The Supermagic Square in Characteristic 3 and Jordan Superalgebras

Recently, the classical Freudenthal Magic Square has been extended over fields of characteristic 3 with two more rows and columns filled with (mostly simple) Lie superalgebras specific of this characteristic. This Supermagic Square will be reviewed and some of the simple Lie superalgebras that appear will be shown to be isomorphic to the Tits-Kantor-Koecher Lie superalgebras of some Jordan superalgebras. Introduction The classical Freudenthal Magic Square, which contains in characteristic 0 the exceptional simple finite dimensional Lie algebras, other than G2, is usually constructed based on two ingredients: a unital composition algebra and a central simple degree 3 Jordan algebra (see [Sch95, Chapter IV]). This construction, due to Tits, does not work in characteristic 3. A more symmetric construction, based on two unital composition algebras, which play symmetric roles, and their triality Lie algebras, has been given recently by several authors ([AF93], [BS], [LM02] [LM04]). Among other things, this construction has the advantage of being valid too in characteristic 3. Simpler formulas for triality appear if symmetric composition algebras are used, instead of the more classical unital composition algebras ([Eld04, Eld07a]). But the characteristic 3 presents an exceptional feature, as only over fields of this characteristic there are nontrivial composition superalgebras, which appear in dimensions 3 and 6. The unital such composition superalgebras were discovered by Shestakov [She97]. This fact allows to extend Freudenthal Magic Square ([CE07a]) with the addition of two further rows and columns, filled with (mostly simple) Lie superalgebras, specific of characteristic 3, which had appeared first (with one exception) in [Eld06] and [Eld07b]. Most of the Lie superalgebras in characteristic 3 that appear in the Supermagic Square have been shown to be related to degree three simple Jordan algebras in [CE07b]. The aim of this paper is to show that some of the Lie superalgebras in the Supermagic Square are isomorphic to the Tits-Kantor-Koecher Lie superalgebras of some distinguished Jordan superalgebras. More specifically, let Si denote the split para-Hurwitz algebra of dimension i = 1, 2 or 4, and let S be the para-Hurwitz superalgebra associated to the unital composition superalgebra C (see Section 1 for definitions and notations). Let g(Si, S) be the corresponding entry in the Supermagic Square. Then g(S1, S) was shown in [CE07b] to be isomorphic to the Lie superalgebra of derivations of the Date: February 22, 2008. ⋄ Supported by CMUC, Department of Mathematics, University of Coimbra. ⋆ Supported by the Spanish Ministerio de Educación y Ciencia and FEDER (MTM 200767884-C04-02) and by the Diputación General de Aragón (Grupo de Investigación de Álgebra). 1 2 ISABEL CUNHA AND ALBERTO ELDUQUE Jordan superalgebra J = H3(C) of hermitian 3 × 3 matrices over C. Here the following results will be proved: (i) The Lie superalgebras g(S2, S) in the second row of the Supermagic Square will be shown to be isomorphic to the projective structure superalgebras of the Jordan superalgebras J = H3(C). Here the structure superalgebra is str(J) = LJ ⊕ der(J) and the projective structure superalgebra pstr(J) is the quotient of str(J) by its center. (See Theorem 3.3 and Corollary 3.5.) (ii) The Lie superalgebras g(S4, S) in the third row of the Supermagic Square will be shown to be isomorphic to the Tits-Kantor-Koecher Lie superalgebras of the Jordan superalgebras J = H3(C). (See Theorem 3.8 and Corollary 3.10.) (iii) The Lie superalgebra g(S1,2, S1,2) will be shown to be isomorphic to the Tits-Kantor-Koecher Lie superalgebra of the nine dimensional Kac Jordan superalgebra K9. Note that the ten dimensional Kac Jordan superalgebra K10 is no longer simple in characteristic 3, but contains a nine dimensional simple ideal, which is K9. (See Theorem 4.2.) (iv) The Lie superalgebra g(S1, S1,2) will be shown to be isomorphic to the Tits-Kantor-Koecher Lie superalgebra of the three dimensional Kaplansky superalgebra K3. (See Corollary 4.3.) The paper is structured as follows. In Section 1 the construction of the Supermagic Square in terms of two symmetric composition superalgebras will be reviewed. Then the relationship of the Lie superalgebras in the first row of the Supermagic Square with the Lie superalgebras of derivations of the Jordan superalgebras J = H3(C) above, proven in [CE07b], will be reviewed in Section 2. Section 3 will be devoted to the Lie superalgebras in the second and third rows of the Supermagic Square, while Section 4 will deal with the Lie superalgebra g(S1,2, S1,2) and the nine dimensional Kac Jordan superalgebra K9. It was Shestakov [She96] who first noticed that K9 is isomorphic to the tensor product (in the graded sense) of two copies of the three dimensional Kaplansky Jordan superalgebra K3 (this was further developed in [BE02]), and this is the key for the results in Section 4. Unless otherwise stated, all the vector spaces and superspaces considered will be assumed to be finite dimensional over a ground field k of characteristic 6= 2. 1. The Supermagic Square A quadratic superform on a Z2-graded vector space U = U0̄ ⊕ U1̄ over a field k is a pair q = (q0̄, b) where q0̄ : U0̄ → k is a quadratic form, and b : U × U → k is a supersymmetric even bilinear form such that b|U0̄×U0̄ is the polar form of q0̄: b(x0̄, y0̄) = q0̄(x0̄ + y0̄)− q0̄(x0̄)− q0̄(y0̄) for any x0̄, y0̄ ∈ U0̄. The quadratic superform q = (q0̄, b) is said to be regular if the bilinear form b is nondegenerate. Then a superalgebra C = C0̄ ⊕ C1̄ over k, endowed with a regular quadratic superform q = (q0̄, b), called the norm, is said to be a composition superalgebra (see [EO02]) in case q0̄(x0̄y0̄) = q0̄(x0̄)q0̄(y0̄), (1.1a) b(x0̄y, x0̄z) = q0̄(x0̄)b(y, z) = b(yx0̄, zx0̄), (1.1b) b(xy, zt) + (−1)b(zy, xt) = (−1)b(x, z)b(y, t), (1.1c) THE SUPERMAGIC SQUARE AND JORDAN SUPERALGEBRAS 3 for any x0̄, y0̄ ∈ C0̄ and homogeneous elements x, y, z, t ∈ C. (As we are working in characteristic 6= 2, it is enough to consider equation (1.1c).) As usual, the expression (−1) equals −1 if the homogeneous elements y and z are both odd, otherwise, it equals 1. The unital composition superalgebras are termed Hurwitz superalgebras, while 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. Hurwitz algebras are the well-known algebras that generalize the classical real division algebras of the real and complex numbers, quaternions and octonions. Over any algebraically closed field k, there are exactly four of them: k, k × k, Mat2(k) and C(k) (the split Cayley algebra), with dimensions 1, 2, 4 and 8. Only over fields of characteristic 3 there appear nontrivial Hurwitz superalgebras (see [EO02]): • Let V be a two dimensional vector space over a field k, endowed with a nonzero alternating bilinear form 〈.|.〉 (that is 〈v|v〉 = 0 for any v ∈ V ). Consider the superspace B(1, 2) (see [She97]) with B(1, 2)0̄ = k1, and B(1, 2)1̄ = V, (1.2) endowed with the supercommutative multiplication given by 1x = x1 = x and uv = 〈u|v〉1 for any x ∈ B(1, 2) and u, v ∈ V , and with the quadratic superform q = (q0̄, b) given by: q0̄(1) = 1, b(u, v) = 〈u|v〉, (1.3) for any u, v ∈ V . If the characteristic of k is equal to 3, then B(1, 2) is a Hurwitz superalgebra ([EO02, Proposition 2.7]). • Moreover, with V as before, let f 7→ f̄ be the associated symplectic involution on Endk(V ) (so 〈f(u)|v〉 = 〈u|f̄(v)〉 for any u, v ∈ V and f ∈ Endk(V )). Consider the superspace B(4, 2) (see [She97]) with B(4, 2)0̄ = Endk(V ), and B(4, 2)1̄ = V, (1.4) with multiplication given by the usual one (composition of maps) in Endk(V ), and by v · f = f(v) = f̄ · v ∈ V, u · v = 〈.|u〉v ∈ Endk(V ) for any f ∈ Endk(V ) and u, v ∈ V , where 〈.|u〉v denotes the endomorphism w 7→ 〈w|u〉v; and with quadratic superform q = (q0̄, b) such that q0̄(f) = det(f), b(u, v) = 〈u|v〉, for any f ∈ Endk(V ) and u, v ∈ V . If the characteristic is equal to 3, B(4, 2) is a Hurwitz superalgebra ([EO02, Proposition 2.7]). Given any Hurwitz superalgebra C with norm q = (q0̄, b), its standard involution is given by x 7→ x̄ = b(x, 1)1− x. A new product can be defined on C by means of x • y = x̄ȳ. (1.5) The resulting superalgebra, denoted by C̄, is called the para-Hurwitz superalgebra attached to C, and it turns out to be a symmetric composition superalgebra. 4 ISABEL CUNHA AND ALBERTO ELDUQUE Given a symmetric composition superalgebra S, its triality Lie superalgebra tri(S) = tri(S)0̄ ⊕ tri(S)1̄ is defined by: tri(S )̄i = {(d0, d1, d2) ∈ osp(S, q) 3 ī : d0(x • y) = d1(x) • y + (−1) x • d2(y) ∀x, y ∈ S0̄ ∪ S1̄}, where ī = 0̄, 1̄, and osp(S, q) denotes the associated orthosymplectic Lie superalgebra. The bracket in tri(S) is given componentwise. Now, given two symmetric composition superalgebras S and S, one can form (see [CE07a, §3], or [Eld04] for the non-super situation) the Lie superalgebra: g = g(S, S) = ( tri(S)⊕ tri(S) ) ⊕ ( ⊕i=0ιi(S ⊗ S ) ) , (1.6) where ιi(S ⊗ S ) is just a copy of S ⊗ S (i = 0, 1, 2), with bracket given by: • the Lie bracket in tri(S)⊕tri(S), which thus becomes a Lie subsuperalgebra of g, • [(d0, d1, d2), ιi(x ⊗ x )] = ιi (