PRE-PUBLICACIONES del seminario matematico

“garcia de galdeano”

Freudenthal’s magic supersquare in characteristic 3

2007

Isabel Cunha Alberto Elduque

n. 7

seminario matemático

garcía de galdeano Universidad de Zaragoza

FREUDENTHAL’S MAGIC SUPERSQUARE IN CHARACTERISTIC 3 ISABEL CUNHA AND ALBERTO ELDUQUE? Abstract. A family of simple Lie superalgebras over fields of characteristic 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 excepcional 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 defining 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

2

ISABEL CUNHA AND ALBERTO ELDUQUE

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 construction based on two Hurwitz algebras C, C 0 , their Lie algebras of triality tri(C), tri(C 0 ), and three copies of their tensor product: ιi (C ⊗C 0 ), i = 0, 1, 2. The Jordan algebra J = H3 (C 0 ), its subspace of trace zero elements and its derivation algebra can be split naturally as: J = H3 (C 0 ) ∼ = k 3 ⊕ ⊕2 ιi (C 0 ) , i=0

J0 ∼ = k 2 ⊕ ⊕2i=0 ιi (C 0 ) , der J ∼ = tri(C 0 ) ⊕ ⊕2i=0 ιi (C 0 ) , and the above mentioned symmetric constructions are obtained by rearranging Tits construction as follows: T (C, J) = der C ⊕ (C0 ⊗ J0 ) ⊕ der J ∼ = der C ⊕ (C0 ⊗ k 2 ) ⊕ ⊕2i=0 C0 ⊗ ιi (C 0 ) ⊕ tri(C 0 ) ⊕ (⊕2i=0 ιi (C 0 )) ∼ = tri(C) ⊕ tri(C 0 ) ⊕ ⊕2i=0 ιi (C ⊗ C 0 ) . This construction, besides its symmetry, has the advantage of being valid too in characteristic 3. Simpler formulas are obtained if symmetric composition 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 algebra. The classification of symmetric composition algebras was given by Elduque, 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, nevertheless, 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.

FREUDENTHAL’S MAGIC SUPERSQUARE IN CHARACTERISTIC 3

3

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: 0 if dim S 2-dim’l abelian if dim S tri(S) = so(S0 , q)3 if dim S so(S, q) if dim S

= 1, = 2, = 4, = 8.

The construction given by Elduque in [Eld04] starts with two symmetric composition algebras S, S 0 and considers the Z2 × Z2 -graded algebra g(S, S 0 ) = tri(S) ⊕ tri(S 0 ) ⊕ ⊕2i=0 ιi (S ⊗ S 0 ) , where ιi (S ⊗S 0 ) is just a copy of S ⊗S 0 , with anticommutative multiplication given by: • tri(S) ⊕ tri(S 0 ) is a Lie subalgebra of g(S, S 0 ), • [(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 ⊗ y 0 )] = ιi+2 (x ∗ y) ⊗ (x0 ∗ y 0 ) (indices modulo 3), • [ιi (x ⊗ x0 ), ιi (y ⊗ y 0 )] = q 0 (x0 , y 0 )θi (tx,y ) + q(x, y)θ0i (t0x0 ,y0 ), 0 ). for any x, y ∈ S, x0 , y 0 ∈ S 0 , (d0 , d1 , d2 ) ∈ tri(S) and (d00 , d01 , d02 ) ∈ tri(S The triple tx,y = q(x, .)y − q(y, .)x, 12 q(x, y)1 − rx ly , 21 q(x, y)1 − lx ry 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(S 0 ).

With this multiplication, g(S, S 0 ) is a Lie algebra and, if the characteristic of the ground field is 6= 2, 3, Freudenthal’s Magic Square is recovered. dim S 0

dim S

g(S, S 0 )

1

2

4

8

1

A1

A2

C3

F4

2

A2 A2 ⊕ A2 A5 E6

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.

4

ISABEL CUNHA AND ALBERTO ELDUQUE

dim S 0 g(S, S 0 ) 1 2 dim S

4 8

• • •

1

2 A˜2

4

8

C3 F4 A1 ˜6 A˜2 A˜2 ⊕ A˜2 A˜5 E C3 A˜5 D6 E7 ˜ F4 E6 E7 E8

A˜2 denotes a form of pgl3 , so [A˜2 , A˜2 ] is a form of psl3 . A˜5 denotes a form of pgl6 , so [A˜5 , A˜5 ] is a form of psl6 . ˜6 is not simple, but [E ˜6 , E ˜6 ] is a codimension 1 simple ideal. E

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, specific to this characteristic, constructed in terms of orthogonal and symplectic triple systems, which are defined in terms of central simple degree three Jordan 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 superform q = (q¯0 , b), called the norm, is said to be a composition superalgebra (see [EO02]) in case: q¯0 (x¯0 y¯0 ) = q¯0 (x¯0 )q¯0 (y¯0 ), b(x¯0 y, x¯0 z) = 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 superalgebras: 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,

FREUDENTHAL’S MAGIC SUPERSQUARE IN CHARACTERISTIC 3

5

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→ λ = B(1, 2) with the v, is an automorphism of B(1, 2) verifying ϕ3 = 1. S1,2 same norm but new product x ∗ y = ϕ(¯ x)ϕ2 (¯ y ) is a symmetric composition 0 superalgebra. For λ = 0, S1,2 = S1,2 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 λ or S , in characteristic 3. S1,2 4,2 The symmetric composition superalgebras can be plugged into the superization of the construction g(S, S 0 ) in [Eld04]. Given two symmetric composition superalgebras S and S 0 over a field k of characteristic 6= 2, one can form (see [CEa, §3]) the Lie superalgebra: g = g(S, S 0 ) = tri(S) ⊕ tri(S 0 ) ⊕ ⊕2i=0 ιi (S ⊗ S 0 ) , where ιi (S ⊗ S 0 ) is just a copy of S ⊗ S 0 (i = 0, 1, 2), with bracket given by: • tri(S) ⊕ tri(S 0 ) is a Lie subsuperalgebra of g, • [(d0 , d1 , d2 ), ιi (x ⊗ x0 )] = ιi di (x) ⊗ x0 , 0 • [(d00 , d01 , d02 ), ιi (x ⊗ x0 )] = (−1)|di ||x| ιi x ⊗ d0i (x0 ) ,

6

ISABEL CUNHA AND ALBERTO ELDUQUE

0 • [ιi (x ⊗ x0 ), ιi+1 (y ⊗ y 0 )] = (−1)|x ||y| ιi+2 (x • y) ⊗ (x0 • y 0 ) (indices modulo 3), 0

0

0

• [ιi (x ⊗ x0 ), ιi (y ⊗ y 0 )] = (−1)|x||x |+|x||y |+|y||y | b0 (x0 , y 0 )θi (tx,y ) 0 +(−1)|y||x | b(x, y)θ0i (t0x0 ,y0 ), for any i = 0, 1, 2 and homogeneous x, y ∈ S, x0 , y 0 ∈ S 0 , (d0 , d1 , d2 ) ∈ tri(S), and (d00 , d01 , d02 ) ∈ tri(S 0 ). Here θ denotes the natural automorphism θ : (d0 , d1 , d2 ) 7→ (d2 , d0 , d1 ) in tri(S), θ0 the analogous automorphism of tri(S 0 ), and tx,y = σx,y , 21 b(x, y)1 − rx ly , 12 b(x, y)1 − lx ry (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 θ0 and t0x0 ,y0 denote the analogous elements for tri(S 0 ). 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, S 0 ) S1 S1 S2 S4 S8 S1,2 S4,2

A1

S2 A˜2 A˜2 ⊕ A˜2

S4

S8

S1,2

S4,2

C3 F4 (6,8) ˜6 (11,14) A˜5 E

(21,14)

D6 E7 (24,26)

(66,32)

(35,20)

E8 (55,50) (133,56) (21,16)

(36,40) (78,64)

The table above presents only the entries above the diagonal, since the ˜ indicates that in characteristic 3 the square is symmetric. The notation X 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 superalgebras 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]).

FREUDENTHAL’S MAGIC SUPERSQUARE IN CHARACTERISTIC 3 AS,S1,2 1 2 −1 0 @−1 0 1A 0 −1 2

0

0 g(S1 , S1,2 ) ∼ = g AS1 ,S1,2 , {2} /c

0

g(S2 , S1,2 ) ∼ = g AS2 ,S1,2 , {3} /c

0 g(S8 , S1,2 ) ∼ = g AS8 ,S1,2 , {5}

α1 ◦

1

0 g(S1,2 , S1,2 ) ∼ = g AS1,2 ,S1,2 , {4} /c

g(S1,2 , S4,2 ) ∼ = g AS1,2 ,S4,2 , {1, 3, 4}

α3 ◦

◦... α1

◦ α2

α1 ◦

α2 ◦

α3 ◦

◦

α4 ◦

α5 ◦ ×

α1 ◦

α2 α3 ◦ > ◦

α4 ◦ ×

α1 ◦ ×

α2 α3 α4 ◦ > • < × ◦

Then, for instance, the superalgebra g(S1 , S1,2 ) is isomorphic to the cen0 terless derived contragredient Lie superalgebra g 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:

g(S1 , S4,2 ) ∼ = g AS1 ,S4,2 , {2, 3}

g(S2 , S4,2 ) ∼ = g AS2 ,S4,2 , {3, 5} /c

g(S4 , S4,2 ) ∼ = g AS4 ,S4,2 , {4, 6}

g(S1,2 , S4,2 ) ∼ = g AS8 ,S4,2 , {6, 7}

g(S4,2 , S4,2 ) ∼ = g AS4,2 ,S4,2 , {1, 2, 4, 6}

AS,S4,2 1 2 −1 0 @1 2 1A 0 1 0 0 2 −1 0 0 2 −1 0 B−1 B 0 −1 0 −1 B @ 0 0 −1 2 0 0 1 0 0 2 −1 0 0 0 B−1 2 −1 0 0 B −1 2 −1 0 B 0 B 0 0 1 0 1 B @ 0 0 0 −1 2 0 0 0 −1 0 0 2 0 −1 0 0 B 0 2 0 −1 0 B 0 2 −1 0 B−1 B 0 −1 −1 2 −1 B B 0 0 0 −1 2 B @ 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 B1 0 −1 0 0 B 2 −1 0 B0 −1 B0 0 −1 0 −1 B @0 0 0 −1 2 0 0 0 1 0

0

α1 ◦

>

α1 ◦

α2 ◦

α2 •

“garcia de galdeano”

Freudenthal’s magic supersquare in characteristic 3

2007

Isabel Cunha Alberto Elduque

n. 7

seminario matemático

garcía de galdeano Universidad de Zaragoza

FREUDENTHAL’S MAGIC SUPERSQUARE IN CHARACTERISTIC 3 ISABEL CUNHA AND ALBERTO ELDUQUE? Abstract. A family of simple Lie superalgebras over fields of characteristic 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 excepcional 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 defining 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

2

ISABEL CUNHA AND ALBERTO ELDUQUE

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 construction based on two Hurwitz algebras C, C 0 , their Lie algebras of triality tri(C), tri(C 0 ), and three copies of their tensor product: ιi (C ⊗C 0 ), i = 0, 1, 2. The Jordan algebra J = H3 (C 0 ), its subspace of trace zero elements and its derivation algebra can be split naturally as: J = H3 (C 0 ) ∼ = k 3 ⊕ ⊕2 ιi (C 0 ) , i=0

J0 ∼ = k 2 ⊕ ⊕2i=0 ιi (C 0 ) , der J ∼ = tri(C 0 ) ⊕ ⊕2i=0 ιi (C 0 ) , and the above mentioned symmetric constructions are obtained by rearranging Tits construction as follows: T (C, J) = der C ⊕ (C0 ⊗ J0 ) ⊕ der J ∼ = der C ⊕ (C0 ⊗ k 2 ) ⊕ ⊕2i=0 C0 ⊗ ιi (C 0 ) ⊕ tri(C 0 ) ⊕ (⊕2i=0 ιi (C 0 )) ∼ = tri(C) ⊕ tri(C 0 ) ⊕ ⊕2i=0 ιi (C ⊗ C 0 ) . This construction, besides its symmetry, has the advantage of being valid too in characteristic 3. Simpler formulas are obtained if symmetric composition 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 algebra. The classification of symmetric composition algebras was given by Elduque, 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, nevertheless, 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.

FREUDENTHAL’S MAGIC SUPERSQUARE IN CHARACTERISTIC 3

3

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: 0 if dim S 2-dim’l abelian if dim S tri(S) = so(S0 , q)3 if dim S so(S, q) if dim S

= 1, = 2, = 4, = 8.

The construction given by Elduque in [Eld04] starts with two symmetric composition algebras S, S 0 and considers the Z2 × Z2 -graded algebra g(S, S 0 ) = tri(S) ⊕ tri(S 0 ) ⊕ ⊕2i=0 ιi (S ⊗ S 0 ) , where ιi (S ⊗S 0 ) is just a copy of S ⊗S 0 , with anticommutative multiplication given by: • tri(S) ⊕ tri(S 0 ) is a Lie subalgebra of g(S, S 0 ), • [(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 ⊗ y 0 )] = ιi+2 (x ∗ y) ⊗ (x0 ∗ y 0 ) (indices modulo 3), • [ιi (x ⊗ x0 ), ιi (y ⊗ y 0 )] = q 0 (x0 , y 0 )θi (tx,y ) + q(x, y)θ0i (t0x0 ,y0 ), 0 ). for any x, y ∈ S, x0 , y 0 ∈ S 0 , (d0 , d1 , d2 ) ∈ tri(S) and (d00 , d01 , d02 ) ∈ tri(S The triple tx,y = q(x, .)y − q(y, .)x, 12 q(x, y)1 − rx ly , 21 q(x, y)1 − lx ry 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(S 0 ).

With this multiplication, g(S, S 0 ) is a Lie algebra and, if the characteristic of the ground field is 6= 2, 3, Freudenthal’s Magic Square is recovered. dim S 0

dim S

g(S, S 0 )

1

2

4

8

1

A1

A2

C3

F4

2

A2 A2 ⊕ A2 A5 E6

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.

4

ISABEL CUNHA AND ALBERTO ELDUQUE

dim S 0 g(S, S 0 ) 1 2 dim S

4 8

• • •

1

2 A˜2

4

8

C3 F4 A1 ˜6 A˜2 A˜2 ⊕ A˜2 A˜5 E C3 A˜5 D6 E7 ˜ F4 E6 E7 E8

A˜2 denotes a form of pgl3 , so [A˜2 , A˜2 ] is a form of psl3 . A˜5 denotes a form of pgl6 , so [A˜5 , A˜5 ] is a form of psl6 . ˜6 is not simple, but [E ˜6 , E ˜6 ] is a codimension 1 simple ideal. E

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, specific to this characteristic, constructed in terms of orthogonal and symplectic triple systems, which are defined in terms of central simple degree three Jordan 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 superform q = (q¯0 , b), called the norm, is said to be a composition superalgebra (see [EO02]) in case: q¯0 (x¯0 y¯0 ) = q¯0 (x¯0 )q¯0 (y¯0 ), b(x¯0 y, x¯0 z) = 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 superalgebras: 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,

FREUDENTHAL’S MAGIC SUPERSQUARE IN CHARACTERISTIC 3

5

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→ λ = B(1, 2) with the v, is an automorphism of B(1, 2) verifying ϕ3 = 1. S1,2 same norm but new product x ∗ y = ϕ(¯ x)ϕ2 (¯ y ) is a symmetric composition 0 superalgebra. For λ = 0, S1,2 = S1,2 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 λ or S , in characteristic 3. S1,2 4,2 The symmetric composition superalgebras can be plugged into the superization of the construction g(S, S 0 ) in [Eld04]. Given two symmetric composition superalgebras S and S 0 over a field k of characteristic 6= 2, one can form (see [CEa, §3]) the Lie superalgebra: g = g(S, S 0 ) = tri(S) ⊕ tri(S 0 ) ⊕ ⊕2i=0 ιi (S ⊗ S 0 ) , where ιi (S ⊗ S 0 ) is just a copy of S ⊗ S 0 (i = 0, 1, 2), with bracket given by: • tri(S) ⊕ tri(S 0 ) is a Lie subsuperalgebra of g, • [(d0 , d1 , d2 ), ιi (x ⊗ x0 )] = ιi di (x) ⊗ x0 , 0 • [(d00 , d01 , d02 ), ιi (x ⊗ x0 )] = (−1)|di ||x| ιi x ⊗ d0i (x0 ) ,

6

ISABEL CUNHA AND ALBERTO ELDUQUE

0 • [ιi (x ⊗ x0 ), ιi+1 (y ⊗ y 0 )] = (−1)|x ||y| ιi+2 (x • y) ⊗ (x0 • y 0 ) (indices modulo 3), 0

0

0

• [ιi (x ⊗ x0 ), ιi (y ⊗ y 0 )] = (−1)|x||x |+|x||y |+|y||y | b0 (x0 , y 0 )θi (tx,y ) 0 +(−1)|y||x | b(x, y)θ0i (t0x0 ,y0 ), for any i = 0, 1, 2 and homogeneous x, y ∈ S, x0 , y 0 ∈ S 0 , (d0 , d1 , d2 ) ∈ tri(S), and (d00 , d01 , d02 ) ∈ tri(S 0 ). Here θ denotes the natural automorphism θ : (d0 , d1 , d2 ) 7→ (d2 , d0 , d1 ) in tri(S), θ0 the analogous automorphism of tri(S 0 ), and tx,y = σx,y , 21 b(x, y)1 − rx ly , 12 b(x, y)1 − lx ry (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 θ0 and t0x0 ,y0 denote the analogous elements for tri(S 0 ). 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, S 0 ) S1 S1 S2 S4 S8 S1,2 S4,2

A1

S2 A˜2 A˜2 ⊕ A˜2

S4

S8

S1,2

S4,2

C3 F4 (6,8) ˜6 (11,14) A˜5 E

(21,14)

D6 E7 (24,26)

(66,32)

(35,20)

E8 (55,50) (133,56) (21,16)

(36,40) (78,64)

The table above presents only the entries above the diagonal, since the ˜ indicates that in characteristic 3 the square is symmetric. The notation X 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 superalgebras 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]).

FREUDENTHAL’S MAGIC SUPERSQUARE IN CHARACTERISTIC 3 AS,S1,2 1 2 −1 0 @−1 0 1A 0 −1 2

0

0 g(S1 , S1,2 ) ∼ = g AS1 ,S1,2 , {2} /c

0

g(S2 , S1,2 ) ∼ = g AS2 ,S1,2 , {3} /c

0 g(S8 , S1,2 ) ∼ = g AS8 ,S1,2 , {5}

α1 ◦

1

0 g(S1,2 , S1,2 ) ∼ = g AS1,2 ,S1,2 , {4} /c

g(S1,2 , S4,2 ) ∼ = g AS1,2 ,S4,2 , {1, 3, 4}

α3 ◦

◦... α1

◦ α2

α1 ◦

α2 ◦

α3 ◦

◦

α4 ◦

α5 ◦ ×

α1 ◦

α2 α3 ◦ > ◦

α4 ◦ ×

α1 ◦ ×

α2 α3 α4 ◦ > • < × ◦

Then, for instance, the superalgebra g(S1 , S1,2 ) is isomorphic to the cen0 terless derived contragredient Lie superalgebra g 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:

g(S1 , S4,2 ) ∼ = g AS1 ,S4,2 , {2, 3}

g(S2 , S4,2 ) ∼ = g AS2 ,S4,2 , {3, 5} /c

g(S4 , S4,2 ) ∼ = g AS4 ,S4,2 , {4, 6}

g(S1,2 , S4,2 ) ∼ = g AS8 ,S4,2 , {6, 7}

g(S4,2 , S4,2 ) ∼ = g AS4,2 ,S4,2 , {1, 2, 4, 6}

AS,S4,2 1 2 −1 0 @1 2 1A 0 1 0 0 2 −1 0 0 2 −1 0 B−1 B 0 −1 0 −1 B @ 0 0 −1 2 0 0 1 0 0 2 −1 0 0 0 B−1 2 −1 0 0 B −1 2 −1 0 B 0 B 0 0 1 0 1 B @ 0 0 0 −1 2 0 0 0 −1 0 0 2 0 −1 0 0 B 0 2 0 −1 0 B 0 2 −1 0 B−1 B 0 −1 −1 2 −1 B B 0 0 0 −1 2 B @ 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 B1 0 −1 0 0 B 2 −1 0 B0 −1 B0 0 −1 0 −1 B @0 0 0 −1 2 0 0 0 1 0

0

α1 ◦

>

α1 ◦

α2 ◦

α2 •