteA→B
2
ifλ∈O(µ). NotethatinthiscasewehaveO(λ)⊂O(µ). Hence,thedefinitionofadegenerationdoesn’tdependonthechoice
of µ and λ. If A 6∼= B, then the assertion A → B is called a properdegeneration. We write A 6→ B if λ 6∈ O(µ). Let now
A(∗) := {A(α)} beasetof2-dimensionalalgebrasandµ(α) ∈ A representA(α)forα ∈ I. Ifλ ∈ {O(µ(α))} ,then
α∈I 2 α∈I
wewriteA(∗)→B andsaythatA(∗)degeneratestoB. IntheoppositecasewewriteA(∗)6→B.
LetA(∗), B, µ(α) (α ∈ I) andλbe asabove. Letck (i,j,k = 1,2)be thestructureconstantsof λin the basise ,e . If
ij 1 2
we constructaj : k∗ → k (i,j = 1,2) and f : k∗ → I such that a1(t)e +a2(t)e and a1(t)e +a2(t)e form a basis of
i 1 1 1 2 2 1 2 2
V foranyt ∈ k∗, andthe structure constantsof µ in thisbasis are such polynomialsck(t) ∈ k[t] thatck(0) = ck, then
f(t) ij ij ij
A(∗) → B. In this case (a1(t)e +a2(t)e ,a1(t)e +a2(t)e ) and f(t) are called a parametrizedbasis and a parametrized
1 1 1 2 2 1 2 2
indexforA(∗) → B respectively. Thecaseofdegenerationbetweentwoalgebrascorrespondstothecase|I| = 1. Inthiscase
weneedonlyparametrizedbasis,becausef(t)istheuniqueelementofI foranyt∈k∗.
Wetaketheideasforprovingnon-degenerationsfrom[25]. LetQbeasetofpolynomialequationsinvariablesxk (i,j,k=
i,j
1,2).SupposethatQsatisfiesthefollowingproperty.Ifxk =ck isasolutiontoallequationsinQ,thenxk =c˜k isasolution
i,j ij i,j ij
toallequationsinQtoointhefollowingcases:
(1) thereareα ,α ∈k∗ suchthatc˜k = αiαjck;
(2) thereisα∈1ks2uchthat ij αk ij
c˜1 =c1 +α(c1 +c1 )+α2c1 , c˜1 =c1 +αc1 , c˜1 =c1 +αc1 , c˜1 =c1 ,
11 11 12 21 22 21 21 22 12 12 22 22 22
c˜2 =c2 +α(c2 +c2 −c1 )+α2(c2 −c1 −c1 )−α3c1 ,
11 11 12 21 11 22 12 21 22
c˜2 =c2 +α(c2 −c1 )−α2c1 , c˜2 =c2 +α(c2 −c1 )−α2c1 , c˜2 =c2 −αc1 .
21 21 22 21 22 12 12 22 12 22 22 22 22
LetR ⊂ A bea setofallalgebrastructureswhosestructureconstantssatisfy allequationsinQ. We willcallsucha setR a
2
closedupperinvariantset. Let{A(α)} beasetof2-dimensionalalgebrassuchthatA(α)canberepresentedbyastructure
α∈I
fromRforanyα∈I. LetBbea2-dimensionalalgebrarepresentedbythestructureλ∈A . IfO(λ)∩R =∅,thenA(∗)6→B.
2
InthiscasewecallRaseparatingsetforA(∗)6→B.
3
Let us recall two more tools for proving degenerations and non-degenerations. Firstly, if A → B, then dimAut(A) <
dimAut(B). Note that if A(∗) 6→ B, then either dimAut(A(α)) = dimAut(B) for infinitely many α ∈ I or
dimAut(A(α))<dimAut(B)forsomeα∈I,butitispossiblethatdimAut(A(α))≥dimAut(B)forallα∈I. Notealso
thatdimAut(A)=dimDer(A). Secondly,ifA→C andC →BthenA→B. IfthereisnoC suchthatA→CandC →B
areproperdegenerations,thentheassertionA→B iscalledaprimarydegeneration.IftherearenoC andDsuchthatC →A,
B → D, C 6→ D andoneoftheassertionsC → AandB → D isaproperdegeneration,thentheassertionA 6→ B iscalled
aprimarynon-degeneration.Itsufficestoproveonlyprimarydegenerationsandnon-degenerationstodescribedegenerationsin
thevarietyunderconsideration.Notealsothatanyalgebradegeneratestothealgebrawithzeromultiplication.
3. ALGEBRAIC CLASSIFICATION
Thefirstofouraimsistoclassifyall2-dimensionalalgebrasoverkmoduloisomorphism. Ourclassificationisbasedonthe
followinglemma.
Lemma 1. Let A be a 2-dimensionalalgebra. Then there exists a non-zero element x ∈ A such that x and x2 are linearly
dependent.
Proof. Ifc1 =0,wecantakex=e . Supposethatc1 6=0. Letusdefinex =e +te fort∈k. Since
22 2 22 t 1 2
x2 =(c1 +(c1 +c1 )t+c1 t2)e +(c2 +(c2 +c2 )t+c2 t2)e ,
t 11 12 21 22 1 11 12 21 22 2
x andx2 arelinearlydependentiff
t t
c1 +(c1 +c1 )t+c1 t2 c2 +(c2 +c2 )t+c2 t2
0= 11 12 21 22 11 12 21 22 =c1 t3+At2+Bt+C,
(cid:12) 1 t (cid:12) 22
(cid:12) (cid:12)
whereA,B,C ∈ kare(cid:12)(cid:12) someelementsdependingonlyonthestructureconstants.(cid:12)(cid:12)Sincec1 6= 0andkisalgebraicallyclosed,
22
thereexiststsuchthatx andx2 arelinearlydependent.
t t
✷
Notethatifx ∈ Aandx2 arelinearlydependent,theneitherx2 = 0orx = αeforsomeα ∈ k∗ andsomee ∈ Asuchthat
e2 =e. Ifx2 =0,thenxiscalleda2-nilelement.Anelementesuchthate2 =eiscalledanidempotent.
Corollary2. Any2-dimensionalk-algebrabelongstooneofthefollowingdisjointclasses:
A. algebrasthatdon’thavenonzeroidempotentsandhaveaunique1-dimensionalsubspaceof2-nilelements;
B. algebrasthatdon’thavenonzeroidempotentsandhavetwolinearlyindependent2-nilelements;
C. algebrasthathaveauniquenonzeroidempotentanddon’thavenonzero2-nilelements;
D. algebrasthathaveauniquenonzeroidempotentandanonzero2-nilelement;
E. algebrasthathavetwodifferentnonzeroidempotents.
Proof. The fact thatthe classes are disjointis obvious. The fact thatany 2-dimensionalalgebra belongsto one of the classes
followseasilyfromLemma1andtheremarkafterit.
✷
To give the classification of 2-dimensionalalgebras we have to introduce some notation. Let us consider the action of the
cyclicgroupC = hρ | ρ2ionkdefinedbytheequalityρα = −αforα ∈ k. Letusfixsomesetofrepresentativesoforbits
2
underthisactionanddenoteitbyk≥0. Forexample,ifk = C, thenonecantakeC≥0 = {α ∈ C | Re(α) > 0}∪{α ∈ C |
Re(α)=0,Im(α)≥0}.
LetusalsoconsidertheactionofC onk2definedbytheequalityρ(α,β)=(1−α+β,β)for(α,β)∈k2. Letusfixsome
2
setofrepresentativesoforbitsunderthisactionanddenoteitbyU. LetusalsodefineT ={(α,β)∈k2 |α+β =1}.
Given(α,β,γ,δ) ∈ k4,wedefineD(α,β,γ,δ) = (α+γ)(β+δ)−1. WedefineC (α,β,γ,δ) = (β,δ),C (α,β,γ,δ) =
1 2
(γ,α),andC (α,β,γ,δ) = βγ−(α−1)(δ−1),αδ−(β−1)(γ−1) for(α,β,γ,δ)suchthatD(α,β,γ,δ) 6= 0. Letusconsiderthe
3 D(α,β,γ,δ) D(α,β,γ,δ)
(cid:16) (cid:17)
set C (Γ),C (Γ),C (Γ) |Γ∈k4,D(Γ)6=0,C (Γ),C (Γ)6∈T ⊂ (k2)3. OnecanshowthatthesymmetricgroupS acts
1 2 3 1 2 3
ont(cid:8)h(cid:0)issetbytheequalityσ(cid:1) C (Γ),C (Γ),C (Γ) = C (Γ),C(cid:9) (Γ),C (Γ) forσ ∈S . Notethatthereexistsa
1 2 3 σ−1(1) σ−1(2) σ−1(3) 3
(cid:0) (cid:1) (cid:0) (cid:1)
4
setofrepresentativesoforbitsunderthisactionV˜ suchthatif(C ,C ,C ) ∈ V˜ andC 6= C ,thenC 6= C ,C . Letusfixsuch
1 2 3 1 2 3 1 2
V˜ anddefine
V={Γ∈k4 |D(Γ)6=0;C (Γ),C (Γ)6∈T, C (Γ),C (Γ),C (Γ) ∈V˜}.
1 2 1 2 3
ForΓ∈V,wealsodefineC(Γ)={C (Γ),C (Γ),C (Γ)}⊂k2. (cid:0) (cid:1)
1 2 3
Let us consider the action of the cyclic group C on k∗ \{1} defined by the equality ρα = α−1 for α ∈ k∗ \{1}. Let
2
us fix some set of representativesof orbitsunderthis actionand denoteit by k∗ . For example, if k = C, then onecan take
>1
C∗ ={α∈C∗ ||α|>1}∪{α∈C∗ ||α|=1,0<arg(α)≤π}. For(α,β,γ)∈k2×k∗ wedefine
>1 >1
β 1−β
C(α,β,γ)= αγ,(1−α)γ , , ⊂k2.
(cid:26) (cid:18)γ γ (cid:19)(cid:27)
(cid:0) (cid:1)
LetF ⊂A bethesetformedbythefollowingalgebrastructuresonV:
2
A (α),α∈k e e =e +e , e e =αe , e e =(1−α)e , e e =0
1 1 1 1 2 1 2 2 2 1 2 2 2
A e e =e , e e =e , e e =−e , e e =0
2 1 1 2 1 2 2 2 1 2 2 2
A e e =e , e e =0, e e =0, e e =0
3 1 1 2 1 2 2 1 2 2
A4(α),α∈k≥0 e1e1 =αe1+e2, e1e2 =e1+αe2, e2e1 =−e1, e2e2 =0
B (α),α∈k e e =0, e e =(1−α)e +e , e e =αe −e , e e =0
1 1 1 1 2 1 2 2 1 1 2 2 2
B (α),α∈k e e =0, e e =(1−α)e , e e =αe , e e =0
2 1 1 1 2 1 2 1 1 2 2
B e e =0, e e =e , e e =−e , e e =0
3 1 1 1 2 2 2 1 2 2 2
C(α,β),(α,β)∈k×k≥0 e1e1 =e2, e1e2 =(1−α)e1+βe2, e2e1 =αe1−βe2, e2e2 =e2
D (α,β),(α,β)∈U e e =e , e e =(1−α)e +βe , e e =αe −βe , e e =0
1 1 1 1 1 2 1 2 2 1 1 2 2 2
D (α,β),(α,β)∈k2\T e e =e , e e =αe , e e =βe , e e =0
2 1 1 1 1 2 2 2 1 2 2 2
D (α,β),(α,β)∈k2\T e e =e , e e =e +αe , e e =−e +βe , e e =0
3 1 1 1 1 2 1 2 2 1 1 2 2 2
E (α,β,γ,δ),(α,β,γ,δ)∈V e e =e , e e =αe +βe , e e =γe +δe , e e =e
1 1 1 1 1 2 1 2 2 1 1 2 2 2 2
E (α,β,γ),
2 e e =e , e e =(1−α)e +βe , e e =αe +γe , e e =e
(α,β,γ)∈k3\k×T 1 1 1 1 2 1 2 2 1 1 2 2 2 2
E (α,β,γ),
3 e e =e , e e =(1−α)γe + βe , e e =αγe + 1−βe , e e =e
(α,β,γ)∈k2×k∗ 1 1 1 1 2 1 γ 2 2 1 1 γ 2 2 2 2
>1
E e e =e , e e =e +e , e e =0, e e =e
4 1 1 1 1 2 1 2 2 1 2 2 2
E (α),α∈k e e =e , e e =(1−α)e +αe , e e =αe +(1−α)e , e e =e
5 1 1 1 1 2 1 2 2 1 1 2 2 2 2
Thissectionisdevotedtotheproofofthefollowingtheoremthatgivesanalgebraicclassificationof2-dimensionalalgebras
overk.
Theorem3. Anynon-trivial2-dimensionalk-algebracanberepresentedbyauniquestructurefromF.
Inotherwords,Theorem3statesthatA = O(µ)∪{k2}andthat,ifµ,λ∈Faredifferentstructures,thenO(µ)∩O(λ) =
2
µS∈F
∅. Whenever an algebra named A appears in this section, we suppose that it is represented by some structure from A with
2
structureconstantsck (i,j,k =1,2). TherestofthissectionisdevotedtotheproofofTheorem3. AccordingtoCorollary2,it
ij
sufficestoconsidereachoftheclassesA–Eseparately.
Lemma4. IfAbelongstotheclassA,thenitcanberepresentedbyauniquestructurefromtheset
(1) {A1(α)}α∈k∪{A2}∪{A3}∪{A4(α)}α∈k≥0.
Proof. Let us represent the algebra A by a structure such that e e = 0. It is easy to see that A belongs to the class A iff
2 2
x =e +te andx2 arelinearlyindependentforanyt∈k. Since
t 1 2 t
x2 =(c1 +(c1 +c1 )t)e +(c2 +(c2 +c2 )t)e ,
t 11 12 21 1 11 12 21 2
x andx2 arelinearlyindependentiff
t t
c1 +(c1 +c1 )t c2 +(c2 +c2 )t
06= 11 12 21 11 12 21 =(c1 +c1 )t2+(c1 −c2 −c2 )t−c2 .
(cid:12) 1 t (cid:12) 12 21 11 12 21 11
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
5
Sincebyourassumptionx andx2arelinearlyindependentforanyt∈k,wehavec1 +c1 =0,c1 =c2 +c2 ,andc2 6=0.
t t 12 21 11 12 21 11
Nowwehavefourcases:
• c112 =0,c111 6=0. Consideringthebasis ce11 , (cc2111e)22 ofV,onecancheckthatAcanberepresentedbyA1 cc1221 .
11 11 (cid:16) 11(cid:17)
• c1 =0,c2 =−c2 6=0. Consideringthebasis e1 , c211e2 ofV,onecancheckthatAcanberepresentedbyA .
12 12 21 c2 (c2 )2 2
12 12
• c1 =c2 =c2 =0. Consideringthebasise ,c2 e ofV,onecancheckthatAcanberepresentedbyA .
12 12 21 1 11 2 3
• c112 6= 0. Leta ∈ k∗ besuchthatc211c112a2 = 1andc111a ∈ k≥0. Consideringthebasisa e1− cc2211e2 , ce12 ofV,one
(cid:16) 12 (cid:17) 12
cancheckthatAcanberepresentedbyA (c1 a).
4 11
It remains to prove that any two different structures from the set (1) represent non-isomorphicalgebras. Firstly, note that
dim(A )2 = dim(A )2 = 1whiledim(A (α))2 = dim(A (α))2 = 2foranyα ∈ k. WehavealsoA 6∼= A ,becauseA
2 3 1 4 2 3 3
hasanonzeroannihilator.
SupposethatAisrepresentedbythestructureA (α)forsomeα∈k. Thenthereexistsx∈Asuchthatx2 =0,xA+Ax⊂
1
hxi, and αxy = (1−α)yx for any y ∈ A. Such an element doesn’t exist in A (β) for any β ∈ k and in A (β) for any
4 1
β ∈k\{α}.
SupposethatA is representedby the structure A4(α) forsome α ∈ k≥0. Supposethatthe structure constantsof A in the
basisE1, E2 equalthestructureconstantsofA4(β)forsomeβ ∈ k≥0. SinceE2E2 = 0andE2E1 = −E1, itiseasytosee
thatE =e andE = ae forsomea∈ k∗. ThenweobtainfromtheequalityE E =βE +E thata =±1andβ =±α.
2 2 1 1 1 1 1 2
Sinceα,β ∈k≥0,wehaveβ =α.
✷
Lemma5. IfAbelongstotheclassB,thenitcanberepresentedbyauniquestructurefromtheset
(2) {B1(α)}α∈k∪{B2(α)}α∈k∪{B3}.
Proof. LetusrepresentthealgebraAbyastructuresuchthate e = e e = 0. Fors,t ∈ k, letusdefinex = se +te .
1 1 2 2 s,t 1 2
Supposethatthereare s,t ∈ k∗ suchthat0 = x2 = st(e e +e e ). Thene e +e e = 0 andA isanticommutative. It
s,t 1 2 2 1 1 2 2 1
iseasytoseethatany2-dimensionalanticommutativealgebraeitherhasthetrivialmultiplicationorcanberepresentedbyB
3
(notethatbyourdefinitionAisanticommutativeiffx2 =0foranyx∈A).
Supposenowthatx2 6= 0 foranys,t ∈ k∗. SinceAdoesn’thaveidempotents,x andx2 arelinearlyindependentfor
s,t s,t s,t
s,t ∈ k∗. Itiseasytocheckthatx andx2 arelinearlydependentfors = c2 +c2 ,t = c1 +c1 . Hence,c1 +c1 = 0
s,t s,t 12 21 12 21 12 21
or c2 +c2 = 0. Without loss of generality we may assume that c2 +c2 = 0. Since A is not anticommutative, we have
12 21 12 21
c1 +c1 6=0inthiscase.
12 21
If c2 6= 0, then, considering the basis e1 , e2 of V, one can check that A can be represented by B c121 . If
12 c2 c1 +c1 1 c1 +c1
12 12 21 (cid:16) 12 21(cid:17)
c2 =0,then,consideringthebasise , e2 ofV,onecancheckthatAcanberepresentedbyB c121 .
12 1 c1 +c1 2 c1 +c1
It remains to prove that any two diffe1r2ent21structures from the set (2) represent non-isomorphical(cid:16)ge1b2ras2.1(cid:17)Since B is anti-
3
commutative,it isnotisomorphicto otheralgebrasfrom(2). Notealso thatdim(B (α))2 = 2 > 1 = dim(B (β))2 forany
1 2
α,β ∈k.
SupposethatAisrepresentedbythestructureB (α)forsomeα∈kandi=1,2. SupposethatthestructureconstantsofA
i
inthebasisE ,E equalthestructureconstantsofB (β)forsomeβ ∈k. SinceE E =E E =0,wehaveeitherE =ae ,
1 2 i 1 1 2 2 1 1
E = be orE = ae ,E = be forsomea,b ∈ k∗. SinceE E +E E = E ,wehaveE = ae andE = e . Thenwe
2 2 1 2 2 1 1 2 2 1 1 1 1 2 2
getβ =αfromtheequalityE E =(1−β)E +(2−i)E .
1 2 1 2
✷
Lemma6. IfAbelongstotheclassC,thenitcanberepresentedbyC(α,β)forauniquepair(α,β)∈k×k≥0.
Proof. Let us represent the algebra A by a structure such that e e = e . It is easy to see that A belongs to the class C iff
2 2 2
x =e +te andx2 arelinearlyindependentforanyt∈k. Since
t 1 2 t
x2 =(c1 +(c1 +c1 )t)e +(c2 +(c2 +c2 )t+t2)e ,
t 11 12 21 1 11 12 21 2
6
x andx2 arelinearlyindependentiff
t t
c1 +(c1 +c1 )t c2 +(c2 +c2 )t+t2
06= 11 12 21 11 12 21 =(c1 +c1 −1)t2+(c1 −c2 −c2 )t−c2 .
(cid:12) 1 t (cid:12) 12 21 11 12 21 11
(cid:12) (cid:12)
Sincebyourassu(cid:12)(cid:12)mptionxtandx2t arelinearlyindependentforan(cid:12)(cid:12)yt∈k,wehavec112+c121 =1,c111 =c212+c221,andc211 6=0.
Letabesuchanelementofk∗ thatc211a2 =1anda(c212−c111c121)∈k≥0. Consideringthebasisa(e1−c111e2),e2ofV,one
cancheckthatAcanberepresentedbyC c1 ,a(c2 −c1 c1 ) .
21 12 11 21
SupposethatAisrepresentedbythestr(cid:0)uctureC(α,β)forso(cid:1)mepair(α,β)∈k×k≥0. Supposethatthestructureconstants
ofAinthebasisE1, E2 equalthestructureconstantsofC(γ,δ)forsome(γ,δ) ∈ k×k≥0. SinceE2E2 = E2 andC(α,β)
hasauniqueidempotent,wehaveE =e . WegetE =±e fromtheequalityE E = E . Thenγ = αandδ = ±β. Since
2 2 1 1 1 1 2
β,δ ∈k≥0,wehave(γ,δ)=(α,β).
✷
Lemma7. IfAbelongstotheclassD,thenitcanberepresentedbyauniquestructurefromtheset
(3) {D1(α,β)}(α,β)∈U∪{D2(α,β)}(α,β)∈k2\T∪{D3(α,β)}(α,β)∈k2\T.
Proof. LetusrepresentthealgebraAbyastructuresuchthate e =e ande e =0.
1 1 1 2 2
Letusconsiderthefollowingcases:
• c1 +c1 6=0. Ifc2 +c2 6=0,thenonecancheckthat 1 e + c212+c221−1e isanidempotentthatisnotequal
12 21 12 21 c2 +c2 1 c1 +c1 2
12 21 (cid:16) 12 21 (cid:17)
toe . Thus,c2 +c2 = 0. If c112 ,c2 ∈ U,then,consideringthebasise , e2 ofV,onecancheckthatA
1 12 21 c1 +c1 12 1 c1 +c1
(cid:16) 12 21 (cid:17) 12 21
canberepresentedbyD1 c1c+112c1 ,c212 . If c1c+112c1 ,c212 6∈ U, then c1c+121c1 +c212,c212 ∈ Uand,consideringthe
(cid:16) 12 21 (cid:17) (cid:16) 12 21 (cid:17) (cid:16) 12 21 (cid:17)
basise ,e − e2 ofV,onecancheckthatAcanberepresentedbyD c121 +c2 ,c2 .
1 1 c1 +c1 1 c1 +c1 12 12
12 21 (cid:16) 12 21 (cid:17)
• c1 = −c1 6= 0. Consideringthebasise , e2 ofV,onecancheckthatAcanberepresentedbyD (c2 ,c2 ). Since
12 21 1 c1 3 12 21
12
e +e isnotidempotent,(c2 ,c2 )6∈T.
1 2 12 21
• c1 =c1 6=0.ThenonecancheckthatAisrepresentedbyD (c2 ,c2 ).Sincee +e isnotidempotent,(c2 ,c2 )6∈T.
12 21 2 12 21 1 2 12 21
Itremainstoprovethatanytwodifferentstructuresfromtheset(3)representnon-isomorphicalgebras.
Suppose that A is represented by the structure D (α,β) for some pair (α,β) ∈ U. Note that e e = (e − e )2 = 0
1 2 2 1 2
in D (α,β) while the structures D (γ,δ) and D (γ,δ) have a unique 1-dimensionalsubspace of 2-nilelements for any pair
1 2 3
(γ,δ)∈k2. SupposenowthatthestructureconstantsofAinthebasisE ,E equalthestructureconstantsofD (γ,δ)forsome
1 2 1
pair(γ,δ)∈U. SinceE isanidempotentandD (α,β)hasauniqueidempotent,wehaveE =e . SinceE E =0,wehave
1 1 1 1 2 2
eitherE = ae orE = a(e −e )forsomea ∈ k∗. Weobtaina = 1inbothcasesfromtheequalityE E +E E = E .
2 2 2 1 2 1 2 2 1 1
Thenwehaveδ =β andeitherγ =αorγ =1−α+β. Since(α,β),(γ,δ)∈U,wehave(γ,δ)=(α,β).
SupposethatAisrepresentedbythestructureD (α,β)forsomepair(α,β) ∈ k2. NotethatAhasanelementxsuchthat
2
x2 = 0 and xA+Ax ⊂ hxi while D (γ,δ) doesn’t have such an element for any pair (γ,δ) ∈ k2. Suppose now that the
3
structureconstantsofAinthebasisE ,E equalthestructureconstantsofD (γ,δ)forsomepair(γ,δ) ∈ k2. SinceE isan
1 2 2 1
idempotentandD (α,β)hasauniqueidempotent,wehaveE = e . SinceE E = 0,wehaveE = ae forsomea ∈ k∗.
2 1 1 2 2 2 2
Thenitiseasytoseethat(γ,δ)=(α,β).
Finally, suppose that A is represented by the structure D (α,β) for some pair (α,β) ∈ k2. Suppose that the structure
3
constantsofAinthebasisE ,E equalthestructureconstantsofD (γ,δ)forsomepair(γ,δ)∈k2. SinceE isanidempotent
1 2 3 1
andD (α,β)hasauniqueidempotent,wehaveE =e . SinceE E =0,wehaveE =ae forsomea∈k∗. Thenitiseasy
3 1 1 2 2 2 2
toseethata=1and(γ,δ)=(α,β).
✷
AsaconsequenceoftheproofsofLemmas4–7wecandescribetheautomorphismgroupsofthealgebrasoftheclassesA–D.
Corollary8. 1.Aut(A (α))∼=Aut(A )isisomorphictotheadditivegroupofk.
1 2
a 0
2.Aut(A )isisomorphictothesubgroupofGL (k)formedbymatricesoftheform ,wherea∈k∗ andb∈k.
3 2 (cid:18) b a2 (cid:19)
7
3.Aut(A (α))∼=C ifα=0andchark6=2;Aut(A (α))istrivialifeitherα∈k∗ orα=0andchark=2.
4 2 4
4.Aut(B (α))istrivial;Aut(B (α))∼=k∗.
1 2
1 0
5.Aut(B )isisomorphictothesubgroupofGL (k)formedbymatricesoftheform ,wherea∈k∗ andb∈k.
3 2 (cid:18) b a (cid:19)
6.Aut(C(α,β))∼=C ifβ =0andchark6=2;Aut(C(α,β))istrivialifeitherα∈k∗ orα=0andchark=2.
2
7.Aut(D (α,β))∼=C ifβ =2α−1andAut(C(α,β))istrivialifβ 6=2α−1.
1 2
8.Aut(D (α,β))∼=k∗ andAut(D (α,β))istrivialifα+β 6=1.
2 3
Inparticular,
dimAut(A (α))=dimAut(B (α))=dimAut(C (α,β))=dimAut(D (α,β))=dimAut(D (α,β))=0,
4 1 1 1 3
dimAut(A (α))=dimAut(A )=dimAut(B (α))=dimAut(D (α,β))=1anddimAut(A )=dimAut(B )=2.
1 2 2 2 3 3
Proof. 1–3. AnystructureoftheclassAhasauniquesubspaceof2-nilelementsgeneratedbye . Thus,anyautomorphismof
2
suchanalgebrasendse ande toae +be andce respectively,wherea,c∈k∗ andb∈k. Itiseasytocheckthata=c=1
1 2 1 2 2
forA (α)andA ;c=a2forA ;a=±1,b=0andc=1forA (0);anda=c=1,b=0forA (α)ifα6=0.
1 2 3 4 4
4. ItfollowsfromtheproofofLemma5thatanyautomorphismofthealgebraB (α), wherei ∈ {1,2},sendse ande to
i 1 2
ae ande respectivelyforsomea∈k∗. Itiseasytoseethata=1fori=1andacanbearbitraryfori=2.
1 2
5. Since B (V,V) is generatedby e , any automorphismof B sends e and e to ae +be and ce respectively,where
3 2 3 1 2 1 2 2
a,c∈k∗ andb∈k. Itiseasytoseethatsuchamapisanautomorphismiffa=1.
6. It follows from the proof of Lemma 6 that any automorphism of the algebra C(α,β) sends e and e to ±e and e
1 2 1 2
respectively. It is easy to see that the map that sends e and e to −e and e respectively is an automorphismiff β = 0 or
1 2 1 2
chark=2.
7. ItfollowsfromtheproofofLemma7thatanyautomorphismofthealgebraD (α,β)sendse toe andsendse eitherto
1 1 1 2
e ore −e . Itiseasytoseethatthemapthatsendse ande toe ande −e respectivelyisanautomorphismiffβ =2α−1.
2 1 2 1 2 1 1 2
8. It follows from the proof of Lemma 7 that any automorphism of the algebra D (α,β) sends e and e to e and ae
2 1 2 1 2
respectivelyforsomea∈k∗. ItfollowsfromthesameproofthatanyautomorphismofthealgebraD (α,β)istrivial.
3
✷
WewillfinishtheproofofTheorem3inthenextsectiondevotedtothealgebrasoftheclassE.
4. ALGEBRAS OF THECLASSE
InthissectionweconsiderthealgebrasoftheclassE. ItisclearthatsuchanalgebraisisomorphictoE (Γ)forsomeΓ∈k4.
1
Firstly,wedescribeisomorphismsinsidethissetand,thus,finishtheproofofTheorem3.
Lemma9. E (Γ )∼=E (Γ )iffoneofthefollowingconditionsholds:
1 1 1 2
• Γ =Γ ;
1 2
• C (Γ )=C (Γ )andC (Γ )=C (Γ );
1 1 2 2 2 1 1 2
• C (Γ ),C (Γ ),C (Γ ),C (Γ )∈T,C (Γ )6=C (Γ ),C (Γ )6=C (Γ );
1 1 1 2 2 1 2 2 1 1 2 1 1 2 2 2
• C (Γ ),C (Γ )6∈T,D(Γ )6=0,andthereissomeσ ∈S suchthatC (Γ )=C (Γ )fori∈{1,2,3}.
1 2 2 2 2 3 i 1 σ(i) 2
Proof. Supposethatg ∈GL(V)issuchthatg∗E (Γ )=E (Γ ). Thenge andge aretwolinearlyindependentidempotents
1 1 1 2 1 2
ofE (Γ ). Letusdescribeallnonzeroidempotentsofthisalgebra. LetΓ = (α,β,γ,δ)andu = xe +ye besomeelement
1 2 2 1 2
ofV. ThenE (Γ )(u,u)=uiffx=x2+(α+γ)xyandy =(β+δ)xy+y2.Letusconsiderthefollowingcases:
1 2
• C (Γ )=C (Γ )∈T. InthiscaseE (Γ )(u,u)=uiffeitherx=y =0orx+y =1. Thus,ge =ae +(1−a)e
1 2 2 2 1 2 1 1 2
andge =be +(1−b)e fortwodifferenta,b∈k. OnecancheckthatinthiscaseΓ =Γ .
2 1 2 1 2
• C (Γ ),C (Γ ) ∈ T, C (Γ ) 6= C (Γ ), i.e. α + γ = β + δ = 1, α 6= δ. As in the previous case, we have
1 2 2 2 1 2 2 2
ge = ae +(1−a)e andge = be +(1−b)e fortwodifferenta,b ∈ k. Thus,E (Γ ) ∼= E (Γ )iffthereexist
1 1 2 2 1 2 1 1 1 2
a,b∈k,a6=bsuchthat
Γ =((1−b)α+bδ,aβ+(1−a)γ,bβ+(1−b)γ,(1−a)α+aδ).
1
ItiseasytoseethatsuchaandbexistiffC (Γ ),C (Γ )∈TandC (Γ )6=C (Γ ).
1 1 2 1 1 1 2 1
8
• oneofthefollowingthreeconditionsholds:
1. C (Γ )∈T,C (Γ )6∈T;
1 2 2 2
2. C (Γ )6∈T,C (Γ )∈T;
1 2 2 2
3. C (Γ ),C (Γ )6∈T,D(Γ )=0.
1 2 2 2 2
Itiseasytoseethat,inallofthesecases,E (Γ )(u,u) = uonlyforu ∈ {0,e ,e }. Thus,eitherge = e ,ge = e
1 2 1 2 1 1 2 2
andΓ =Γ orge =e ,ge =e ,C (Γ )=C (Γ )andC (Γ )=C (Γ ).
1 2 1 2 2 1 1 1 2 2 2 1 1 2
• C (Γ ),C (Γ ) 6∈ T, D(Γ ) 6= 0. In this case, E (Γ )(u,u) = u for u ∈ {0,e ,e ,e }, where e = α+γ−1e +
1 2 2 2 2 1 2 1 2 3 3 D(Γ2) 1
β+δ−1e . Thus, there is σ ∈ S such that ge = e for i ∈ {1,2}. Then direct calculations show that C (Γ ) =
D(Γ2) 2 3 i σ(i) i 1
C (Γ )fori∈{1,2,3}.
σ(i) 2
✷
NowwecanfinishtheproofofTheorem3.
ProofofTheorem3. ByCorollary2,thealgebraAbelongstooneoftheclassesA–EandtheclasscontainingAisunique. If
AbelongstooneoftheclassesA–D,thenthestatementofthetheoremfollowsfromLemmas4–7.
SupposethatAbelongstotheclassE. ThenAcanberepresentedbyE (Γ)forsomeΓ=(α,β,γ,δ)∈k4. Nowwehave
1
• ifC (Γ)=C (Γ)∈T,thenE (Γ)=E (β);
1 2 1 4
• ifC (Γ),C (Γ)∈TandC (Γ)6=C (Γ),thenE (Γ)∼=E (1,1,0,0)=E byLemma9;
1 2 1 2 1 1 4
• ifC (Γ)∈TandC (Γ)6∈T,thenE (Γ)∼=E (1−β,γ,β,α)=E (β,γ,α)byLemma9;
1 2 1 1 2
• ifC (Γ)6∈TandC (Γ)∈T,thenE (Γ)=E (γ,β,δ);
1 2 1 2
• ifC (Γ),C (Γ)6∈T,D(Γ)=0andα+γ ∈k∗ ,thenE (Γ)=E γ(β+δ),β(α+γ),α+γ ;
1 2 >1 1 3
• ifC (Γ),C (Γ) 6∈ T,D(Γ) = 0andα+γ 6∈ k∗ ,thenE (Γ) ∼= E(cid:0)(δ,γ,β,α) = E β(α+γ)(cid:1),γ(β+δ),β+δ by
1 2 >1 1 1 3
Lemma9; (cid:0) (cid:1)
• ifC (Γ),C (Γ)6∈TandD(Γ)6=0,thenthereisauniqueσ ∈S suchthatσ−1(C (Γ),C (Γ),C (Γ))∈V˜ andwehave
1 2 3 1 2 3
E1(Γ)∼=E1(Γ′)byLemma9,whereΓ′ ∈VissuchthatCi(Γ′)=Cσ(i)(Γ)fori∈{1,2,3}.
ByLemma9,thestructuresfromtheset
{E1(Γ)}Γ∈V∪{E2(α,β,γ)}(α,β,γ)∈k3\k×T∪{E3(α,β,γ)}(α,β,γ)∈k2×k∗ ∪{E4}∪{E5(α)}α∈k
>1
arepairwisenon-isomorphic.
✷
AsaconsequenceoftheproofofLemma9wecandescribetheautomorphismgroupsofalgebrasoftheclassE.
Corollary 10. 1. For Γ ∈ V, Aut(E (Γ)) is trivial if C (Γ) 6= C (Γ), Aut(E (Γ)) ∼= C if C (Γ) = C (Γ) 6= (−1,−1),
1 1 2 1 2 1 2
Aut(E (−1,−1,−1,−1))∼=S ifchark6=3.
1 3
2.Aut(E )andAut(E (α,β,γ))aretrivialfor(α,β,γ)∈k3\k×T.
4 2
3.For(α,β,γ)∈k2×k∗ ,Aut(E (α,β,γ))∼=C ifγ =−1andα=β andAut(E (α,β,γ))istrivialotherwise.
>1 3 2 3
a b
4.Aut(E (α))isisomorphictothesubgroupofGL (k)formedbymatricesoftheform ,wherea,b∈k,
5 2 (cid:18) 1−a 1−b (cid:19)
a6=b.
In particular, we have dimAut(E (Γ)) = dimAut(E (α,β,γ)) = dimAut(E (α,β,γ)) = dimAut(E ) = 0 and
1 2 3 4
dimAut(E (α))=2.
5
Proof. 1.AnyautomorphismofE (Γ)hastosende ande toe ande respectivelyforsomeσ ∈S ,wheree isdefined
1 1 2 σ(1) σ(2) 3 3
intheproofofLemma9. SuchamapisanautomorphismiffC (Γ)=C (Γ)fori=1,2.IfC (Γ)6=C (Γ),thenwehavealso
i σ(i) 1 2
C (Γ) 6= C (Γ),C (Γ)and,hence,onlyidenticalelementofS determinesanautomorphism. IfC (Γ) = C (Γ) 6= (−1,−1),
3 1 2 3 1 2
then one can check that C (Γ) 6= C (Γ),C (Γ) and, hence, only identical element of S and the element that swaps e and
3 1 2 3 1
e determine automorphisms. If C (Γ) = C (Γ) = (−1,−1), then any σ ∈ S determines an automorphism. Note that
2 1 2 3
(−1,−1,−1,−1)∈Viffchark6=3.
2. FollowsdirectlyfromtheproofofLemma9.
9
3. ItfollowsfromtheproofofLemma9thatanautomorphismofE (α,β,γ)iseitheristrivialorswapse ande . Thelast
3 1 2
mentionedmapisanautomorphismiffγ =−1andα=β.
4. ItfollowsdirectlyfromtheproofofLemma9thatautomorphismsofE (α)areexactlythelinearmapsthatsende and
5 1
e toae +(1−a)e andbe +(1−b)e fortwodifferenta,b∈k.
2 1 2 1 2
✷
NowwearegoingtodiscusssomefactsaboutdegenerationsoftheformA→B,whereAisanalgebraoftheclassE. First
ofall,letusprovethefollowinglemma.
Lemma11. 1.ForanyΓ∈Vand(β,γ)∈C(Γ)thereexistsadegenerationE (Γ)→D (β,γ).
1 2
2.Forany(α,β,γ)∈k3\k×TthereexistsadegenerationE (α,β,γ)→D (β,γ).
2 2
3.Forany(α,δ,ǫ)∈k2×k∗ and(β,γ)∈C(α,δ,ǫ)thereexistsadegenerationE (α,δ,ǫ)→D (β,γ).
>1 3 2
Proof. TheparametrizedbasisEt = e ,Et = te givesthedegenerationE (Γ) → D C (Γ) foranyΓ ∈ k4. IfΓ ∈ Vand
1 1 2 2 1 2 1
1≤i≤3,then,byLemma9,thereexistsΓ′ ∈k4 suchthatE1(Γ)∼=E1(Γ′)andCi(Γ)(cid:0)=C1(Γ(cid:1)′). Hence,E1(Γ)∼=E1(Γ′)→
D C (Γ) . WealsohaveE (α,β,γ)=E (1−α,β,α,γ)→D (β,γ)for(α,β,γ)∈k3\k×Tand
2 i 2 1 2
(cid:0) (cid:1)
δ 1−δ 1−δ δ δ 1−δ
E (α,δ,ǫ)=E (1−α)ǫ, ,αǫ, ∼=E ,αǫ, ,(1−α)ǫ →D αǫ,(1−α)ǫ) ,D ,
3 1(cid:18) ǫ ǫ (cid:19) 1(cid:18) ǫ ǫ (cid:19) 2 2(cid:18)ǫ ǫ (cid:19)
(cid:0) (cid:1)
for(α,δ,ǫ)∈k2×k∗ .
>1
✷
Letusdefine,forΓ=(α,β,γ,δ)∈k4,thefollowingsubsetofA :
2
c1 =0, c1 =γc2 , c1 =αc2 ,
22 21 22 12 22
 (cid:12) 1−γ−δ(α+γ) c212− 1−α−β(α+γ) c221 = β(1−γ)−δ(1−α) c111, 
G(Γ)=µ(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0)(cid:0)11−−γα−−δβ((αα++γγ))(cid:1)(cid:1)22cc212111cc2222(cid:0)22==((δβcc111111−−cc222112))(cid:0)DD(cid:1)((ΓΓ))cc222112(cid:0)++((((βα−−11))((γδ−−11))−−α(cid:1)βδγ))cc111111(cid:1),, .
(cid:12) (cid:0)1−α−β(α+γ)(cid:1) 1−γ−δ(α+γ) c2(cid:0)c2 =(βc1 −c2 ) D(Γ)c2 +((β−(cid:1)1)(γ−1)−αδ)c1 ,
 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0)(cid:0)1−α−β(α+γ)(cid:1)(cid:1)(cid:0)(cid:0)1−γ−δ(α+γ)(cid:1)(cid:1)c12111c22222 =(δc11111−c22112)(cid:0)(cid:0)D(Γ)c21221+((α−1)(δ−1)−βγ)c11111(cid:1)(cid:1) 
HereandfurtherinadefinitionofasubsetofA wealwaysassumebydefaultthatµ∈A andck (i,j,k ∈{1,2})arestructure
2 2 ij
constantsof µ. The followinglemma will allow us to use G(Γ) as a separatingset for some non-degenerations. Its proofis a
directcalculationandsoitislefttoareader.
Lemma12. ThesetG(Γ)isclosedupperinvariantandcontainsE (Γ)foranyΓ∈k4.
1
5. DEGENERATIONSOF2-DIMENSIONALALGEBRAS
In thissection we describeall degenerationsof 2-dimensionalalgebras. Note that the resultsare valid for algebrasoveran
algebraicallyclosedfieldofarbitrarycharacteristic.
Theorem13. A hasthefollowinggraphofprimarydegenerations:
2
10
0 A4(α) B1(γ) C(α,β) D1(α,β) D3(β,γ) E1(Γ) E2(α,β,γ) E3(α,δ,ǫ) E4
γ∈{α,1−α+β}
(β,γ)∈C(Γ) (β,γ)∈C(α,δ,ǫ)
β+γ=0
γ= 1−δ−(1−α)ǫ
1−ǫ
1 A1(α) A2 B2(γ) D2(β,γ)
2 A3 B3 E5(α)
4 k2
Proof. Allprimarydegenerationsthatdon’tfollowfromLemma11arepresentedinthefollowingtable:
degenerations parametrizedbases
A (α)→A Et =te Et =t2e
1 3 1 1 2 2
A (α)→E (α) Et =e Et =e +t−1e
1 5 1 1 2 1 2
A →A Et =te Et =t2e
2 3 1 1 2 2
A →B Et =e Et =t−1e
2 3 1 1 2 2
A (α)→A Et =te −e Et =t2e
4 2 1 1 2 2 2
B (γ)→A Et =e +te Et =−t2e
1 2 1 1 2 2 2
B (γ)→B (γ) Et =te Et =e
1 2 1 1 2 2
B (γ)→A Et =e +te Et =te
2 3 1 1 2 2 1
C(α,β)→A (α) Et =te +e Et =t2e
1 1 1 2 2 2
D (α,β)→B (α) Et =te Et =e
1 2 1 1 2 2
D (α,β)→B (1−α+β) Et =te Et =e −e
1 2 1 2 2 1 2
D (α,β)→D (β,−β) Et =e Et =te
1 2 1 1 2 2
D (β,γ)→A Et =te +te Et =t2e +(β+γ)t2e
2 3 1 1 2 2 1 2
D (β,γ)→A Et = t e −e Et =te
3 2 1 1−β−γ 1 2 2 2
D (β,γ)→D (β,γ) Et =e Et =te
3 2 1 1 2 2
E (α,β,γ)→A (α) Et =te +e Et =(1−β−γ)t2e
2 1 1 1 2 2 1
E (α,δ,ǫ)→B 1−δ−(1−α)ǫ Et =te Et = ǫe1−e2
3 2 1−ǫ 1 1 2 ǫ−1
E →B (cid:16) (cid:17) Et =e −e Et =te
4 3 1 1 2 2 2
E →E (α) Et =αe +(1−α)e Et =(α−t)e +(1−α+t)e
4 5 1 1 2 2 1 2
Thenexttabledescribesseparatingsetsforallrequirednon-degenerationsand,thus,finishestheproofofthetheorem.