“garcia de galdeano”
garcía de galdeano
seminario matemático
n. 7
PRE-PUBLICACIONES del
seminario matematico 2007
Isabel Cunha Alberto Elduque
Freudenthal’s magic supersquare in
characteristic 3
Universidad de Zaragoza
CHARACTERISTIC 3
ISABEL CUNHAAND ALBERTO ELDUQUE?
Abstract. A family of simple Lie superalgebras over fields of charac- teristic 3, with no counterpart in Kac’s classification in characteristic 0 have been recently obtained related to an extension of the classical Freudenthal’s Magic Square. This article gives a survey of these new simple Lie superalgebras and the way they are obtained.
1. Introduction
Over the years, many different constructions have been given of the excep- cional simple Lie algebras in Killing-Cartan’s classification, involving some nonassociative algebras or triple systems. Thus the Lie algebra G2 appears as the derivation algebra of the octonions (Cartan 1914), while F4 appears as the derivation algebra of the Jordan algebra of 3 × 3 hermitian matrices over the octonions and E6 as an ideal of the Lie multiplication algebra of this Jordan algebra (Chevalley-Schafer 1950).
In 1966 Tits [Tit66] gave a unified construction of the exceptional simple Lie algebras, valid over arbitrary fields of characteristic 6= 2, 3 which uses a couple of ingredients: a unital composition algebra (or Hurwitz algebra) C, and a central simple Jordan algebra J of degree 3:
T (C, J ) = der C ⊕ (C0⊗ J0) ⊕ der J,
where C0 and J0 denote the sets of trace zero elements in C and J . By defin- ing a suitable Lie bracket on T (C, J ), Tits obtained Freudenthal’s Magic Square ([Sch95, Fre64]):
T (C, J ) H3(k) H3(k × k) H3(Mat2(k)) H3(C(k))
k A1 A2 C3 F4
k × k A2 A2⊕ A2 A5 E6
Mat2(k) C3 A5 D6 E7
C(k) F4 E6 E7 E8
At least in the split cases, this is a construction which depends on two unital composition algebras, since the Jordan algebra involved consists of the 3 × 3 hermitian matrices over a unital composition algebra. Even though
Supported by Centro de Matem´atica da Universidade de Coimbra – FCT.
? Supported by the Spanish Ministerio de Educaci´on y Ciencia and FEDER (MTM 2004-081159-C04-02) and by the Diputaci´on General de Arag´on (Grupo de Investigaci´on de ´Algebra).
1
the construction is not symmetric on the two Hurwitz algebras involved, the result (the Magic Square) is symmetric.
Over the years, several symmetric constructions of Freudenthal’s Magic Square based on two Hurwitz algebras have been proposed: Vinberg [Vin05], Allison and Faulkner [AF93] and more recently, Barton and Sudbery [BS, BS03], and Landsberg and Manivel [LM02, LM04] provided a different con- struction based on two Hurwitz algebras C, C0, their Lie algebras of triality tri(C), tri(C0), and three copies of their tensor product: ιi(C ⊗C0), i = 0, 1, 2.
The Jordan algebra J = H3(C0), its subspace of trace zero elements and its derivation algebra can be split naturally as:
J = H3(C0) ∼= k3⊕ ⊕2i=0ιi(C0), J0 ∼= k2⊕ ⊕2i=0ιi(C0),
derJ ∼= tri(C0) ⊕ ⊕2i=0ιi(C0),
and the above mentioned symmetric constructions are obtained by rearrang- ing Tits construction as follows:
T (C, J ) = der C ⊕ (C0⊗ J0) ⊕ der J
∼= der C ⊕ (C0⊗ k2) ⊕ ⊕2i=0C0⊗ ιi(C0) ⊕ tri(C0) ⊕ (⊕2i=0ιi(C0))
∼= tri(C) ⊕ tri(C0) ⊕ ⊕2i=0ιi(C ⊗ C0).
This construction, besides its symmetry, has the advantage of being valid too in characteristic 3. Simpler formulas are obtained if symmetric composi- tion algebras are used, instead of the more classical Hurwitz algebras. This led the second author to reinterpret the above construction in terms of two symmetric composition algebras [Eld04].
An algebra endowed with a nondegenerate quadratic form (S, ∗, q) is said to be a symmetric composition algebra if it satisfies
( q(x ∗ y) = q(x)q(y), q(x ∗ y, z) = q(x, y ∗ z).
for any x, y, z ∈ S, where q(x, y) = q(x + y) − q(x) − q(y) is the polar of q.
Any Hurwitz algebra C with norm q, standard involution x 7→ ¯x = q(x, 1)1 − x, but with new multiplication
x ∗ y = ¯x¯y,
is a symmetric composition algebra, called the associated para-Hurwitz al- gebra.
The classification of symmetric composition algebras was given by El- duque, Okubo, Osborn, Myung and P´erez-Izquierdo (see [EM93, EP96, KMRT98]). In dimension 1,2 or 4, any symmetric composition algebra is a para-Hurwitz algebra, with a few exceptions in dimension 2 which are, nev- ertheless, forms of para-Hurwitz algebras; while in dimension 8, apart from the para-Hurwitz algebras, there is a new family of symmetric composition algebras termed Okubo algebras.
If (S, ∗, q) is a symmetric composition algebra, the subalgebra of so(S, q)3 defined by
tri(S) = {(d0, d1, d2) ∈ so(S, q)3 : d0(x ∗ y) = d1(x) ∗ y + x ∗ d2(y)∀x, y ∈ S}
is the triality Lie algebra of S, which satisfies:
tri(S) =
0 if dim S = 1,
2-dim’l abelian if dim S = 2, so(S0, q)3 if dim S = 4, so(S, q) if dim S = 8.
The construction given by Elduque in [Eld04] starts with two symmetric composition algebras S, S0 and considers the Z2× Z2-graded algebra
g(S, S0) = tri(S) ⊕ tri(S0) ⊕ ⊕2i=0ιi(S ⊗ S0),
where ιi(S ⊗S0) is just a copy of S ⊗S0, with anticommutative multiplication given by:
• tri(S) ⊕ tri(S0) is a Lie subalgebra of g(S, S0),
• [(d0, d1, d2), ιi(x ⊗ x0)] = ιi di(x) ⊗ x0,
• [(d00, d01, d02), ιi(x ⊗ x0)] = ιi x ⊗ d0i(x0),
• [ιi(x ⊗ x0), ιi+1(y ⊗ y0)] = ιi+2 (x ∗ y) ⊗ (x0∗ y0) (indices modulo 3),
• [ιi(x ⊗ x0), ιi(y ⊗ y0)] = q0(x0, y0)θi(tx,y) + q(x, y)θ0i(t0x0,y0),
for any x, y ∈ S, x0, y0 ∈ S0, (d0, d1, d2) ∈ tri(S) and (d00, d01, d02) ∈ tri(S0).
The triple tx,y = q(x, .)y − q(y, .)x,12q(x, y)1 − rxly,12q(x, y)1 − lxry is in tri(S) and θ : (d0, d1, d2) 7→ (d2, d0, d1) is the triality automorphism in tri(S);
and similarly for t0 and θ0 in tri(S0).
With this multiplication, g(S, S0) is a Lie algebra and, if the characteristic of the ground field is 6= 2, 3, Freudenthal’s Magic Square is recovered.
dim S0
g(S, S0) 1 2 4 8
1 A1 A2 C3 F4
2 A2 A2⊕ A2 A5 E6 dim S
4 C3 A5 D6 E7
8 F4 E6 E7 E8
In characteristic 3, some attention has to be paid to the second row (or column), where the Lie algebras obtained are not simple but contain a simple codimension 1 ideal.
dim S0
g(S, S0) 1 2 4 8
1 A1 A˜2 C3 F4 2 A˜2 A˜2⊕ ˜A2 A˜5 E˜6
dim S
4 C3 A˜5 D6 E7
8 F4 E˜6 E7 E8
• ˜A2 denotes a form of pgl3, so [ ˜A2, ˜A2] is a form of psl3.
• ˜A5 denotes a form of pgl6, so [ ˜A5, ˜A5] is a form of psl6.
• ˜E6 is not simple, but [ ˜E6, ˜E6] is a codimension 1 simple ideal.
The characteristic 3 presents also another exceptional feature. Only over fields of this characteristic there are nontrivial composition superalgebras, which appear in dimensions 3 and 6. This fact allows to extend Freudenthal’s Magic Square with the addition of two further rows and columns, filled with (mostly simple) Lie superalgebras.
A precise description of those superalgebras can be done as contragredient Lie superalgebras (see [CEa] and [BGL]).
Most of the Lie superalgebras in the extended Freudenthal’s Magic Square in characteristic 3 are related to some known simple Lie superalgebras, spe- cific to this characteristic, constructed in terms of orthogonal and symplectic triple systems, which are defined in terms of central simple degree three Jor- dan algebras.
2. The extended Freudenthal’s Magic Square in characteristic 3
A superalgebra C = C¯0⊕ C¯1 over k, endowed with a regular quadratic su- perform q = (q¯0, b), called the norm, is said to be a composition superalgebra (see [EO02]) in case:
q¯0(x¯0y¯0) = q¯0(x¯0)q¯0(y¯0),
b(x¯0y, x¯0z) = q¯0(x¯0)b(y, z) = b(yx¯0, zx¯0), b(xy, zt) + (−1)|x||y|+|x||z|+|y||z|
b(zy, xt) = (−1)|y||z|b(x, z)b(y, t), for any x¯0, y¯0 ∈ C¯0 and homogeneous elements x, y, z, t ∈ C (where |x|
denotes the parity of the homogeneous element x).
The unital composition superalgebras are termed Hurwitz superalgebras.
A composition superalgebra is said to be symmetric in case its bilinear form is associative, that is, b(xy, z) = b(x, yz), for any x, y, z.
Only over fields of characteristic 3 there appear nontrivial Hurwitz super- algebras:
Example 2.1. Let V be a two dimensional vector space over a field k of characteristic 3, endowed with a nonzero alternating bilinear form h.|.i. The Jordan superalgebra
B(1, 2) = k1 ⊕ V,
with the multiplication given by
1x = x1 = x and uv = hu|vi1
for any x ∈ B(1, 2) and u, v ∈ V , and with the quadratic superform q = (q¯0, b) such that
q¯0(1) = 1, b(u, v) = hu|vi, for any u, v ∈ V , is a Hurwitz superalgebra.
Fix a symplectic basis {u, v} of V and λ ∈ k. ϕ : 1 7→ 1, u 7→ u + λv, v 7→
v, is an automorphism of B(1, 2) verifying ϕ3 = 1. S1,2λ = B(1, 2) with the same norm but new product x ∗ y = ϕ(¯x)ϕ2(¯y) is a symmetric composition superalgebra. For λ = 0, S1,2 = S1,20 denotes the para-Hurwitz associated to B(1, 2).
Example 2.2. With V as before, let f 7→ ¯f be the associated symplectic involution on Endk(V ) (so hf (u)|vi = hu| ¯f (v)i for any u, v ∈ V and f ∈ Endk(V )). The superspace
B(4, 2) = Endk(V ) ⊕ V with multiplication given by:
• the usual multiplication (composition of maps) in Endk(V ),
• v · f = f (v) = ¯f · v
• u · v = h.|uiv w 7→ hw|uiv
∈ Endk(V ) , and with quadratic superform
q¯0(f ) = det f, b(u, v) = hu|vi,
for any f ∈ Endk(V ) and u, v ∈ V , is a Hurwitz superalgebra.
S4,2 will denote the associated para-Hurwitz superalgebra.
The classification of the composition superalgebras appears in [EO02, Theorem 4.3]. Any unital composition superalgebra is either: a Hurwitz algebra, or a Z2-graded Hurwitz algebra in characteristic 2, or isomorphic to either B(1, 2) or B(4, 2) in characteristic 3. Any symmetric composition superalgebra is either: a symmetric composition algebra, or a Z2-graded symmetric composition algebra in characteristic 2, or isomorphic to either S1,2λ or S4,2, in characteristic 3.
The symmetric composition superalgebras can be plugged into the super- ization of the construction g(S, S0) in [Eld04].
Given two symmetric composition superalgebras S and S0 over a field k of characteristic 6= 2, one can form (see [CEa, §3]) the Lie superalgebra:
g= g(S, S0) = tri(S) ⊕ tri(S0) ⊕ ⊕2i=0ιi(S ⊗ S0),
where ιi(S ⊗ S0) is just a copy of S ⊗ S0 (i = 0, 1, 2), with bracket given by:
• tri(S) ⊕ tri(S0) is a Lie subsuperalgebra of g,
• [(d0, d1, d2), ιi(x ⊗ x0)] = ιi di(x) ⊗ x0,
• [(d00, d01, d02), ιi(x ⊗ x0)] = (−1)|d0i||x|ιi x ⊗ d0i(x0),
• [ιi(x ⊗ x0), ιi+1(y ⊗ y0)] = (−1)|x0||y|ιi+2 (x • y) ⊗ (x0 • y0) (indices modulo 3),
• [ιi(x ⊗ x0), ιi(y ⊗ y0)] = (−1)|x||x0|+|x||y0|+|y||y0|b0(x0, y0)θi(tx,y) +(−1)|y||x0|b(x, y)θ0i(t0x0,y0),
for any i = 0, 1, 2 and homogeneous x, y ∈ S, x0, y0 ∈ S0, (d0, d1, d2) ∈ tri(S), and (d00, d01, d02) ∈ tri(S0). Here θ denotes the natural automorphism θ : (d0, d1, d2) 7→ (d2, d0, d1) in tri(S), θ0 the analogous automorphism of tri(S0), and
tx,y = σx,y,12b(x, y)1 − rxly,12b(x, y)1 − lxry
(2.3)
(with lx(y) = x • y, rx(y) = (−1)|x||y|y • x, σx,y(z) = (−1)|y||z|b(x, z)y − (−1)|x|(|y|+|z|)b(y, z)x for homogeneous x, y, z ∈ S), while θ0and t0x0,y0 denote the analogous elements for tri(S0).
Then an extension in characteristic 3 is obtained of Freudenthal’s Magic Square, in which Lie superalgebras appear. (Here Sr denotes a symmetric composition algebra of dimension r.)
g(S, S0) S1 S2 S4 S8 S1,2 S4,2
S1 A1 A˜2 C3 F4 (6,8) (21,14) S2 A˜2⊕ ˜A2 A˜5 E˜6 (11,14) (35,20)
S4 D6 E7 (24,26) (66,32)
S8 E8 (55,50) (133,56)
S1,2 (21,16) (36,40)
S4,2 (78,64)
The table above presents only the entries above the diagonal, since the square is symmetric. The notation ˜X indicates that in characteristic 3 the algebra obtained is not simple, but contains a simple codimension 1 ideal of type X. Besides, only the pairs (dim g¯0, dim g¯1) are displayed for the nontrivial Lie superalgebras that appear in the “Supersquare”. With just a single exception, which corresponds to the pair (6, 8), these Lie superalgebras have no counterpart in characteristic 0, and they are either simple or contain a simple codimension 1 ideal (this happens again only in the second row).
As can be seen in [CEa], over an algebraically closed field all these superal- gebras can be described in a unified way, as contragredient Lie superalgebras.
The following tables summarize and give a Cartan matrix for each one of them, as well as the associated Dynkin diagram (according the conventions in [Kac77, Tables IV and V]).
AS,S1,2
g(S1, S1,2) ∼= g0 AS1,S1,2, {2} /c
0
@2 −1 0
−1 0 1
0 −1 2 1
A ◦ ×◦ ◦
α1 α2 α3
g(S2, S1,2) ∼= g AS2,S1,2, {3} /c
0
@ 2 −1 −1
−1 2 −1
1 1 0
1
A ◦ ◦
◦
×
................ .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .
α1 α2
α3
g(S4, S1,2) ∼= g AS4,S1,2, {4}
0 B@
2 −1 0 0
−1 2 −2 −1
0 −1 2 0
0 1 0 0
1
CA ◦ ◦ ◦
◦
×
<
α1 α2 α3
α4
g(S8, S1,2) ∼= g AS8,S1,2, {5} 0 BB B@
2 −1 0 0 0
−1 2 −1 0 0
0 −2 2 −1 0
0 0 −1 2 −1
0 0 0 1 0
1 CC
CA ◦ ◦ > ◦ ◦ ×◦ α1 α2 α3 α4 α5
g(S1,2, S1,2) ∼= g0 AS1,2,S1,2, {4}
/c 0 B@
2 −1 0 0
−1 2 −1 0
0 −2 2 −1
0 0 1 0
1
CA ◦ ◦ > ◦ ×◦ α1 α2 α3 α4
g(S1,2, S4,2) ∼= g AS1,2,S4,2, {1, 3, 4} 0 B@
0 1 0 0
−1 2 −1 0
0 −2 2 −2
0 0 1 0
1
CA ×◦ ◦ > • < ×◦ α1 α2 α3 α4
Then, for instance, the superalgebra g(S1, S1,2) is isomorphic to the cen- terless derived contragredient Lie superalgebra g0 AS1,S1,2, {2} and g(S2, S1,2) is isomorphic to the centerless contragredient Lie superalgebra g AS2,S1,2, {3}/c.
Similarly, for g(S, S4,2) we have:
AS,S4,2
g(S1, S4,2) ∼= g AS1,S4,2, {2, 3} 0
@2 −1 0
1 2 1
0 1 0
1
A ◦ • ×◦
α1 α2 α3
> <
g(S2, S4,2) ∼= g AS2,S4,2, {3, 5} /c
0 BB B@
2 −1 0 0 0
−1 2 −1 0 0
0 −1 0 −1 1
0 0 −1 2 0
0 0 1 0 0
1 CC
CA ◦ ◦ ◦
◦
◦
×
×
............
......
α1 α2 α3 α4 α5
g(S4, S4,2) ∼= g AS4,S4,2, {4, 6}
0 BB BB B@
2 −1 0 0 0 0
−1 2 −1 0 0 0
0 −1 2 −1 0 0
0 0 1 0 1 −1
0 0 0 −1 2 0
0 0 0 −1 0 0
1 CC CC
CA ◦ ◦ ◦ ◦
◦
◦
×
×
............
......
α1 α2 α3 α4 α5
α6
g(S1,2, S4,2) ∼= g AS8,S4,2, {6, 7} 0 BB BB BB B@
2 0 −1 0 0 0 0
0 2 0 −1 0 0 0
−1 0 2 −1 0 0 0
0 −1 −1 2 −1 0 0
0 0 0 −1 2 −1 0
0 0 0 0 1 0 −1
0 0 0 0 0 −1 0
1 CC CC CC CA
◦ ◦ ◦ ◦ ◦ ◦
◦
× × α1
α2
α3 α4 α5 α6 α7
g(S4,2, S4,2) ∼= g AS4,2,S4,2, {1, 2, 4, 6} 0 BB BB B@
0 1 0 0 0 0
1 0 −1 0 0 0
0 −1 2 −1 0 0
0 0 −1 0 −1 −1
0 0 0 −1 2 0
0 0 0 1 0 0
1 CC CC
CA ◦ ◦ ◦ ◦
◦
◦
× × ×
×
............
......
α1 α2 α3 α4 α5 α6
The computation of the even and odd components of the Lie superalgebras involved are performed in [CEa]. The following table is obtained. (It must be remarked that g(S1,2λ , S) is isomorphic to g(S1,2, S) for any λ.)
S1,2 S4,2
S1 psl2,2 sp6⊕(14)
S2 sl2⊕ pgl3 ⊕ (2) ⊗ psl3
pgl6⊕(20) S4 sl2⊕ sp6 ⊕ (2) ⊗ (13) so12⊕ spin12 S8 sl2⊕f4 ⊕ (2) ⊗ (25) e7⊕ (56) S1,2 so7⊕2 spin7 sp8⊕(40) S4,2 sp8⊕(40) so13⊕ spin13
The even and odd parts of the nontrivial superalgebras in the table are displayed, spin denotes the spin module for the corresponding orthogonal Lie algebra, while (n) denotes an irreducible module of dimension n. Thus, for example, g(S4, S1,2) is a Lie superalgebra whose even part is (isomorphic to) the direct sum of the symplectic Lie algebra sp6 and of sl2, while its odd part is the tensor product of a 13 dimensional module for sp6 and the natural 2 dimensional module for sl2.
3. The Supersquare and Jordan algebras
It turns out that the Lie superalgebras g(Sr, S1,2) and g(Sr, S4,2), for r = 1, 4 and 8, and their derived subalgebras for r = 2, are precisely the simple Lie superalgebras defined in [Eld06b] in terms of orthogonal and symplectic triple systems and strongly related to simple Jordan algebras of degree 3.
Let S be a para-Hurwitz algebra. Then,
g(S1,2, S) = tri(S1,2) ⊕ tri(S) ⊕ ⊕2i=0ιi(S1,2⊗ S)
= (sp(V ) ⊕ V ) ⊕ tri(S) ⊕ ⊕2i=0ιi(1 ⊗ S) ⊕ ⊕2i=0ιi(V ⊗ S).
Consider the Jordan algebra of 3×3 hermitian matrices over the associated Hurwitz algebra:
J = H3( ¯S) =
α0 ¯a2 a1
a2 α1 ¯a0
¯
a1 a0 α2
: α0, α1, α2 ∈ k, a0, a1, a2 ∈ S
∼= k3⊕ ⊕2i=0ιi(S).
In [CEb, Theorem 4.9.] it is proved that:
g(S1,2, S)¯0∼= sp(V ) ⊕ der J (as Lie algebras), g(S1,2, S)¯1∼= V ⊗ ˆJ (as modules for the even part).
Therefore, some results concerning Z2-graded Lie superalgebras and or- thogonal triple systems ([Eld06b]) allow us to conclude that ˆJ = J0/k1 is an orthogonal triple system with product given by
[ˆxˆy ˆz] = x ◦ (y ◦ z) − y ◦ (x ◦ z)ˆ
(ˆx = x + k1).
Orthogonal triple systems were first considered by Okubo [Oku93].
Theorem 3.1 (Elduque-Cunha [CEb]). The Lie superalgebra g(S1,2, S) is the Lie superalgebra associated to the orthogonal triple system ˆJ = J0/k1, for J = H3( ¯S).
In order to analyze the Lie superalgebras g(S4,2, S), let V be, as before, a two dimensional vector space endowed with a nonzero alternating bilinear form. Assume that the ground field is algebraically closed. Then (see [CEb]) it can be checked that g(S8, S) can be identified to a Z2-graded Lie algebra
g(S8, S) ' sp(V ) ⊕ g(S4, S) ⊕ V ⊗ T (S), for a suitable g(S4, S)-module T (S) of dimension 8 + 6 dim S.
In a similar vein, one gets that g(S4,2, S) can be identified with g(S4,2, S) ' g(S4, S) ⊕ T (S).
Hence, according to [CEb, Theorem 5.3, Theorem 5.6]:
Corollary 3.2 (Elduque-Cunha). Let S be a para-Hurwitz algebra, then:
(i) T (S) above is a symplectic triple system.
(ii) g(S4,2, S) ∼= inder T ⊕ T is the Lie superalgebra attached to this triple system.
Remark 3.3. Symplectic triple systems are closely related to the so called Freudenthal triple systems (see [YA75]). The classification of the simple finite dimensional symplectic triple systems in characteristic 3 appears in [Eld06b, Theorem 2.32] and it follows from this classification that the sym- plectic system triple T (S) above is isomorphic to a well-known triple system defined on the set of 2 × 2-matrices k J
J k
, with J = H3( ¯S).
The next table summarizes the previous arguments:
g S1 S2 S4 S8
S1,2 Lie superalgebras attached to orthogonal triple systems ˆJ = J0/k1
S4,2 Lie superalgebras attached to symplectic triple systems k J
J k
(J a degree 3 central simple Jordan algebra)
4. Final remarks
In the previous section, the Lie superalgebras g(Sr, S1,2) and g(Sr, S4,2) (r = 1, 2, 4, 8) in the extended Freudenthal’s Magic Square have been shown to be related to Lie superalgebras which had been constructed in terms of orthogonal and symplectic triple systems in [Eld06b]. Let us have a look here at the remaining Lie superalgebras in the supersquare:
The result in [CEa, Corollary 5.20] shows that the even part of g(S1,2, S1,2) is isomorphic to the orthogonal Lie algebra so7(k), while its odd part is the direct sum of two copies of the spin module for so7(k), and therefore, over any algebraically closed field of characteristic 3, g(S1,2, S1,2) is isomorphic to the simple Lie superalgebra in [Eld06b, Theorem 4.23(ii)], attached to a simple null orthogonal triple system.
For g(S4,2, S4,2), as shown in [CEa, Proposition 5.10 and Corollary 5.11], its even part is isomorphic to the orthogonal Lie algebra so13(k), while its odd part is the spin module for the even part, and hence that g(S4,2, S4,2) is the simple Lie superalgebra in [Elda, Theorem 3.1(ii)] for l = 6.
Only the simple Lie superalgebra g(S1,2, S4,2) has not previously appeared in the literature. Its even part is isomorphic to the symplectic Lie algebra sp8(k), while its odd part is the irreducible module of dimension 40 which appears as a subquotient of the third exterior power of the natural module for sp8(k) (see [CEa, §5.5]).
Also g(S1,2, S1,2) and g(S1,2, S4,2) are related to some orthosymplectic triple systems [CEb], which are triple systems of a mixed nature.
In conclusion, notice that
the main feature of Freudenthal’s Magic Supersquare is that among all the Lie superalgebras involved, only g(S1,2, S1) ∼= psl2,2 has a counterpart in Kac’s classification in characteristic 0. The other Lie superalgebras in Freudenthal’s Magic Supersquare, or their derived algebras, are new simple Lie superalgebras, specific of characteristic 3.
References
[AF93] B. N. Allison and J. R. Faulkner, Nonassociative coefficient algebras for Stein- berg unitary Lie algebras, J. Algebra 161 (1993), no. 1, 1–19.
[BS] C. H. Barton and A. Sudbery, Magic Squares of Lie Algebras, arXiv:math.RA/0001083.
[BS03] C. H. Barton and A. Sudbery, Magic squares and matrix models of Lie algebras, Adv. Math. 180 (2003), no. 2, 596–647.
[BL06] Sofiane Bouarroudj and Dimitri Leites, Simple Lie superalgebras and noninte- grable distributions in characteristic p, Zap. Nauchn. Sem. S.-Peterburg. Otdel.
Mat. Inst. Steklov. (POMI) 331 (2006), no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 14, 15–29, 221.
[BGL] Sofiane Bouarroudj, Pavel Grozman and Dimitri Leites, Cartan matrices and presentations of Cunha and Elduque superalgebras, arXiv:math.RT/0611391.
[Cun07] Isabel Cunha, Extens˜ao do Quadrado M´agico de Freudenthal em caracter´ıstica 3, Doctoral Thesis, Universidade da Beira Interior.
[CEa] Isabel Cunha and Alberto Elduque, An extended Freudenthal Magic Square in characteristic 3, arXiv:math.RA/0605379.
[CEb] , The extended Freudenthal Magic Square and Jordan Algebras, arXiv:math.RA/0608191.
[Eld04] Alberto Elduque, The magic square and symmetric compositions, Rev. Mat.
Iberoamericana 20 (2004), no. 2, 475–491.
[Eld06a] , A new look at Freudenthal’s magic square, Non-associative algebra and its applications, L. Sabinin, L.V. Sbitneva, and I. P. Shestakov, eds., Lect. Notes Pure Appl. Math., vol. 246, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 149–165.
[Eld06b] , New simple Lie superalgebras in characteristic 3, J. Algebra 296 (2006), no. 1, 196–233.
[Elda] , Some new simple modular Lie superalgebras, Pacific J. Math. (to appear), arXiv:math.RA/0512654.
[Eldb] , The Magic Square and Symmetric Compositions II, Rev. Mat.
Iberoamericana (to appear), arXiv:math.RT/0507282.
[EM93] Alberto Elduque and Hyo Chul Myung, Flexible composition algebras and Okubo algebras, Comm. Algebra 21 (1993), no. 7, 2481–2505.
[EO02] Alberto Elduque and Susumu Okubo, Composition superalgebras, Comm. Al- gebra 30 (2002), no. 11, 5447–5471.
[EP96] Alberto Elduque and Jos´e Mar´ıa P´erez, Composition algebras with associative bilinear form, Comm. Algebra 24 (1996), no. 3, 1091–1116.
[Fre64] H. Freudenthal, Lie groups in the foundations of geometry, Adv. Math. 1 (1964), 145–190.
[Kac77] V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96.
[KMRT98] Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tig- nol, The book of involutions, American Mathematical Society Colloquium Pub- lications, vol. 44, American Mathematical Society, Providence, RI, 1998.
[LM02] J. M. Landsberg and L. Manivel, Triality, exceptional Lie algebras and Deligne dimension formulas, Adv. Math. 171 (2002), no. 1, 59–85.
[LM04] Joseph M. Landsberg and Laurent Manivel, Representation theory and projec- tive geometry, Algebraic transformation groups and algebraic varieties, Ency- clopaedia Math. Sci., vol. 132, Springer, Berlin, 2004, pp. 71–122.
[Oku93] Susumu Okubo, Triple products and Yang-Baxter equation. I. Octonionic and quaternionic triple systems, J. Math. Phys. 34 (1993), no. 7, 3273–3291.
[Sch95] Richard D. Schafer, An introduction to nonassociative algebras, Dover Publi- cations Inc., New York, 1995.
[Tit66] J. Tits, Alg`ebres alternatives, alg`ebres de Jordan et alg`ebres de Lie exception- nelles. I. Construction, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag.
Math. 28 (1966), 223–237.
[Vin05] E.B. Vinberg, Construction of the exceptional simple Lie algebras., Vinberg, Ernest (ed.), Lie groups and invariant theory. Providence, RI: American Math- ematical Society (AMS). Translations. Series 2. American Mathematical Soci- ety 213. Advances in the Mathematical Sciences 56 (2005), 241–242.
[YA75] Kiyosi Yamaguti and Hiroshi Asano, On the Freudenthal’s construction of exceptional Lie algebras, Proc. Japan Acad. 51 (1975), no. 4, 253–258.
Departamento de Matem´atica, Universidade da Beira Interior, 6200 Covilh˜a, Portugal
E-mail address: [email protected]
Departamento de Matem´aticas, Universidad de Zaragoza, 50009 Zaragoza, Spain
E-mail address: [email protected]
