On Kac's Jordan superalgebra

0.14 MB·

by  Alejandra S. Cordova-Martinez

ON KAC’S JORDAN SUPERALGEBRA
ALEJANDRAS.CO´RDOVA-MART´INEZ⋆,ABBASDAREHGAZANI⋄,
ANDALBERTOELDUQUE⋆
Abstract. The group-scheme of automorphisms of the ten-dimensional ex-
7 ceptionalKac’sJordansuperalgebraisshowntobeisomorphictothesemidi-
1 rectproductofthedirectproductoftwocopiesofSL2 bytheconstantgroup
0 schemeC2.
2 This is usedto revisit, extend, andsimplify,known results onthe classifi-
cationofthetwistedformsofthissuperalgebraandofitsgradings.
n
a
J
4 Kac’s ten-dimensional superalgebra K10 is an exceptional Jordan superalgebra
2 whichappearedforthe firsttimeinKac’sclassification[Kac77]ofthe finite dimen-
sionalsimpleJordansuperalgebrasoveranalgebraicallyclosedfieldofcharacteristic
]
A 0. It was constructed by Lie theoretical terms from a 3-grading of the exceptional
simple Lie superalgebra F(4).
R
A more conceptual definition was given in [BE02] over an arbitrary field F of
.
h characteristicnottwo(theseassumptionson thegroundfieldwill bekept throughout
at the paper), using thethree-dimensionalKaplanskysuperalgebraK3. Theevenpart
m (K3)¯0 is a copy of the ground field F: (K3)¯0 = Fa, with a2 = a; while the odd
partis a two-dimensionalvectorspaceW endowedwith anonzeroskew-symmetric
[
bilinear form (|). The multiplication is determined as follows:
1
v 1
a2 =a, av = v =va, vw =(v|w)a,
8
2
9
7 for any v,w ∈W. We extend (|) to a supersymmetric bilinear form on K3: (K3)¯0
.06 andTh(Ken3)K¯11a0re=oFrt1h⊕ogoKn3a⊗l,FwKit3h,(wa|iath)=(K2110.)¯0 =F1⊕ K3⊗FK3 ¯0 =F1⊕F(a⊗a)⊕
01 (gWive⊗nbWy)i,mapnodsi(nKg1t0h)(cid:0)¯1at=1iKst3h⊗e(cid:1)FuKni3ty¯1, a=nd(Wfor⊗hao)m⊕og((cid:0)ean⊗eoWus)e.leT(cid:1)mheenmtsuxlt,iyp,lizc,att∈ionKis,
7 (cid:0) (cid:1) 3
1 3
: (x⊗y)(z⊗t)=(−1)yz xz⊗yt− (x|z)(y |t)1 . (0.1)
v 4
i (cid:16) (cid:17)
X Ifthecharacteristicis3,thenK9 :=K3⊗FK3 isasimpleidealinK10. Otherwise
r K10 is simple.
a
It must be remarked that an ‘octonionic’ construction of K has been given in
10
[RZ15].
Date:January23,2017.
2010 Mathematics Subject Classification. Primary17C40; Secondary17C70,17C30.
Keywords and phrases. Kac’sJordansuperalgebra;group-schemeofautomorphisms,twisted
forms,gradings.
⋆SupportedbytheSpanishMinisteriodeEconom´ıayCompetitividad—FondoEuropeodeDe-
sarrolloRegional(FEDER)MTM2013-45588-C3-2-P,andbytheDiputaci´onGeneraldeArago´n—
FondoSocialEuropeo(GrupodeInvestigacio´ndeA´lgebra). A.S.Co´rdova-Mart´ınezalsoacknowl-
edgessupportfromConsejoNacionaldeCienciayTecnolog´ıa(CONACyT,M´exico)throughgrant
420842/262964.
⋄ This research was in part supported by a grant from IPM (No. 94160221) and partially
carriedoutinIPM-IsfahanBranch.
1
2 A.S.CO´RDOVA-MART´INEZ,A.DAREHGAZANI,ANDA.ELDUQUE
Usingthe constructionaboveofK intermsofK ,the groupofautomorphisms
10 3
Aut K wascomputedin[ELS07]. Thiswasusedin[CDM10]toclassifygradings
10
on K over algebraically closed fields of characteristic zero.
10
(cid:0) (cid:1)
Our goal here is to compute the group-scheme of automorphisms of K , and
10
to use it to revisit, extend, and simplify drastically, the known results on twisted
forms and on gradings on K .
10
Here we follow the functorial approach (see, for instance, [Wat79]). An affine
group-schemeoverFisarepresentablegroup-valuedfunctordefinedonthecategory
of unital, commutative, associative algebras over F: AlgF. Thus the group-scheme
Aut K10 is the functor AlgF →Grp that takes any object R in AlgF to the group
AutR(cid:0)K10(cid:1)⊗FR)(the groupofautomorphismsofthe R-superalgebraK10⊗FR, i.e.,
the groupofR-linearisomorphismspreservingthe multiplicationandthe grading),
(cid:0)
with the natural definition on morphisms.
ItturnsoutthatAut K isisomorphictoasemidirectproduct SL ×SL ⋊
10 2 2
C (Theorem 1.1), where C is the constant group scheme attached to the cyclic
2 2
group of two elements (a(cid:0)lso d(cid:1)enoted by C ). This extends the result(cid:0)in [ELS07](cid:1).
2
Moreover,the canonical projection π :Aut K →C induces an isomorphism
10 2
of pointed sets H1 F,Aut K → H1(F,C ) (Theorem 2.1). This associates
10 (cid:0)2 (cid:1)
to the isomorphism class of any twisted form of K , the isomorphism class of a
10
quadratic´etalealgeb(cid:0)raover(cid:0)F,w(cid:1)h(cid:1)ichallowstogeteasilytheclassificationoftwisted
forms of K (Corollary 2.2). Twisted forms of K were classified, in a completely
10 10
differentway,in[RZ03]and[EO00]. We believethatourapproachismorenatural.
A simple observation shows that SL ×SL ⋊C is also the automorphism
2 2 2
group-schemeofK ×K ,orofJ(W)×J(W),whereJ(W)istheJordansuperalgebra
3 3
of the ‘superform’ (|) on W = W¯1 a(cid:0)bove. That i(cid:1)s, J(W)¯0 = F1, J(W)¯1 = W and
the multiplication is given by
1x=x1=x, vw =(v |w)1,
for any x ∈ J(W), v,w ∈ W. This observation will be used in the proof of the
bijection H1 F,Aut K ≃H1(F,C ).
10 2
Finally,givenanabeliangroupG, aG-gradingona superalgebraA corresponds
(cid:0) (cid:0) (cid:1)(cid:1)
to a homomorphism of group schemes
GD −→Aut(A),
whereGD istheCartierdualtotheconstantgroupschemegivenbyG. (See[EK13]
for the basic definitions and facts on gradings.)
TheclassificationofG-gradingsuptoisomorphismonK ×K (orJ(W)×J(W))
3 3
is an easy exercise (Proposition3.2), and it follows at once fromthis the classifica-
tion of G-gradings, up to isomorphism, on K (Theorem 3.5), thus extending and
10
simplifying widely the results in [CDM10].
1. The group-scheme of automorphisms
Consider the two-dimensional vector space W = (K3)¯1, which is endowed with
the nonzero skew-symmetric bilinear form (|) : W ×W → F. The special linear
group-scheme SL(W) coincides with the symplectic group scheme Sp(W), which
is, up to isomorphism, the group-scheme of automorphisms of K (or of J(W)).
3
The constant group scheme C acts on SL(W)×SL(W) by swapping the argu-
2
ments,andhencewegetanaturalsemidirectproduct SL(W)×SL(W) ⋊C . The
2
groupisomorphismΦ in [ELS07, p. 3809]extends naturallyto a homomorphismof
(cid:0) (cid:1)
ON KAC’S JORDAN SUPERALGEBRA 3
affine group-schemes:
Φ: SL(W)×SL(W) ⋊C −→Aut K
2 10
(cid:0) (cid:1) (cid:0) (cid:1) 1 7→ 1,
a⊗a7→a⊗a,
(f,g) 7→ Φ(f,g) :va⊗⊗av 7→7→fa(⊗v)g⊗(va),, (1.1)
v⊗w7→f(v)⊗g(w),
generator of C 7→ τ : 17→ 1,
2 (x⊗y 7→(−1)xyy⊗x,
for any R in AlgF, f,g ∈ SL(W)(R) (i.e., f,g ∈ EndR(W ⊗F R) ≃ M2(R) of
determinant 1), v,w ∈ WR := W ⊗FR, x,y ∈ (K3)R. (Note that a representation
of a constant group scheme ρ : G → GL(V) is determined by its behavior over F:
ρF :G→GL(V).)
Theorem 1.1. Φ is an isomorphism of affine group-schemes.
Proof. If F denotes an algebraic closure of F, then Φ is a group isomorphism
F
[ELS07, Theorem 3.3]. Since SL(W) × SL(W) ⋊ C ≃ SL × SL ⋊ C is
2 2 2 2
smooth, it is enough to prove that the differential dΦ is bijective (see, for instance
[EK13, Theorem A.50]). The L(cid:0)ie algebra of SL((cid:1)W)×SL((cid:0)W) ⋊C is(cid:1)sl(W)×
2
sl(W),whiletheLiealgebraofAut K isthe evenpartofitsLiesuperalgebraof
10 (cid:0) (cid:1)
derivations, which is again, up to isomorphism, sl(W)×sl(W) identified naturally
with a subalgebra of EndF K10 (se(cid:0)e [B(cid:1)E02, Theorem 2.8]). Moreover, with the
natural identifications, dΦ is the identity map. (cid:3)
(cid:0) (cid:1)
Thelastresultinthissectionisthesimpleobservationthat SL(W)×SL(W) ⋊
C is also the group-scheme of automorphisms of the Jordansuperalgebra K ×K
2 3 3
and J(W) × J(W). This will be instrumental in the next s(cid:0)ection. Its proof(cid:1)is
straightforward.
Proposition 1.2. The natural transformations defined by:
Ψ1 : SL(W)×SL(W) ⋊C −→Aut K ×K
2 3 3
(cid:0) (cid:1) (cid:0) (a,(cid:1)0)7→(a,0),
(f,g) 7→ Ψ1 : (0,a)7→(0,a),
(f,g) 
(v,w)7→ f(v),g(w) ,
generator of C2 7→ τ :(x,y)7→(y,x), (cid:0) (cid:1)
for v,w∈W and x,y ∈K , and
3
Ψ2 : SL(W)×SL(W) ⋊C −→Aut J(W)×J(W)
2
(cid:0) (cid:1) (cid:0) (1,0)7→(cid:1)(1,0),
(f,g) 7→ Ψ2 : (0,1)7→(0,1),
(f,g) 
(v,w)7→ f(v),g(w) ,
generator of C2 7→ τ :(x,y)7→(y,x), (cid:0) (cid:1)
for v,w∈W and x,y ∈J(W), are isomorphisms of group-schemes.
4 A.S.CO´RDOVA-MART´INEZ,A.DAREHGAZANI,ANDA.ELDUQUE
2. Twisted forms
The set of the isomorphism classes of twisted forms of K is identified with
10
the pointed set H1 F,Aut K . (This is H1 F/F,Aut(K in the notation of
10 10
[Wat79].)
The key to the c(cid:0)lassifica(cid:0)tion(cid:1)o(cid:1)f twisted forms(cid:0)of K is the f(cid:1)o(cid:1)llowing result:
10
Theorem 2.1. The canonical projection π : SL(W)×SL(W) ⋊C →C induces
2 2
a bijection of pointed spaces π∗ :H1 F,Aut((cid:0)K10 →H1(F,C2(cid:1)).
Proof. First,thenaturalsectionι:C(cid:0) → SL(W(cid:1))(cid:1)×SL(W) ⋊C satisfiesπ◦ι=id,
2 2
so π∗◦ι∗ =id, and π∗ is surjective.
(cid:0) (cid:1)
Second, the short exact sequence
1−→SL(W)×SL(W)−→ SL(W)×SL(W) ⋊C −→C −→1
2 2
induces an exact sequence (see [Wat(cid:0)79, §18.1]) in cohom(cid:1) ology:
1→SL(W)×SL(W)→ SL(W)×SL(W) ⋊C →C
2 2
→H1(F,SL(W)×SL(W(cid:0)))→H1 F, SL((cid:1)W)×SL(W) ⋊C2 →H1(F,C2).
Since H1(F,SL ) is trivial, this gives(cid:0)an(cid:0)exact sequence (cid:1) (cid:1)
2
1−→H1 F, SL(W)×SL(W) ⋊C →H1(F,C ),
2 2
but, in principle, this do(cid:0)es(cid:0)not prove that π∗(cid:1)is a b(cid:1)ijection. It just says that the
only element in H1 F, SL(W)×SL(W) ⋊C2 that is sent by π∗ to the trivial
element in H1(F,C ) is the trivial element.
2(cid:0) (cid:0) (cid:1) (cid:1)
Through the isomorphisms in the previous section, it is enough to prove that π
induces a bijection of pointed sets π∗ :H1 F,Aut J(W)×J(W) →H1(F,C2).
Write J = J(W) × J(W) and identify SL(W) × SL(W) ⋊ C with Aut(J)
(cid:0) (cid:0) (cid:1)(cid:1) 2
by means of Ψ2 in Proposition 1.2. The(cid:0)n J¯0 = F × F, an(cid:1)d π corresponds to
the restriction map Aut(J) → Aut(J¯0), and the induced map in cohomology
π∗ : H1(F,Aut(J)) → H1(F,Aut(J¯0)) corresponds to the map that takes the
isomorphism class of a twisted form H of J to the isomorphism class of H¯0. If H¯0
is trivial (H¯0 ≃F×F) then, by the above, H is isomorphic to J. Otherwise, H¯0 is
a quadratic separable field extension of F, so that H¯1 is a two-dimensional vector
space over H¯0, and the multiplication H¯1×H¯1 →H¯0 is H¯0-bilinear and anticom-
mutative, and hence uniquely determined. It turns out that H is isomorphic to
J(W)⊗FH¯0. This proves the injectivity of π∗. (cid:3)
Corollary 2.2. Any twisted form of K splits after a quadratic separable field
10
extension, and for any such field extension K of F, there is a unique (up to isomor-
phism) nontrivial twisted form of K which splits over K.
10
If K is a quadratic separable field extension of F, the unique nontrivial twisted
formofK thatsplitsoverKcorrespondstothenontrivialelementinH1(K/F,C ),
10 2
and hence if the Galois group Gal(K/F) is generated by σ (σ2 = id), this twisted
form is given by
K10,K ={X ∈K10⊗FK:(τ ⊗id)(X)=(id⊗σ)(X)}
(2.1)
={X ∈K10⊗FK:(τ ⊗σ)(X)=X},
where τ is given in (1.1).
ON KAC’S JORDAN SUPERALGEBRA 5
Hence, with K = F(α), α2 ∈ F, σ(α) = −α, so if {u,v} is a symplectic basis of
W =(K3)¯1,
K10,K ¯0 =F-span 1⊗1,(a⊗a)⊗1,(u⊗u)⊗α,(v⊗v)⊗α,
(cid:0) (cid:1) D (u⊗v−v⊗u)⊗1,(u⊗v+v⊗u)⊗α , (2.2)
E
K10,K ¯1 =F-span (a⊗x+x⊗a)⊗1,(a⊗x+x⊗a)⊗α:x∈{u,v} .
(cid:0) (cid:1) D E
Remark 2.3.
• BothK andJ(W)arerigid: anytwistedformofanyofthesesuperalgebras
3
is isomorphic to it. This is because Aut(K ) ≃ Aut(J(W)) ≃ SL(W) ≃
3
SL .
2
• The proofofTheorem2.1showsthatthe twistedformsofK ×K (respec-
3 3
tively J(W)×J(W)) are, up to isomorphism, the superalgebras K3 ⊗F K
(resp. J(W)⊗F K), where K is quadratic ´etale algebra over F, and any
two such forms are isomorphic if and only if so are the corresponding´etale
algebras.
3. Gradings
Given an abelian group G, a G-grading on a superalgebra A = A¯0 ⊕A¯1 is a
decomposition into a direct sum of subspaces: A = A , such that A A ⊆
g∈G g g h
A foranyg,h∈G,andeachhomogeneouscomponentisasubspaceinthe‘super’
gh
L
senseofA: Ag =(Ag∩A¯0)⊕(Ag∩A¯1). Wewillwritedeg(x)=gincase06=x∈Ag.
Two G-gradings Γ : A = A and Γ′ = A′, are isomorphic if there is
g∈G g g∈G g
an automorphism ϕ∈Aut(A) such that ϕ(A )=A′ for any g ∈G.
L gL g
AgradingbyGisequivalenttoahomomorphismofaffinegroupschemesGD →
Aut(A)(see[EK13]),andthisshowsthattwosuperalgebraswithisomorphicgroup-
schemes of automorphisms have equivalent classifications of G-gradings up to iso-
morphism. Therefore, in order to classify gradings on K it is enough to classify
10
gradings on K ×K (or J(W)×J(W)), and this is straightforward.
3 3
Actually, fix a symplectic basis {u,v} of W =(K3)¯1.
Definition 3.1. Given an abelian group G, consider the following gradings (e
denotes the neutral element of G):
• For g ,g ∈G, denote by Γ1(G;g ,g ) the G-grading given by:
1 2 1 2
deg(x)=e for any x∈ K3×K3)¯0,
deg(u,0)=g1 =deg(v,(cid:0)0)−1, deg(0,u)=g2 =deg(0,v)−1.
• For g,h ∈ G with h2 = e 6= h, denote by Γ2(G;g,h) the G-grading given
by:
deg(a,a)=e, deg(a,−a)=h,
deg(u,u)=g =deg(v,v)−1, deg(u,−u)=gh=deg(v,−v)−1.
Proposition 3.2. Any grading on K ×K by the abelian group G is isomorphic
3 3
to either Γ1(G;g ,g ) or to Γ2(G;g,h) (for some g ,g or g,h in G).
1 2 1 2
Moreover, no grading of the first type (Γ1(G;g ,g )) is isomorphic to a grading
1 2
of the second type (Γ2(G;g,h)), and
• Γ1(G,g ,g ) is isomorphic to Γ1(G;g′,g′) if and only if the sets
1 2 1 2
{g ,g−1,g ,g−1} and {g′,(g′)−1,g′,(g′)−1} coincide.
1 1 2 2 1 1 2 2
6 A.S.CO´RDOVA-MART´INEZ,A.DAREHGAZANI,ANDA.ELDUQUE
• Γ2(G;g,h) is isomorphic to Γ2(G;g′,h′) if and only if h′ = h and g′ ∈
{g,gh,g−1,g−1h}.
Proof. Any G-gradingonJ:=K3×K3 inducesa G-gradingonJ¯0,whichis isomor-
phic to F×F, and hence we are left with two cases:
(1) ThegradingonJ¯0istrivial,i.e.,J¯0iscontainedinthehomogeneouscompo-
nentJe. Then,withW =(K3)¯1,bothW×0=(a,0)J¯1and0×W =(0,a)J¯1
aregradedsubspacesofJ¯1. Hencewecantakebases{ui,vi}ofW,i=1,2,
such that {(u1,0),(v1,0),(0,u2),(0,v2)} is a basis of J¯1 consisting of ho-
mogeneous elements. We can adjust v , i = 1,2, so that (u | v ) = 1. If
i i i
deg(u ,0)=g , i=1,2, the grading is isomorphic to Γ1(G;g ,g ).
i i 1 2
(2) The grading on J¯0 is not trivial. Then there is an element h ∈ G of order
2 such that deg(a,a) = e and deg(a,−a) = h. (Note that (a,a) is the
unity element of J¯0(≃ F×F), so it is always homogeneous of degree e.)
As J¯0 = (J¯1)2, there are homogeneous elements (u1,u2),(v1,v2) ∈ J¯1 such
that (u ,u )(v ,v )=(a,a). If g =deg(u ,u ), this grading is isomorphic
1 2 1 2 1 2
to Γ2(G;g,h).
The conditions for isomorphism are clear. (cid:3)
Any grading Γ1(G;g ,g ) is a coarsening of the grading Γ1 Z2;(1,0),(0,1) ,
1 2
whileanygradingΓ2(G;g,h)isacoarseningofthegradingΓ2 Z×Z ;(1,¯0),(0,¯1) ,
(cid:0) 2 (cid:1)
where Z =Z/2Z. As an immediate consequence, we obtain the next result.
2
(cid:0) (cid:1)
Corollary 3.3. Up to equivalence, there are exactly two fine gradings on K ×K :
3 3
Γ1 Z2;(1,0),(0,1) and Γ2 Z×Z ;(1,¯0),(0,¯1) .
2
(cid:0)Proposition3.2a(cid:1)ndCoro(cid:0)llary3.3havecomple(cid:1)telysimilarcounterpartsforJ(W)×
J(W).
In order to transfer these results to Kac’s superalgebra K , take into account
10
that K10 is generated by its odd part, as (K10)¯1 2 = (K10)¯0, so any grading is
determined by its restriction to the odd part, and use the commutativity of the
(cid:0) (cid:1)
diagram
Aut K ×K oo Ψ1 SL(W)×SL(W) ⋊C Φ // Aut K
3(cid:127)_ 3 2 (cid:127)_ 10
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
(cid:15)(cid:15) (cid:15)(cid:15)
GL(W ×W) ≃ //GL (W ⊗a)⊕(a⊗W)
where the vertical arrows are given by the restrictions to(cid:0)the odd parts, and t(cid:1)he
bottom isomorphism is given by the natural identification W ×W → (W ⊗a)⊕
(a⊗W), (v,w)7→v⊗a+a⊗w.
Thus Definition 3.1 transfers to K as follows:
10
Definition 3.4. Given an abelian group G, consider the following G-gradings on
K :
10
• For g ,g ∈G, denote by Γ1 (G;g ,g ) the G-grading determined by:
1 2 K10 1 2
deg(u⊗a)=g =deg(v⊗a)−1, deg(a⊗u)=g =deg(a⊗v)−1.
1 2
• For g,h ∈ G with h2 = e 6= h, denote by Γ2 (G;g,h) the G-grading
K10
determined by:
deg(u⊗a+a⊗u)=g =deg(v⊗a+a⊗v)−1,
deg(u⊗a−a⊗u)=gh=deg(v⊗a−a⊗v)−1.
And Proposition 3.2 and Corollary 3.3 are now transferred easily to K :
10
ON KAC’S JORDAN SUPERALGEBRA 7
Theorem 3.5. Any grading on K by the abelian group G is isomorphic to either
10
Γ1 (G;g ,g ) or to Γ2 (G;g,h) (for some g ,g or g,h in G).
K10 1 2 K10 1 2
No grading of the first type is isomorphic to a grading of the second type, and
• Γ1 (G;g ,g ) is isomorphic to Γ1 (G;g′,g′) if and only if the sets
K10 1 2 K10 1 2
{g ,g−1,g ,g−1} and {g′,(g′)−1,g′,(g′)−1} coincide.
1 1 2 2 1 1 2 2
• Γ2 (G;g,h) is isomorphic to Γ2 (G;g′,h′) if and only if h′ =h and g′ ∈
K10 K10
{g,gh,g−1,g−1h}.
Moreover, there are exactly two fine gradings on K up to equivalence, namely
10
Γ1 Z2;(1,0),(0,1) and Γ2 Z×Z ;(1,¯0),(0,¯1) .
K10 K10 2
(cid:0) (cid:1) (cid:0) (cid:1)
Remark 3.6. Any grading on K by an abelian group G extends naturally to a
10
gradingbyeitherZ×GorZ2×GontheexceptionalsimpleLiesuperalgebraF(4),
2
whichisobtainedfromK usingthewell-knownTits-Kantor-Koecherconstruction.
10
However, not all gradings on F(4) are obtained in this way.
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Departamento de Matema´ticas e Instituto Universitario de Matema´ticas y Aplica-
ciones, Universidadde Zaragoza,50009Zaragoza,Spain
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Departamento de Matema´ticas e Instituto Universitario de Matema´ticas y Aplica-
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E-mail address: [email protected]