Algebra and Applications 1
Non-associative Algebras and Categories
Coordinated by
Abdenacer Makhlouf
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ERC code:
PE1 Mathematics
PE1_2 Algebra
PE1_5 Lie groups, Lie algebras
PE1_12 Mathematical physics
Foreword .......................................xi
AbdenacerM AKHLOUF
Chapter1.JordanSuperalgebras ......................1
ConsueloM ARTINEZ andEfimZ ELMANOV
1.1.Introduction.................................1
1.2.Tits–Kantor–Koecherconstruction.....................4
1.3.Basicexamples(classicalsuperalgebras).................5
1.4.Brackets....................................8
1.5.Cheng–Kacsuperalgebras.........................10
1.6.FinitedimensionalsimpleJordansuperalgebras.............11
1.6.1.Case F isalgebraicallyclosedand char F =0 ............11
1.6.2.Case char F = p> 2,theevenpart J0 issemisimple........11
1.6.3.Case char F = p> 2,theevenpart J0 isnotsemisimple......13
1.6.4.Non-unitalsimpleJordansuperalgebras................13
1.7.Finitedimensionalrepresentations.....................14
1.7.1.Superalgebrasofrank ≥ 3 .......................16
1.7.2.Superalgebrasofrank ≤ 2 .......................17
1.8.Jordansuperconformalalgebras......................21
1.9.References..................................23
Chapter2.CompositionAlgebras ......................27 AlbertoE LDUQUE
2.1.Introduction.................................27
2.2.Quaternionsandoctonions.........................28
2.2.1.Quaternions...............................28
2.2.2.Rotationsinthree-(andfour-)dimensionalspace..........31
2.2.3.Octonions.................................33
2.3.Unitalcompositionalgebras........................35
2.3.1.TheCayley-Dicksondoublingprocessandthegeneralized Hurwitztheorem................................37
2.3.2.IsotropicHurwitzalgebras.......................41
2.4.Symmetriccompositionalgebras......................43
2.5.Triality....................................50
2.6.Concludingremarks.............................54
2.7.Acknowledgments..............................55
2.8.References..................................55
Chapter3.Graded-DivisionAlgebras ....................59 YuriB AHTURIN ,MikhailK OCHETOV andMikhailZ AICEV
3.1.Introduction.................................59
3.2.Backgroundongradings..........................62
3.2.1.Gradingsinducedbyagrouphomomorphism............62
3.2.2.Weakisomorphismandequivalence..................63
3.2.3.Basicpropertiesofdivisiongradings.................63
3.2.4.Gradedpresentationsofassociativealgebras.............64
3.2.5.Tensorproductsofdivisiongradings.................68
3.2.6.Loopconstruction............................70
3.2.7.Anotherconstructionofgraded-simplealgebras...........72
3.3.Graded-divisionalgebrasoveralgebraicallyclosedfields........75
3.4.Realgraded-divisionassociativealgebras.................77
3.4.1.Simplegraded-divisionalgebras....................77
3.4.2.Pauligradings..............................80
3.4.3.Commutativecase............................80
3.4.4.Non-commutativegraded-divisionalgebraswith one-dimensionalhomogeneouscomponents.................82
3.4.5.Equivalenceclassesofgraded-divisionalgebraswith one-dimensionalhomogeneouscomponents.................84
3.4.6.Graded-divisionalgebraswithnon-central two-dimensionalidentitycomponents....................90
3.4.7.Graded-divisionalgebraswithfour-dimensionalidentity components...................................94
3.4.8.Classificationofrealgraded-divisionalgebras,upto isomorphism...................................95
3.5.Realloopalgebraswithanon-splitcentroid...............96
3.6.Alternativealgebras.............................98
3.6.1.Cayley–Dicksondoublingprocess...................99
3.6.2.Gradingsonoctonionalgebras.....................100
3.6.3.Graded-simplerealalternativealgebras................101
3.6.4.Graded-divisionrealalternativealgebras...............102
3.7.Gradingsoffields..............................106
3.8.References..................................107
Chapter4.Non-associative C∗ -algebras ..................111 ÁngelR ODRÍGUEZ PALACIOS andMiguelC ABRERA G ARCÍA
4.1.Introduction.................................111
4.2. JB-algebras..................................111
4.3.Thenon-associativeVidav–PalmerandGelfand–Naimarktheorems..116
4.4. JB∗ -triples..................................128
4.5.Past,presentandfutureofnon-associative C ∗ -algebras.........141
4.6.Acknowledgments..............................145
4.7.References..................................145
Chapter5.Structureof H -algebras .....................155 JoséAntonioC UENCA M IRA
5.1.Introduction.................................155
5.2.Preliminaries:aspectsofthegeneraltheory................156
5.3.Ultraproductsof H -algebras........................164
5.4.Quadratic H -algebras............................166
5.5.Associative H -algebras...........................167
5.6.Flexible H -algebras............................173
5.7.Non-commutativeJordan H -algebras..................175
5.8.Jordan H -algebras.............................178
5.9.Moufang H -algebras............................182
5.10.Lie H -algebras...............................184
5.11.TopicscloselyrelatedtoLie H -algebras................188
5.12.Two-graded H -algebras.........................190
5.13.Othertopics:beyondthe H -algebras..................194
5.14.Acknowledgments.............................194
5.15.References.................................194
Chapter6.Krichever–NovikovTypeAlgebras:Definitions andResults .....................................199 MartinS CHLICHENMAIER
6.1.Introduction.................................199
6.2.TheVirasoroalgebraanditsrelatives...................201
6.3.Thegeometricpicture............................204
6.3.1.ThegeometricrealizationsoftheWittalgebra............204
6.3.2.Arbitrarygenusgeneralizations....................204
6.3.3.Meromorphicforms...........................206
6.4.Algebraicstructures.............................209
6.4.1.Associativestructure..........................209
6.4.2.LieandPoissonalgebrastructure...................210
6.4.3.ThevectorfieldalgebraandtheLiederivative............210
6.4.4.Thealgebraofdifferentialoperators..................211
6.4.5.Differentialoperatorsofalldegrees..................212
6.4.6.Liesuperalgebrasofhalfforms....................213
6.4.7.Jordansuperalgebra...........................213
6.4.8.Highergenuscurrentalgebras.....................214
6.4.9.KN-typealgebras............................215
6.5.Almost-gradedstructure..........................215
6.5.1.Definitionofalmost-gradedness....................215
6.5.2.SeparatingcycleandKNpairing....................216
6.5.3.Thehomogeneoussubspaces......................217
6.5.4.Thealgebras...............................219
6.5.5.Triangulardecompositionandfiltrations...............221
6.6.Centralextensions..............................221
6.6.1.Centralextensionsandcocycles....................222
6.6.2.Geometriccocycles...........................223
6.6.3.Uniquenessandclassificationofcentralextensions.........226
6.7.Examplesandgeneralizations.......................229
6.7.1.Thegenuszeroandthree-pointsituation...............229
6.7.2.Genuszeromultipointalgebras–integrablesystems........231
6.7.3.Deformations...............................232
6.8.Laxoperatoralgebras............................232
6.9.FermionicFockspace............................235
6.9.1.Semi-infiniteformsandfermionicFockspacerepresentations...235
6.9.2. b – c systems...............................237
6.10.Sugawararepresentation..........................237
6.11.Applicationtomodulispace........................240
6.12.Acknowledgments.............................240
6.13.References.................................240
Chapter7.AnIntroductiontoPre-LieAlgebras .............245 ChengmingB AI
7.1.Introduction.................................245
7.1.1.Explanationofnotions.........................245
7.1.2.Twofundamentalproperties......................246
7.1.3.Somesubclasses.............................247
7.1.4.Organizationofthischapter......................248
7.2.Someappearancesofpre-Liealgebras...................249
7.2.1.Left-invariantaffinestructuresonLiegroups:ageometric interpretationof“left-symmetry”.......................249
7.2.2.Deformationcomplexesofalgebrasandright-symmetric algebras.....................................250
7.2.3.Rootedtreealgebras:freepre-Liealgebras..............251
7.2.4.ComplexstructuresonLiealgebras..................251
7.2.5.SymplecticstructuresonLiegroupsandLiealgebras,phase spacesofLiealgebrasandKählerstructures.................252
7.2.6.Vertexalgebras..............................254
7.3.Somebasicresultsandconstructionsofpre-Liealgebras........255
7.3.1.Somebasicresultsofpre-Liealgebras................255
7.3.2.Constructionsofpre-Liealgebrasfromsomeknownstructures..258
7.4.Pre-LiealgebrasandCYBE........................261
7.4.1.Theexistenceofacompatiblepre-LiealgebraonaLiealgebra..261
7.4.2.CYBE:unificationoftensorandoperatorforms...........262
7.4.3.Pre-Liealgebras, O-operatorsandCYBE...............264
7.4.4.Analgebraicinterpretationof“left-symmetry”:construction fromLiealgebrasrevisited...........................265
7.5.Alargerframework:LieanaloguesofLodayalgebras..........266
7.5.1.Pre-Liealgebras,dendriformalgebrasandLodayalgebras.....266
7.5.2.L-dendriformalgebras.........................267
7.5.3.LieanaloguesofLodayalgebras....................269
7.6.References..................................271
Chapter8.Symplectic,ProductandComplexStructureson3-Lie Algebras .......................................275
YunheS HENG andRongTANG
8.1.Introduction.................................275
8.2.Preliminaries.................................278
8.3.Representationsof3-pre-Liealgebras...................280
8.4.Symplecticstructuresandphasespacesof 3-Liealgebras........282
8.5.Productstructureson 3-Liealgebras....................288
8.6.Complexstructureson 3-Liealgebras...................295
8.7.Complexproductstructureson 3-Liealgebras..............304
8.8.Para-Kählerstructureson 3-Liealgebras.................308
8.9.Pseudo-Kählerstructureson 3-Liealgebras................315
8.10.References.................................317
Chapter9.DerivedCategories .........................321
BernhardK ELLER
9.1.Introduction.................................321
9.2.Grothendieck’sdefinition..........................322
9.3.Verdier’sdefinition.............................323
9.4.Triangulatedstructure............................326
9.5.Derivedfunctors...............................331
9.6.DerivedMoritatheory............................332
9.7.Dgcategories.................................334
9.7.1.Dgcategoriesandfunctors.......................334
9.7.2.Thederivedcategory..........................336
9.7.3.Derivedfunctors.............................337
9.7.4.Dgquotients...............................338
9.7.5.Invariants.................................340
9.8.References..................................342
JordanSuperalgebras
ConsueloM ARTINEZ 1 andEfimZ ELMANOV 2
1 DepartmentofMathematics,UniversityofOviedo,Spain
2 DepartmentofMathematics,UniversityofCaliforniaSanDiego,USA
1.1.Introduction
Superalgebrasappearedinaphysicalcontextinordertostudy,inaunifiedway, supersymmetryofelementaryparticles.Jordanalgebrasgrewoutofquantum mechanicsandgainedprominenceduetotheirconnectionstoLietheory.Inthis chapter,wesurveyJordansuperalgebrasfocusingontheirconnectionstoother subjects.Inthissectionweintroducesomebasicdefinitionsandinsection1.2we givetheTits–Kantor–KoecherconstructionthatshowsthewayinwhichLieand Jordanstructuresareconnected.Insection1.3,weshowexamplesofsomebasic superalgebras(theso-calledclassicalsuperalgebras).Section1.4isaboutthenotion ofbracketsandexplainshowtoconstructsuperalgebrasusingdifferenttypesof brackets.Section1.5explainsCheng–Kacsuperalgebras,animportantclassof superalgebrasthatappearedforthefirsttimeinthecontextofsuperconformal algebras.TheclassificationofJordansuperalgebrasisexplainedinsection1.6,andit includesthecasesofanalgebraicallyclosedfieldofzerocharacteristics,thecaseof primecharacteristic,bothforJordansuperalgebraswithsemisimpleevenpartand withnon-semisimpleevenpart,andthecaseofnon-unitalJordansuperalgebras. Finally,insection1.7,wegivesomegeneralideasaboutJordansuperconformal algebras.Throughoutthechapter,allalgebrasareconsideredoverafield F , charF =2
D EFINITION 1.1.– A(linear)Jordanalgebraisavectorspace J withalinearbinary operation (x,y ) → xy satisfyingthefollowingidentities:
(J1) xy = yx (commutativity);
(J2) (x2 y )x = x2 (yx) ∀x, y ∈ J (Jordanidentity).
Insteadof(J2)wecanconsiderthecorrespondinglinearizedidentity:
(J’2) (xy )(zu)+(xz )(yu)+(xu)(yz )=((xy )z )u +((xu)z )y +((yu)z )x ∀x, y , z , u ∈ J
R EMARK 1.1.–ALiealgebra L isavectorspacewithalinearbinaryoperation (x,y ) → [x,y ] satisfyingthefollowingidentities:
(L1) [x,y ]= [y,x] (anticommutativity);
(L2) [[x,y ],z ]+[[y,z ],x]+[[z,x],y ]=0 forarbitraryelements x, y , z ∈ J (Jacobiidentity).
E XAMPLE 1.1.–If A isanassociativealgebra,then (A(+) , ·),where a · b = ab + ba is aJordanalgebra,and (A( ) , [ , ]),where [a,b]= ab ba isaLiealgebra.Both A(+) and A( ) havethesameunderlyingvectorspaceas A.
D EFINITION 1.2.– Asuperalgebra A isanalgebrawitha Z/2Z-grading.So A = A0 + A1 isadirectsumoftwovectorspacesand
Elementsof A0 ∪ A1 arecalledhomogeneouselements.Theparityofa homogeneouselement a,denoted |a|,isdefinedby |a| =0 if a ∈ A0 and |a| =1 if a ∈ A1
Elementsin A0 arecalledevenandelementsin A1 arecalledodd.
Notethat A0 isasubalgebraof A,but A1 isnot,insteaditcanbeseenasa bimoduleover A0
E XAMPLE 1.2.–If V isavectorspaceofcountabledimension,then G = G(V ) denotestheGrassmann(orexterior)algebraover V ,thatis,thequotientofthetensor algebraovertheidealgeneratedbythesymmetrictensors v ⊗ w + w ⊗ v , v , w ∈ V . Thisalgebra G(V ) is Z/2Z-graded.Indeed, G(V )= G(V )0 + G(V )1 ,wherethe “evenpart”isthelinearspanofalltensorsofevenlengthandthe“oddpart” G(V )1 isthelinearspanofalltensorsofoddlength.
G(V ) isanexampleofasuperalgebra.
D EFINITION 1.3.– Consideravarietyofalgebras V definedbyhomogeneous identities(seeJacobson(1968)orZhevlakov etal.(1982)).Wesaythata superalgebra A = A0 + A1 isa V-superalgebraiftheevenpartof A ⊗F G(V ) lies inthevariety,thatis
D EFINITION 1.4.– Thealgebra A0 ⊗ G(V )0 + A1 ⊗ G(V )1 iscalledtheGrassmann envelopeofthesuperalgebra A andwillbedenotedas G(A)
Letusconsider V thevarietyofassociative,commutative,anticommutative,Jordan orLiealgebras,respectively.Thenweget:
E XAMPLE 1.3.–Asuperalgebra A = A0 + A1 isanassociativesuperalgebraifand onlyifitisa Z/2Z-gradedassociativealgebra.
E XAMPLE 1.4.–Asuperalgebra A = A0 + A1 isacommutativesuperalgebraifit satisfies:
xy =( 1)|x||y | yx
forany x, y homogeneouselementsof A.
E XAMPLE 1.5.–Asuperalgebra A isananticommutativesuperalgebraif
xy = ( 1)|x||y | yx
forevery x, y homogeneouselementsof A
E XAMPLE 1.6.–AJordansuperalgebraisasuperalgebrathatiscommutativeand satisfiesthegradedidentity: (xy )(zu)+( 1)|
foreveryhomogeneouselements x, y , z , u ∈ A0 ∪ A1 .
E XAMPLE 1.7.–Ananticommutativesuperalgebra A isaLiesuperalgebraifit satisfies:
forevery x, y , z ∈ A0 + A1 .
D EFINITION 1.5.– If J = J0 + J1 isaJordansuperalgebraand x, y , z ∈ J0 ∪ J1 , thentheirtripleproductisdefinedby: {x,y,z } =(xy )z + x(yz ) ( 1)|x||y | y (xz ).
Notethateveryalgebraisasuperalgebrawiththetrivialgrading,thatis, A = A0 .
1.2.Tits–Kantor–Koecherconstruction
Tits(1962,1966)madeanimportantobservationthatrelatesLieandJordan structures.Let L beaLiesuperalgebrawhoseevenpart L0 containsan sl2 -triple {e,f,h},thatis, [e,f ]= h, [h,e]=2e, [h,f ]= 2f.
D EFINITION 1.6.– An sl2 -triple e, f , h issaidtobe“good”if ad(h): L → L is diagonalizableandtheeigenvaluesareonly 2, 0, 2.
Insuchacase, L = L 2 + L0 + L2 decomposesasadirectsumofeigenspaces. Wecandefineanewproductin L2 by:
Withthisnewproduct, J =(L2 , ◦) becomesaJordansuperalgebra.
Moreover,(Tits1962,1966;Kantor1972)and(Koecher1967)showedthatevery Jordansuperalgebracanbeobtainedinthisway.ThecorrespondingLiesuperalgebra isnotunique,butanytwosuchLiesuperalgebrasarecentrallyisogenous,thatis, theyhavethesamecentralcover.Letusrecalltheconstructionof L =TKK(J ),the universalLiesuperalgebrainthisclass(seeMartinandPiard(1992)).
C ONSTRUCTION .–Consider J aunitalJordansuperalgebra,and {ei }i∈I abasisof J . Let
DefineaLiesuperalgebra K bygenerators {x+ j ,xj }j ∈I andrelations
[x+ i ,x+ j ]=0=[xi ,xj ],
[[x+ i ,xj ],x+ k ]= t γ t ijk x+ t ,
[[xi ,x+ j ],xk ]= t γ t ijk xt , forevery i,j,k ∈ I.
ThisLiesuperalgebrahasashortgrading K = K 1 + K0 + K1 where K 1 = F (xi )i∈I ,K1 = F (x+ i )i∈I ,K0 = F ([xi ,x+ j ])i,j ∈I .
K istheuniversalTits–Kantor–KoecherLiesuperalgebraoftheunitalJordan superalgebra J :
K =TKK(J ).
1.3.Basicexamples(classicalsuperalgebras)
Let A = A0 + A1 beanassociativesuperalgebra.Thenewoperationinthe underlyingvectorspace A givenby: a ◦ b = 1 2 (ab +( 1)|a||b| ba) ∀a,b
definesastructureofaJordansuperalgebraon A thatisdenoted A(+) .
D EFINITION 1.7.– ThoseJordansuperalgebrasthatcanbeobtainedassubalgebras ofasuperalgebra A(+) ,with A anassociativesuperalgebra,arecalledspecial. Superalgebrasthatarenotspecialarecalledexceptional.
R EMARK 1.2.–Ifweconsiderintheoriginalassociativesuperalgebrathenewproduct givenby:
wegetaLiesuperalgebrathatisdenotedas A( ) .
D EFINITION 1.8.– Asuperalgebra A issimpleifitdoesnothavenon-trivialgraded ideals.Agradedidealisanideal I A suchthatforevery a = a0 + a1 ∈ I ,it followsthat a0 , a1 ∈ I .Soeverygradedideal I satisfies I =(A
Wall(1963,1964)provedthatanarbitrarysimplefinitedimensionalsuperalgebra overanalgebraicallyclosedfieldisisomorphictooneofthefollowingtwotypes:
I) A = Mm+n (F )= A0 + A1 ,A0 = ∗ 0 0 ∗ m n ,A1 =
II) A = Q(n)= ab ba a,b ∈ Mn (F ) ≤ Mn+n (F ).
Consequently,wecaneasilygetthefirstexamplesofsimplefinitedimensional Jordansuperalgebrasasexplainedabove.
E XAMPLE 1.8.– J = M (+) m+n (F ), m ≥ 1, n ≥ 1
E XAMPLE 1.9.– J = Q(n)(+) , n ≥ 2.
D EFINITION 1.9.– Let A beanassociativesuperalgebra.Amap ∗ : A → A isa superinvolutionifitsatisfies:
i) (a∗ )∗ = a, ∀a ∈ A; ii) (ab)∗ =( 1)|a||b| b∗ a∗ , ∀a,b ∈ A0 ∪ A1
If ∗ : A → A isasuperinvolutionoftheassociativesuperalgebra A,thenthesetof symmetricelements H (A, ∗) isaJordansuperalgebraof A(+) .Similarly,thesubspace ofskew-symmetricelements K (A, ∗)= {a ∈ A | a∗ = a} isaLiesubsuperalgebra of A( ) .
Thefollowingtwosubsuperalgebrasof M (+) m+n areofthistype.
E XAMPLE 1.10.–Let In , Im betheidentitymatricesandlet t bethetransposition. Letusdenote U = 0 Im Im 0 .
Then U t = U 1 = U ,and ∗ : Mn+2m (F ) → Mn+2m (F ) givenby ab cd ∗ = In 0 0 U at ct
isasuperinvolution.
Thesuperalgebras
ospn,2m = K (Mn+2m (F ), ∗) and Jospn,2m (F )= H (Mn+2m (F ), ∗) aretheLieandJordanorthosymplecticsuperalgebras,respectively.
E XAMPLE 1.11.–Theassociativesuperalgebra Mn+n (F ) hasanother superinvolutiongivenby:
ab cd σ = dt bt ct at .
TheLieandJordansuperalgebras(respectively)thatconsistofskew-symmetric andsymmetricelements,respectively,aredenotedas Pn (F ) and JPn (F ) (andare alsocalled“strangeseries”).
E XAMPLE 1.12.–Thethree-dimensionalKaplanskysuperalgebra K3 = Fe +(Fx + Fy ) withmultiplicationtable:
e 2 = e,ex = 1 2 x,ey = 1 2 y,x 2 = y 2 =0,x · y = y · x = e isasimpleJordansuperalgebra.Notethat K3 isnotunital.
E XAMPLE 1.13.–Theone-parametricfamilyoffour-dimensionalsuperalgebras D (t) definedas D (t)=(Fe1 + Fe2 )+(Fx + Fy ) withtheproduct e 2 i = ei ,e1 e2 =0,ei x = 1 2 x,ei y
Thesuperalgebra D (t) issimpleif t =0.Inthecase t = 1,thesuperalgebra D ( 1) isisomorphicto M1+1 (F )(+) .
E XAMPLE 1.14.–Let V beavectorspacethatis Z/2Z-graded, V = V0 + V1 ,and hasasuperform ( | ): V × V → F ,whichissymmetricon V0 ,skew-symmetricin V1 and (V0 |V1 )=0=(V1 |V0 ).Then J = F 1+ V =(F 1+ V0 )+ V1 isaJordan superalgebra,wheretheproductoftwoarbitraryelements α1+ v and β 1+ w in J is givenby (α1+ v ) (β 1+ w )= αβ 1+(v |w )1+(αw + βv )
forarbitrary v , w ∈ V
Wewillreferto J asthesuperalgebraofasuperform.
J issimpleifandonlyiftheform ( | ) isnon-degenerate.
E XAMPLE 1.15.–Kac(1977a)introduceda 10-dimensionalJordansuperalgebra J whoseevenparthasdimension 6 andsplitsasthedirectsumofasuperalgebraofa superformandaone-dimensionalalgebra.Thus, J0 =(Fe + 1≤i≤4 Fvi ) ⊕ Ff hasthe multiplicationgivenby:
andanyotherproductoftwobasicelementsis 0
Theoddpart J1 hasabasis {x1 ,x2 ,y1 ,y2 } andthefollowingmultiplicationtable:
Finally,theactionof J0 over J1 isgivenby:
v1 xj =0,v1 y1 = x2 ,v1 y2 = x1 ,v2 yj =0,v2 x1 = y2 ,v2 x2 = y1
v3 x1 =0= v3 y2 ,v3 x2 = x1 ,v3 y1 = y2 , v4 x2 =0= v4 y1 ,v4 x1 = x2 ,v4 y2 = y1 .
Thissuperalgebraissimpleif char F =3.Incaseof char F =3,ithasanideal ofdimension 9 thatisspannedby e, vi , 1 ≤ i ≤ 4, xj , yj , 1 ≤ j ≤ 2.Itiscalled degeneratedKacsuperalgebraandisdenotedby K9
InMedvedevandZelmanov(1992),itisprovedthattheKacsuperalgebra K10 is notahomomorphicimageofaspecialJordansuperalgebra.
BenkartandElduque(2002)realized K10 asthespaceof K3 ⊗ K3 +F 1 (witha newproduct)where K3 istheKaplanskysuperalgebra.
Another(octonionic)constructionof K10 wassuggestedinRacineandZelmanov (2015).
1.4.Brackets
D EFINITION 1.10.– Let A beanassociativecommutativesuperalgebra.Abinarymap { , } : A × A → A iscalledaPoissonbracketif 1) (A, { , }) isaLiesuperalgebra;
2) {ab,c} = a{b,c} +( 1)|b||c| {a,c}b forarbitrary a,b,c ∈ A0 ∪ A1 .
E XAMPLE 1.16.–Let F [p1 ,...,pn ,q1 ,...,qn ] beapolynomialalgebrain 2n variables.TheclassicalHamiltonianbracket:
f,g } =
isaPoissonbracket.
E XAMPLE 1.17.–Let
G(n)= ξ1 ,...,ξn betheGrassmannalgebraoveran n-dimensionalvectorspace V .Thenthebracket
forarbitrary f,g ∈ G(V )0 ∪ G(V )1 isaPoissonbracket.
D EFINITION 1.11.– Given A = A0 + A1 anassociativecommutativesuperalgebra, abilinearmap { , } : A × A → A iscalledacontactbracketif:
i) (A, { , }) isaLiesuperalgebra; ii) {ab,c} = a{b,c} +( 1)|b||c| {a,c}b + abD (c) forarbitraryhomogeneous elements a, b, c in A.
NotethataPoissonbracketisacontactbracketwith D =0.
E XAMPLE 1.18.–Let F [t] bethepolynomialalgebra.Thenthebracket {f (t),g (t)} = f (t)g (t) f (t)g (t) isacontactbracket.
E XAMPLE 1.19.–Let Λ(1: n) bethepolynomialsuperalgebrainone(even)Laurent variableand n (odd)Grassmannvariables ξ1 ,...,ξn , Λ(1: n)= F [t,t 1 ,ξ1 ,...,ξn ].Consider D = ∂ ∂t or D = t ∂ ∂t aderivationof F [t,t 1 ].The bilinearmapdefinedongeneratorsof Λ(1: n) by:
[t,ξi ]= 1 2 ξi D (t), [ξi ,ξj ]= δij , 1 ≤ i,j ≤ n
canbeextendedtoacontactbracketon Λ(1: n).
D EFINITION 1.12.– [Kantordouble]Let A beanassociativecommutative superalgebrawithabilinearmap { , } : A × A → A.Assumethat {Ai ,Aj }⊆ Ai+j .Consideradirectsumofvectorspaces KJ(A, { , })= A + Av where |v | =1.Defineanewproductin J (A, { , }) thatcoincideswiththeoriginal onein A andisgivenby:
a(bv )=(ab)v, (bv )a =( 1)|a| (ba)v, (av )(bv )=( 1)|b| {a,b}
Thesuperalgebra KJ(A, { , }) iscalledtheKantordoubleof (A, { , }).
Kantor(1990)provedthatifthebracket { , } isaPoissonbracket,then KJ(A, { , }) isaJordansuperalgebra.
D EFINITION 1.13.– Thebilinearmap { , } iscalledaJordanbracketonthe superalgebra A if KJ(A, { , }) isaJordansuperalgebra(seeKingandMcCrimmon (1992)).
CantariniandKac(2007)showedthatthereisa 1-1 correspondencebetween Jordanbracketsandcontactbrackets.Indeed,if [a,b] isacontactbracketwith derivation D , D (a)=[a, 1],thenthenewbracket
{a,b} =[a,b]+ 1 2 (D (a)b aD (b))
isaJordanbracket.
Applyingthistoexample1.19,wegetthefollowing:
E XAMPLE 1.20.–Thevalues {ξi ,t} =0, {ξi ,ξj } = δij for 1 ≤ i,j ≤ n extendto aJordanbracketof Λ(1: n).ApplyingtheKantordoubleprocesstothisbracket,we getJordansuperalgebras Jn =KJ(Λ(1: n), { , }).
1.5.Cheng–Kacsuperalgebras
Givenanarbitraryunitalassociativecommutative(super)-algebrawithan(even) derivation d : Z → Z ,MartínezandZelmanovconstructed(MartínezandZelmanov 2010)aJordansuperalgebra JCK(Z,d) namedtheCheng–KacJordansuperalgebra.
Theevenpartof JCK(Z,d) isarank 4 freemoduleover Z withbasis {1,w1 ,w2 ,w3 }, J0 = Z + 3 i=1 wi Z ,andmultiplicationgivenby wi wj =0 if 1 ≤ i = j ≤ 3, w 2 1 =1= w 2 2 , w 2 3 = 1.Theoddpartofthissuperalgebraisalsoa rank 4 freemoduleover Z withbasis {x,x1 ,x2 ,x3 }, J1 = xZ + 3 i=1 xi Z .
Theactionoftheevenpartontheoddpartandtheproductsoftwoelements, respectively,aregivenbythefollowingmultiplicationtables:
Thesuperalgebra JCK(Z,d) issimpleifandonlyif Z is d-simple,thatis, Z does notcontainproper d-invariantideals(seeMartínezandZelmanov(2010)).
Letusremarkthatfor Z = C[t,t 1 ] theaboveconstructionleadstotheCheng–Kacsuperconformalalgebra,thatis, CK(6)=TKK(JCK(6)),where
JCK(6)=JCK(C[t,t 1 ],D ) with D = ∂ ∂t (seesection1.8).
1.6.FinitedimensionalsimpleJordansuperalgebras
1.6.1. Case F isalgebraicallyclosedand char F =0
Letusassumenowthat F isalgebraicallyclosedand char F =0.Kacderived theclassificationoffinitedimensionalsimpleJordan F -superalgebrasfromhis classificationofsimplefinitedimensionalLiesuperalgebrasviathe Tits–Kantor–Koecherconstruction.
T HEOREM 1.1(seeKac(1977a)andKantor(1990)).–Let J = J0 + J1 beasimple Jordansuperalgebraoveranalgebraicallyclosedfield F , char F =0.Then J is isomorphictooneofthesuperalgebrasinexamples1.8,1.9and1.10–1.15oritisthe KantordoubleofthePoissonbracketinexample1.17.
R EMARK 1.3.–Wewillassumealwaysinthissectionthat J1 =(0).
1.6.2. Case char F = p> 2,theevenpart J0 issemisimple
Letusassumenextthat char F = p> 2 andtheevenpart J0 isasemisimple Jordanalgebra.
RecallthatasemisimpleJordanalgebraisadirectsumoffinitelymanysimple ideals.
ThiscasewasaddressedinRacineandZelmanov(2003)andtheclassification essentiallycoincideswiththeoneofzerocharacteristic,expectofsomedifferencesif char F =3
E XAMPLE 1.21.–Let H3 (F ), K3 (F ) denotethesymmetricandskew-symmetric 3 × 3 matricesover F , char F =3.Consider J0 = H3 (F ) and J1 = K3 (F ) ⊕ K3 (F ) the sumoftwocopiesof K3 (F ).WehaveaJordansuperalgebrastructureon J = J0 + J1 via a · b = a b in M3 (F )+ if a, b ∈ H3 (F ),thatis,
a · b = ab + ba,
c · d = cd + dc ∈ H3 (F ) for c,d ∈ K3 (F ),
K3 (F ) · K3 (F )=(0)= K3 (F ) · K3 (F ), and ac = ac + ca,ac = ac + ca if a ∈ H3 (F ),c ∈ K3 (F ).
Thissuperalgebraissimple.
E XAMPLE 1.22.–Let B = B0 + B1 with B0 = M2 (F ), B1 = Fm1 + Fm2 ,where F isafield, char F =3.Theproductin B1 isgivenby: m 2 1 = e21 ,m 2 2 = e12 ,m1 m2 = e11 ,m2 m1 = e22 .
Theactionof B0 over B1 isdefinedasfollows: e11 m1 = m1 ,e11 m2 =0= m1 e11 ,m2 e11 = m2 , m1 e22 = m1 ,e22 m1 =0= m2 e22 ,e
,
.
Shestakov(1997)provedthat B isanalternativesuperalgebraandhasanatural involution ∗ givenby (a + m)∗ =¯ a m, a ∈ B0 ,where a → a isthesymplectic involution,and m ∈ B1
If H3 (B, ∗) denotesthesymmetricmatriceswithrespecttotheinvolution ∗,then H3 (B, ∗) isasimpleJordansuperalgebra.Itis i-exceptional,thatis,itisnota homomorphicimageofaspecialJordansuperalgebra.
T HEOREM 1.2(RacineandZelmanov(2003)).–Let J = J0 + J1 beafinite dimensionalcentralsimpleJordansuperalgebraoveranalgebraicallyclosedfield F
of char F = p> 2.If J1 =(0) and J0 issemisimple,then J isisomorphictooneof thesuperalgebrasinexamples1.8,1.9,1.10–1.14or char F =3 and J isthe nine-dimensionaldegenerateKacsuperalgebra(seeexample1.15)or J isisomorphic tooneofthesuperalgebrasinexamples1.21and1.22.
1.6.3. Case char F = p> 2,theevenpart J0 isnotsemisimple
ThiscaseshowssimilaritieswithinfinitedimensionalsuperconformalJordan algebras(seesection1.8)incharacteristic 0
Letusdenote B (m)= F [a1 ,...,am | ap i =0] thealgebraoftruncated polynomialsin m variables.Let B (m,n)= B (m) ⊗ G(n) bethetensorproductof B (m) withtheGrassmannalgebra G(n)= 1,ξ1 ,...,ξn .Then B (m,n) isan associativecommutativesuperalgebra.
T HEOREM 1.3(MartínezandZelmanov(2010)).–Let J = J0 + J1 beafinite dimensionalsimpleunitalJordansuperalgebraoveranalgebraicallyclosedfield F of characteristic p> 2.Iftheevenpart J0 isnotsemisimple,thenthereexistintegers m, n andaJordanbracket { , } on B (m,n) suchthat J = B (m,n)+ B (m,n)v =KJ(B (m,n), { , }) isaKantordoubleof B (m,n) or J isisomorphictoaCheng–KacJordansuperalgebra JCK(B (m),d) forsome derivation d : B (m) → B (m).
1.6.4. Non-unitalsimpleJordansuperalgebras
Finally,letusconsidernon-unitalsimpleJordansuperalgebras.Aswehaveseen, K3 thethree-dimensionalKaplanskysuperalgebraand K9 thenine-dimensional degenerateKacsuperalgebraareexamplesofsuchsuperalgebras.
E XAMPLE 1.23.–Let Z beaunitalassociativecommutativealgebra, D : Z → Z a derivation.Assumethat Z is D -simpleandthattheonlyconstantsareelements α1, α ∈ F .
Letusconsiderin Z thebracket { , } givenby: {a,b} =(aD )b a(bD ) ∀a,b ∈ Z.
TheabovebracketisaJordanbracket,sotheKantordouble V (Z,D )= Z + Zv = KJ(Z, { , }) isasimpleunitalJordansuperalgebra.
Nowwewillchangetheproductin V (Z,D ),modifyingonlytheactionofthe evenpartontheoddpartandpreservingtheproductoftwoeven(respectively,two
odd)elements.Denotewithjuxtapositiontheproducton V (Z,D ).Let a, b ∈ Z .We definethenewproduct · by:
a · b = ab,a · bv = 1 2 (ab)v,av · bv = {a,b} =(av )(bv )
Inthisway,wegetanotherJordansuperalgebra V1/2 (Z,D ) thatissimplebutnot unital.
ItwasprovedinZelmanov(2000)that:
T HEOREM 1.4.–Let J beafinitedimensionalsimplecentralnon-unitalJordan superalgebraoverafield F .Then J isisomorphictooneofthesuperalgebrason thelist:
i)theKaplanskysuperalgebra K3 (example1.12);
ii)thefield F hascharacteristic 3 and J isthedegenerateKacsuperalgebra (example1.15);
iii)asuperalgebra V1/2 (Z,D ) (example1.23).
D EFINITION 1.14.– Let A beaJordansuperalgebraandlet N beitsradical,that is,thelargestsolvableidealof A.Thesuperalgebra A issaidtobesemisimpleif N =(0)
E XAMPLE 1.24.–Let B beasimplenon-unitalJordansuperalgebraandlet H (B )= B + F 1 beitsunitalhull.Then H (B ) isasemisimpleJordansuperalgebrathatisnot simple.
T HEOREM 1.5(Zelmanov(2000)).–Let J beafinitedimensionalJordan superalgebra.Then J issemisimpleifandonlyif
J ∼ = s i=1 (Ji1 ⊕···⊕ Jiri + Ki 1)+ J(1) ⊕···⊕ J(t) where J(1) ,...,J(t) aresimpleJordansuperalgebrasandforevery i =1,...,s,the superalgebras Jij aresimplenon-unitalJordansuperalgebrasoverthefieldextension Ki of F .
1.7.Finitedimensionalrepresentations
Jacobson(1968)developedthetheoryofbimodulesoversemisimplefinite dimensionalJordanalgebras.
Inthissection,wediscussrepresentations(bimodules)offinitedimensionalJordan superalgebras.
D EFINITION 1.15.– TherankofaJordansuperalgebra J isthemaximalnumberof pairwiseorthogonalidempotentsintheevenpart.
Unlessotherwisestatedwewillassume char F =0
D EFINITION 1.16.– Let V bea Z/2Z-gradedvectorspacewithbilinearmappings V × J → V , J × V → V .Wecall V aJordanbimoduleifthesplitnullextension V + J isaJordansuperalgebra.
Recallthatinthesplitnullextensionthemultiplicationextendsthemultiplication on J ,products V · J and J · V aredefinedviathebilinearmappingsaboveand V · V =(0).
Let V = V0 + V1 beaJordanbimoduleover J .Considerthevectorspace V op = V op 1 + V op 0 ,where V op i isacopyof Vi withdifferentparity.Definetheactionof J on V op via av op =( 1)|a| (av )op ,v op a =(va)op
Then V op isalsoaJordanbimoduleover J .Wecallittheoppositemoduleof V .
Let V bethefreeJordan J -bimoduleononefreegenerator.
D EFINITION 1.17.– Theassociativesubsuperalgebra U (J ) of EndF V generatedby alllineartransformations RV (a): V → V , v → va, a ∈ J ,iscalledtheuniversal multiplicativeenvelopingsuperalgebraof J .
EveryJordanbimoduleover J isarightmoduleover U (J )
D EFINITION 1.18.– Abimodule V over J iscalledaone-sidedbimoduleif {J,V,J } =(0).
Let V (1/2) bethefreeone-sidedJordan J -bimoduleononefreegenerator.
D EFINITION 1.19.– Theassociativesubsuperalgebra S (J ) of EndF V (1/2) generatedbyalllineartransformations RV (1/2) (a): V (1/2) → V (1/2), v → va, a ∈ J ,iscalledtheuniversalassociativeenvelopingalgebraof J
Everyone-sidedJordan J -bimoduleisarightmoduleover S (J )
Finally,let J beaunitalJordansuperalgebrawiththeidentityelement e.Let V (1) denotethefreeunital J -bimoduleononefreegenerator.Theassociative subsuperalgebra U1 (J ) of EndF V (1) generatedby {RV (1/2) (a)}a∈J iscalledthe universalunitalenvelopingalgebraof J .
ForanarbitraryJordanbimodule V ,thePeircedecomposition
V = {e,V,e}⊕{e,V, 1 e}⊕{1 e,V, 1 e}
isadecompositionof V intoadirectsumofunitalandone-sidedbimodules.Hence U (J ) ∼ = U1 (J ) ⊕ S (J )
1.7.1. Superalgebrasofrank ≥ 3
Inthissection,weconsiderJordanbimodulesoverfinitedimensionalsimple Jordansuperalgebrasofrank ≥ 3,thatis,superalgebras M (+) m+n , m + n ≥ 3; Josp(n, 2m), n + m ≥ 3; Q(n)(+) , n ≥ 3; JP(n), n ≥ 3.
Inthiscase,theuniversalmultiplicativeenvelopingsuperalgebra U (J ) isfinite dimensionalandsemisimple(MartinandPiard1992).HenceeveryJordanbimodule iscompletelyreducible,asinthecaseofJordanalgebras.
Thesuperalgebras Josp(n, 2m) and JP(n) areofthetype
H (A, ∗)= {a ∈ A | a ∗ = a} where A isasimplefinitedimensionalassociativesuperalgebraand ∗ isaninvolution.
E XAMPLE 1.25.–Anarbitraryrightmoduleover A isaone-sidedmoduleover H (A, ∗).
E XAMPLE 1.26.–Thesubspace K (A, ∗)= {k ∈ A | k ∗ = k } withtheaction k · a = ka + ak ; k ∈ K (A, ∗), a ∈ H (A, ∗) isaunital H (A, ∗)-bimodule.
T HEOREM 1.6(MartinandPiard(1992)).–AnarbitraryirreducibleJordanbimodule over Josp(n, 2m), n + m ≥ 3,or JP(n), n ≥ 3,isabimoduleofexamples1.25and 1.26ortheregularbimodule.
Thesuperalgebras M (+) m+n , Q(n)(+) areofthetype A(+) ,where A isasimple finitedimensionalassociativesuperalgebra.
E XAMPLE 1.27.–Everyrightmoduleover A givesrisetoaone-sidedJordan bimoduleover A(+) .
Supposenowthatthesuperalgebra A isequippedwithaninvolution ∗.
E XAMPLE 1.28.–Thesubspaces H (A, ∗) and K (A, ∗) becomeJordan A(+)bimoduleswiththeactions:
