A NEW CONSTRUCTION OF THE

KAC JORDAN SUPERALGEBRA

Georgia Benkart^{1} Alberto Elduque^{2}

June 19, 2001

To Irving Kaplansky

Abstract. We present an elementary construction of the 10-dimensional simple Jordan superalgebra K_{10} of Kac using the 3-dimensional tiny Kaplansky superalgebra. This new realization of K_{10} enables us to derive many of its properties.

Introduction

The 10-dimensional Jordan superalgebra K_{10} of Kac [K] is the only exceptional finite-dimensional simple Jordan superalgebra over an algebraically closed field of characteristic 0. We give a new, elementary construction of it (over an arbitrary field of characteristic not 2) using the tiny Kaplansky superalgebra K. In our construction, K_{10} appears as a direct sum F 1 ⊕ (K ⊗ K) with unit element 1 and with a suitable multiplication (see (2.1)). In characteristic 3, K_{10} is not simple but possesses a simple ideal K_{9} of dimension 9, which is just the tensor product K ⊗ K (of superalgebras).

Our realization of K_{10} makes it is easy to deduce certain of its known properties: for example, that its Lie superalgebra of superderivations is isomorphic to a direct sum of two copies of the orthosymplectic Lie superalgebra 𝔬𝔰𝔭(1, 2), or that its Lie multiplication superalgebra is 𝔬𝔰𝔭(2, 4) if −1 is a square in the field F.

The Kac superalgebra was originally constructed in [K] by Lie theoretical methods from a 3-grading of the simple Lie superalgebra F (4). No direct proof of the fact that it is a Jordan superalgebra is known. Indeed, in their work on Jordan

Typeset by AMS-TEX

2000 Mathematics Subject Classification. Primary 17A70, 17B60.

1Support from National Science Foundation Grant #DMS–9970119 is gratefully acknowledged.

2Supported by the Spanish DGES (Pb 97-1291-C03-03) and by a grant from the Spanish Dirección General de Enseñanza Superior e Investigación Científica (Programa de Estancias de Investigadores Españoles en Centros de Investigación Extranjeros), while visiting the University of Wisconsin at Madison.

superalgebras with semisimple even part, Racine and Zelmanov [RZ] remark “That K_{10}, and hence K_{9}, is a Jordan superalgebra can be obtained by Lie methods as in [K] but a direct proof would be desirable.” Here we provide such a proof.

The Kaplansky superalgebra

Suppose F is a field of characteristic not 2. The tiny Kaplansky superalgebra ([Kp], [M]) is the 3-dimensional Jordan superalgebra K = K¯0 ⊕ K¯1, with K¯0 = F e and K¯1 = F x ⊕ F y, and with multiplication given by

(1.1)  \begin{matrix}e^{2} = e,  ex = \frac{1}{2}x = xe,  ey = \frac{1}{2}y = ye, \\ xy = e = −yx,  x^{2} = 0 = y^{2}. \end{matrix}

It can be seen using (1.1) that the even supersymmetric bilinear form ( |) defined on K by

(e | e) = \frac{1}{2},  (x | y) = 1,  and  (K¯0 | K¯1) = 0

is associative ( (ab | c) = (a | bc) for all a, b, c ∈ K). Thus,

(1.2)  (ab | c) = (−1)^{ā(b̄+c̄)}(bc | a) = (−1)^{b̄(c̄+ā)}(ca | b)

for all a, b, c ∈ K¯0 ∪ K¯1. (Our convention is that ā = i whenever a ∈ K_{i}.) Hence, since the supertrace, str(Ł_{e}) = tr_{K¯0}(Ł_{e}) − tr_{K¯1}(Ł_{e}), of the left multiplication operator Ł_{e} of e on K is 0, it follows that

Ł_{K_{i}} = {ϕ ∈ End_{F} (K)_{i} | (ϕ(u) | v) = −(−1)^{iū}(u | ϕ(v)) ∀ u, v ∈ K, and str(ϕ) = 0}

for i = ¯0, ¯1. Moreover, [Ł_{K}, Ł_{K}] = DerK = 𝔬𝔰𝔭(K) ≅ 𝔬𝔰𝔭(1, 2).

Proposition 1.3. For all u, v, w ∈ K¯0 ∪ K¯1,

(1.4)  (uv)w = (u | v)w + \frac{1}{2}(u(v | w) + (−1)^{ūv̄}v(u | w))

Proof. Observe first that the expression in (1.4) is supersymmetric in u and v. If u = v = e, then (1.4) reduces to ew = \frac{1}{2}w + (e | w)e, which is valid for all

homogeneous elements w. If u = e and v ∈ K¯1, then (1.4) becomes \frac{1}{2}vw = \frac{1}{2}(e(v |

w) + v(e | w)), which is true. Finally, if u, v ∈ K¯1, then uv = (u | v)e, and in this case, (1.4) says

(u | v)ew = (u | v)w + \frac{1}{2}(u(v | w) + (−1)^{ūv̄}v(u | w)).

For w = e this reduces to (u | v)e = (u | v)e. If instead w ∈ K¯1, then (1.4) holds because (u | v)w + (v | w)u + (w | u)v = 0, as the expression on the left is a skew-symmetric trilinear map on a 2-dimensional vector space (namely K¯1). □

Corollary 1.5. For all u, v ∈ K¯0 ∪ K¯1,

(1.6)  [Ł_{u}, Ł_{v}] = \frac{1}{2}(u(v | −) − (−1)^{ūv̄}v(u | −)) and

(1.7)  Ł_{u} ∘ Ł_{v} = (u | v) I + \frac{3}{2}(u(v | −) + (−1)^{ūv̄}v(u | −)),

where ϕ ∘ ψ = ϕψ + (−1)^{ϕ̄ψ̄}ψϕ for all homogeneous ϕ, ψ ∈ End_{F} (K).

Proof. From (1.4) we have

u(vw) − (−1)^{ūv̄}v(uw) = u(v | w) + \frac{1}{2}((u | v)w + (−1)^{ūv̄}v(u | w))

− (−1)^{ūv̄}(v(u | w) + \frac{1}{2}((v | u)w + (−1)^{ūw̄}(v | w)u))

= \frac{1}{2}(u(v | w) − (−1)^{ūv̄}v(u | w)),

which demonstrates (1.6). The verification of (1.7) is similar. □

The Jordan Superalgebra J

In this section we consider the superalgebra

J = F 1 ⊕ (K ⊗ K)

with unit element 1 and with product defined by

(2.1)  (a ⊗ b)(c ⊗ d) = (−1)^{b̄c̄}(ac ⊗ bd − \frac{3}{4}(a | c)(b | d)1)

for homogeneous elements a, b, c, d ∈ K. (All tensor products are over F unless specified otherwise.) By its definition J is a (super-)commutative superalgebra. Our aim is to show that J is a Jordan superalgebra. First we establish some properties of the multiplication.

Proposition 2.2. For homogeneous elements a, b, c, d ∈ K,

(2.3)

[Ł_{a⊗b}, Ł_{c⊗d}](1) = 0

[Ł_{a⊗b}, Ł_{c⊗d}]|_{K⊗K} = (−1)^{b̄c̄} \frac{1}{2}([Ł_{a}, Ł_{c}] ⊗ (b | d) I + (a | c) I ⊗ [Ł_{b}, Ł_{d}])

Proof. First let us remark that (ϕ⊗ψ)(u⊗v) = (−1)^{ψ̄ū}ϕ(u)⊗ψ(v) for homogeneous ϕ, ψ ∈ End_{F} (K), u, v ∈ K. The first equation above is clear by supercommutativity. As for the second, note that (2.4)

[Ł_{a⊗b}, Ł_{c⊗d}](u ⊗ v)

= (−1)d̄ ; ū(a ⊗ b)(cu ⊗ dv − \frac{3}{4}(c | u)(d | v)1)

− (−1)^{(ā+b̄)(c̄+d̄)}(−1)^{b̄ū}(c ⊗ d)(au ⊗ bv − \frac{3}{4}(a | u)(b | v)1)

= (−1)d̄ ; ū(−1)^{b̄(c̄+ū)}(a(cu) ⊗ b(dv) − \frac{3}{4}(a | cu)(b | dv)1 − \frac{3}{4}a(c | u) ⊗ b(d | v)

− (−1)^{āc̄+b̄d̄}(c(au) ⊗ d(bv) − \frac{3}{4}(c | au)(d | bv)1 − \frac{3}{4}c(a | u) ⊗ d(b | v))).

By associativity (a | cu) = (−1)^{āc̄}(c | au) and (b | dv) = (−1)^{b̄d̄}(d | bv), so that the F 1-component of (2.4) is 0. Moreover,

a(cu) ⊗ b(dv)−(−1)^{āc̄+b̄d̄}c(au) ⊗ d(bv)

= \frac{1}{2}([Ł_{a}, Ł_{c}](u) ⊗ (Ł_{b} ∘ Ł_{d})(v) + (Ł_{a} ∘ Ł_{c})(u) ⊗ [Ł_{b}, Ł_{d}](v)),

and by (1.6),

a(c | u) ⊗ b(d | v)−(−1)^{āc̄+b̄d̄}c(a | u) ⊗ d(b | v)

= \frac{1}{2}(a(c | u) − (−1)^{āc̄}c(a | u)) ⊗ (b(d | v) + (−1)^{b̄d̄}d(b | v))

+ (a(c | u) + (−1)^{āc̄}c(a | u)) ⊗ (b(d | v) − (−1)^{b̄d̄}d(b | v))

= [Ł_{a}, Ł_{c}](u) ⊗ (b(d | v) + (−1)^{b̄d̄}d(b | v))

+ (a(c | u) + (−1)^{āc̄}c(a | v)) ⊗ [Ł_{b}, Ł_{d}](v).

Therefore, it follows from (1.7) that

[Ł_{a⊗b}, Ł_{c⊗d}]|_{K⊗K} = (−1)^{b̄c̄}([Ł_{a}, Ł_{c}] ⊗ (\frac{1}{2} Ł_{b} ∘ Ł_{d} − \frac{3}{4}(b(d | −) + (−1)^{b̄d̄}d(b | −)))

+ (\frac{1}{2} Ł_{a} ∘ Ł_{c} − \frac{3}{4}(a(c | −) + (−1)^{āc̄}c(a | −))) ⊗ [Ł_{b}, Ł_{d}])

= (−1)^{b̄c̄} \frac{1}{2}([Ł_{a}, Ł_{c}] ⊗ (b | d) I + (a | c) I ⊗ [Ł_{b}, Ł_{d}]),

as desired. □

Corollary 2.5. For a, b, c, d ∈ K¯0 ∪ K¯1, [Ł_{a⊗b}, Ł_{c⊗d}] is a superderivation of J.

Proof. Because K is a Jordan superalgebra, for a, b ∈ K¯0∪K¯1, the mapping [Ł_{a}, Ł_{b}] is a superderivation of K. By the associativity of ( |), it is super-skew-symmetric relative to ( |). Consequently, from (2.3) we can conclude [Ł_{a⊗b}, Ł_{c⊗d}] is a superderivation of J. □

Theorem 2.6. J = F 1 ⊕ (K ⊗ K) is a Jordan superalgebra if char F ≠ 2.

Proof. A Jordan algebra over a field F of characteristic not 2 or 3 is characterized by commutativity and the identity ∑_{cyclic}[Ł_{uv}, Ł_{w}] = 0. Hence a Jordan superalgebra over F is characterized by supercommutativity and

∑_{supercyclic}[Ł_{uv}, Ł_{w}] = [Ł_{uv}, Ł_{w}] + (−1)^{ū(v̄+w̄)}[Ł_{vw}, Ł_{u}] + (−1)^{(ū+v̄)w̄}[Ł_{wu}, Ł_{v}] = 0.

In the supercommutative algebra J, we have from (1.2) and (2.3) that for all homogeneous a, b, c, d, u, v ∈ K,

∑_{supercyclic}[Ł_{(a⊗b)(c⊗d)}, Ł_{u⊗v}]

= ∑_{supercyclic}(−1)^{b̄c̄}[Ł_{ac⊗bd}, Ł_{u⊗v}]

= ∑_{supercyclic}(−1)^{b̄(c̄+ū)+d̄v̄} \frac{1}{2}([Ł_{ac}, Ł_{u}] ⊗ (bd | v) I + (ac | u) I ⊗ [Ł_{bd}, Ł_{v}])

= (−1)^{b̄(c̄+ū)+d̄ū} \frac{1}{2}\{( ∑_{supercyclic}[Ł_{ac}, Ł_{u}]) ⊗ (bd | v) I

+ (ac | u) I ⊗ ( ∑_{supercyclic}[Ł_{bd}, Ł_{v}])}

= 0.

Thus, J is a Jordan superalgebra whenever the field has characteristic different from 2 or 3.

If R is the ring of fractions R = S^{−1}ℤ relative to the multiplicative set of integers S = {2^{n} | n = 0, 1, … }, we could have defined K and J over R as above. (Denote the result by J_{R}.) Then J_{ℚ} = ℚ ⊗_{R} J_{R} is a Jordan superalgebra over the rationals ℚ, and so is J_{R} over R. But if F is a field having characteristic not 2, then J = F ⊗_{R} J_{R} is a Jordan superalgebra over F. □

¿From the computations and results above, it is easy to draw the following conclusions about the Jordan superalgebra J. Here we use the associator (u, v, w) = (uv)w − u(vw) in J in the statements.

Proposition 2.7. For the Jordan superalgebra J = F 1 ⊕ (K ⊗ K)

(a) (J, J, J) = K ⊗ K.

(b) J = F 1 ⊕ (J, J, J)

(c) If charF = 3, then K ⊗ K = (J, J, J) is an ideal of J.

(d) e_{1} := −\frac{1}{2}1 + 2e ⊗ e and e_{2} := \frac{3}{2}1 − 2e ⊗ e are orthogonal idempotents in J. Relative to e_{1} the Peirce spaces J^{(λ)} := {u ∈ J | e_{1}u = λu} are given by

J^{(1)} = F e_{1}, J^{(0)} = F e_{2} ⊕ (K¯1 ⊗ K¯1), J^{( \frac{1}{2} )} = (e ⊗ K¯1) ⊕ (K¯1 ⊗ e)

Thus, J¯0 = J^{(1)} ⊕ J^{(0)} and J¯1 = J^{( \frac{1}{2} )}. The space J^{(0)} is the Jordan algebra of a nondegenerate symmetric bilinear form, and e_{2} is its unit element.

Theorem 2.8.

(a) If charF ≠ 2, 3, J = F 1 ⊕ (K ⊗ K) is a simple Jordan superalgebra.

(b) If charF = 3, (J, J, J) = K ⊗ K is a simple Jordan superalgebra.

(c) DerJ = InderJ and it is isomorphic to 𝔬𝔰𝔭(1, 2) ⊕ 𝔬𝔰𝔭(1, 2).

Proof. We begin with (c). Any superderivation of J annihilates F 1 and leaves invariant (J, J, J) = K ⊗ K. Therefore, using the natural identifications and the fact that DerK = InderK ≅ 𝔬𝔰𝔭(1, 2), we have by (2.3) that

(2.9)  DerK ⊕ DerK ≅ (DerK ⊗ I) ⊕ ( I ⊗ DerK) = InderJ.

Now one may check that InderJ = DerJ as follows: First for any D ∈ (DerJ)¯0, D|_{J¯0} ∈ Der(J¯0). Since J¯0 is a direct sum of a 1-dimensional ideal J^{(1)} = F e_{1} and the Jordan algebra J^{(0)} = F e_{2}⊕(K¯1⊗K¯1) of a bilinear form, Der(J¯0) = InderJ¯0. Thus, there is a D̃ ∈ InderJ¯0 with (D − D̃)(J¯0) = 0. It follows easily that D = D̃. Now suppose E ∈ (DerJ)¯1. Then there is a Ẽ ∈ (InderJ)¯1 with E(e_{1}) = Ẽ(e_{1}) because (InderJ)¯1e_{1} = J¯1. For any z ∈ J¯1, e_{1}z = \frac{1}{2}z, so (E − Ẽ)(z) = 2(E − Ẽ)(e_{1}z) = 2e_{1}(E − Ẽ)(z). Thus, (E − Ẽ)(z) ∈ J¯0 ∩ J^{( \frac{1}{2} )} = 0, so that (E − Ẽ)(J¯1) = 0. As a consequence, E = Ẽ and (DerJ)¯1 = (InderJ)¯1.

Since DerK acts irreducibly on K, so does DerJ = InderJ on K ⊗ K = (J, J, J), which is an ideal of J if and only if the characteristic of F is 3. From this (a) and (b) follow. □

The Lie Multiplication Superalgebra of J.

To compute the Lie multiplication superalgebra of J we need some preliminaries. Let H be a unital Jordan superalgebra with a normalized trace, that is, a linear map 𝔱 : H → F such that 𝔱(1) = 1 and 𝔱((H, H, H)) = 0. Assume that (H, H, H) ≠ 0. Let H_{0} = ker 𝔱. Then for any x, y ∈ H_{0}, xy = 𝔱(xy)1+x∗y, where x∗y = xy−𝔱(xy)1. The Lie multiplication superalgebra is the Lie superalgebra spanned by Ł_{H}, which is a direct sum of the central ideal F I and the ideal 𝔏_{0}(H) = Ł_{H_{0}} ⊕ [Ł_{H_{0}}, Ł_{H_{0}}].

For simplicity, let us adopt the notation S = InderH and T = H_{0}. Then 𝔏_{0}(H) is isomorphic to the Lie superalgebra 𝔏 = S ⊕ T with multiplication determined by

(1) the natural Lie multiplication on S,

(2) the natural action of S on T, that is, [s, t] = s(t) for any s ∈ S and t ∈ T, and

(3) [t_{1}, t_{2}] = [Ł_{t_{1}}, Ł_{t_{2}}] ∈ S, for any t_{1}, t_{2} ∈ T.

Also, given a nonzero scalar ν ∈ F, let 𝔏_{ν} be the Lie superalgebra defined over 𝔏, but with the new bracket given by

(i) [x, y]^{ν} = [x, y], for x, y both in S or one in S and the other in T, (ii) [x, y]^{ν} = ν[x, y], for x, y ∈ T.

Lemma 2.10. Under the hypotheses above, if Hom_{S}(T ⊗ T, T) is spanned by u ⊗ v ↦ u ∗ v and if any automorphism of the superalgebra (T, ∗) is an isometry of the supersymmetric bilinear form (u, v) ↦ 𝔱(uv), then for any 0 ≠ μ, ν ∈ F, 𝔏_{μ} is isomorphic to 𝔏_{ν} if and only if μ^{−1}ν ∈ F^{2}.

Proof. It is enough to apply the ideas of ([BDE], proof of Prop. 4.2). We include the argument for completeness.

First, since any isomorphism 𝔏_{μ} → 𝔏_{ν} is also an isomorphism 𝔏 → 𝔏_{μ^{−1}ν}, it is enough to assume μ = 1. Suppose Φ : 𝔏 → 𝔏_{ν} is an isomorphism. Then the map T ⊗T → T defined by u⊗v ↦ Φ^{−1}(Φ(u)∗Φ(v)) is S-invariant. Consequently there exists a nonzero scalar α ∈ F such that αΦ(u) ∗ Φ(v) = Φ(u ∗ v) for any u, v ∈ T. Then ψ : T → T given by ψ(u) = αΦ(u) is an automorphism of (T, ∗) which, by hypothesis, extends by means of ψ(1) = 1 to an automorphism of the Jordan superalgebra H. Now the map Ψ : 𝔏 → 𝔏, such that Ψ(s) = ψsψ^{−1} (composition of maps) for s ∈ S, and Ψ(t) = ψ(t) for t ∈ T is an automorphism of 𝔏.

Furthermore, Φ̃ = ΦΨ^{−1} : 𝔏 → 𝔏_{ν} is an isomorphism such that Φ̃(t) = Φ(ψ^{−1}(t)) = α^{−1}t for t ∈ T. Hence, for any t_{1}, t_{2}, t_{3} ∈ T,

Φ̃([t_{1}, t_{2}]) = [Φ̃(t_{1}), Φ̃(t_{2})]^{ν} = α^{−2}ν[t_{1}, t_{2}],
α^{−1}[[t_{1}, t_{2}], t_{3}] = Φ̃([[t_{1}, t_{2}], t_{3}])
= [[Φ̃(t_{1}), Φ̃(t_{2})]^{ν}, Φ̃(t_{3})]^{ν}
= α^{−3}ν[[t_{1}, t_{2}], t_{3}] ,

so that ν = α^{−2} as required. (Note [[T, T], T] ≠ 0 because (H, H, H) ≠ 0.)

The converse is clear, since the map Φ given by Φ(s) = s for s ∈ S and Φ(t) = αt for t ∈ T provides an isomorphism 𝔏 → 𝔏_{α^{2}}. □

In order to apply the last result to our 10-dimensional Jordan superalgebra J, notice first that J is endowed with a normalized trace 𝔱 such that 𝔱(1) = 1 and 𝔱((J, J, J)) = 0, as (J, J, J) = K ⊗ K does not intersect F 1. Moreover, the associated bilinear form is given by 𝔱((a ⊗ b)(c ⊗ d)) = −(−1)^{b̄c̄} \frac{3}{4}(a | c)(b | d) for any homogeneous a, b, c, d ∈ K, because of (2.1). Then all we need is the following

Lemma 2.11. Any automorphism of (K ⊗K, ∗) is an isometry of the bilinear trace form.

Proof. Write T = J_{0} = K ⊗ K and set ⟨a ⊗ b | c ⊗ d⟩ = (−1)^{b̄c̄}(a | c)(b | d) for any homogeneous elements a, b, c, d ∈ K. Then T¯0 = F (e ⊗ e) ⊕ (K¯1 ⊗ K¯1). Suppose ẽ = e ⊗ e. Then the multiplication in T¯0 is given by

(2.12)  ẽ^{∗2} = ẽ, ẽ ∗ v = \frac{1}{4}v, v ∗ w = ⟨v | w⟩ẽ

for all v, w ∈ K¯1 ⊗ K¯1.

Let φ be an automorphism of (T, ∗) and let us first verify that φ(ẽ) = ẽ. This is clear if the characteristic is 3, because ẽ is then the unit element of T¯0. Otherwise, ẽ is the only idempotent of T¯0 such that the corresponding left multiplication operator has eigenvalue \frac{1}{4} with multiplicity 4. This is because the eigenvalues of Ł_{αẽ+v} (for v ∈ K¯1 ⊗ K¯1, α ∈ F) include \frac{α}{4} with multiplicity at least 3, as any w ∈ K¯1 ⊗ K¯1 which is orthogonal to v relative to ⟨ | ⟩ is a corresponding eigenvector. Moreover, ẽ + v is an idempotent only if v = 0.

Now, φ(K¯1⊗K¯1) ⊆ K¯1⊗K¯1 because K¯1⊗K¯1 = {v ∈ T¯0 | ẽ ∉ F v and v∗v ∈ F ẽ}. Because of (2.12), φ|_{T¯0} is an isometry of the restriction of ⟨ | ⟩ to T¯0.

Finally, (K¯1 ⊗e)∪(e⊗K¯1) = {z ∈ T¯1 | z ∗z ∈ F ẽ}, and K¯1 ⊗e and e⊗K¯1 are the only 2-dimensional subspaces contained in this union. Therefore either φ preserves K¯1 ⊗ e and e ⊗ K¯1 or it interchanges them. Since (u ⊗ e) ∗ (v ⊗ e) = \frac{1}{2}⟨u ⊗ e | v ⊗ e⟩ẽ and (e ⊗ u) ∗ (e ⊗ v) = \frac{1}{2}⟨e ⊗ u | e ⊗ v⟩ẽ for any u, v ∈ K¯1, it follows that φ|_{T¯1} is an isometry too. □

We are ready to determine the Lie multiplication superalgebra 𝔏_{0}(J) for J = F 1 ⊕ (K ⊗ K). When the underlying field F has characteristic 0, the next result is stated in [S] but without the assumption of −1 being a square in F.

Theorem 2.13. 𝔏_{0}(J)_{−1} ≅ 𝔬𝔰𝔭(K ⊕ K) ( ≅ 𝔬𝔰𝔭(2, 4)). In particular, 𝔏_{0}(J) is a form of 𝔬𝔰𝔭(2, 4) and 𝔏_{0}(J) ≅ 𝔬𝔰𝔭(2, 4) if and only if −1 is a square in F.

Proof. On any vector superspace V equipped with a supersymmetric bilinear form ( |), the transformation σ_{u,v} = u(v | −) − (−1)^{ūv̄}v(u | −) belongs to 𝔬𝔰𝔭(V) for all homogeneous u, v ∈ V. Because of (2.3) and (1.7), DerJ is embedded in 𝔬𝔰𝔭(K ⊕K) (with the natural supersymmetric bilinear form on K ⊕ K that makes this an orthogonal sum) by means of

Φ([Ł_{a⊗b}, Ł_{c⊗d}]) = (−1)^{b̄c̄} \frac{1}{4}(σ_{(a,0),(c,0)}(b | d) + (a | c)σ_{(0,b),(0,d)}) ∈ 𝔬𝔰𝔭(K ⊕ K) .

This can be extended to a linear bijection Φ : 𝔏_{0}(J)_{−1} → 𝔬𝔰𝔭(K ⊕ K) by defining

Φ(Ł_{a⊗b}) = \frac{1}{2}σ_{(a,0),(0,b)}

for a, b ∈ K. Since

[σ_{(a,0),(0,b)}, σ_{(c,0),(0,d)}]
= σ_{σ_{(a,0),(0,b)}(c,0),(0,d)} + (−1)^{(ā+b̄)c̄}σ_{(c,0),σ_{(a,0),(0,b)}(0,d)}

= −(−1)^{āb̄}(−1)^{(ā+c̄)b̄}(a | c)σ_{(0,b),(0,d)} + (−1)^{(ā+b̄)c̄}σ_{(c,0),(a,0)}(b | d)

= −(−1^{b̄c̄}((a | c)σ_{(0,b),(0,d)} + σ_{(a,0),(c,0)}(b | d)) ,

it follows that Φ is an isomorphism of Lie superalgebras. The last part is a consequence of Lemmata 2.10 and 2.11. □

The Kac Superalgebra

We may identify the simple Jordan superalgebra J with the 10-dimensional simple Jordan superalgebra K_{10} discovered by Kac [K] (see also [HK]) when the characteristic is not 2 or 3, and (J, J, J) = K ⊗ K with the 9-dimensional degenerate Kac superalgebra K_{9} if charF = 3. This realization may be described explicitly as follows. In the notation of [MZ] or [RZ], we suppose {ě, v_{1}, v_{2}, v_{3}, v_{4}, f̌, x_{1}, y_{1}, x_{2}, y_{2}} is a basis of K_{10} with (K_{10})¯0 = span_{F} {ě, v_{1}, v_{2}, v_{3}, v_{4}, f̌} and (K_{10})¯1 = span_{F} {x_{1}, y_{1}, x_{2}, y_{2}} and with multiplication given by

(3.1)  \begin{matrix}ě^{2} = ě, ěv_{i} = v_{i}, v_{1}v_{2} = 2ě = v_{3}v_{4}, f̌^{2} = f̌ \\ ěx_{j} = \frac{1}{2}x_{j} = f̌x_{j},  ěy_{j} = \frac{1}{2}y_{j} = f̌y_{j} \\ v_{1}y_{1} = x_{2}, v_{1}y_{2} = −x_{1}, v_{2}x_{1} = −y_{2}, v_{2}x_{2} = y_{1} \\ v_{3}x_{2} = x_{1}, v_{3}y_{1} = y_{3}, v_{4}x_{1} = x_{2}, v_{4}y_{2} = y_{1} \\ x_{j}y_{j} = ě − 3f̌, x_{1}x_{3} = v_{1}, x_{1}y_{2} = v_{3}, x_{2}y_{1} = v_{4}, y_{1}y_{2} = v_{2} \end{matrix}

for i = 1, 2, 3, 4 and j = 1, 2. All other products are either 0 or obtained from those in (3.1) by supercommutativity. Then a straightforward computation using (3.1) shows that the assignment

ě ↔ \frac{3}{2}1 − 2e ⊗ e = e_{2}
f̌ ↔ −\frac{1}{2}1 + 2e ⊗ e = e_{1}  x_{1} ↔ 4x ⊗ e
v_{1} ↔ −4x ⊗ x  x_{2} ↔ −4e ⊗ x
v_{2} ↔ −y ⊗ y  y_{1} ↔ −2y ⊗ e
v_{3} ↔ 2x ⊗ y  y_{2} ↔ 2e ⊗ y
v_{4} ↔ −2y ⊗ x

gives an isomorphism between K_{10} and J. Additionally, if charF = 3, then ě ↔ −2e ⊗ e, so that the isomorphism restricts to an isomorphism K_{9} ↔ K ⊗ K.

Remark. As the reader may have guessed, the actual process for deriving our results was the reverse path of the presentation above. First it was verified that 𝔤 := DerK_{10} = 𝔬𝔰𝔭(1, 2) ⊕ 𝔬𝔰𝔭(1, 2), (compare Shtern [S]), and that as module for the Lie superalgebra 𝔤, K_{10} = F 1 ⊕ (V ⊗ V), where V is the natural 3-dimensional module for 𝔬𝔰𝔭(1, 2). The first copy of 𝔬𝔰𝔭(1, 2) operates on the first tensor slot of V ⊗ V, and the second copy on the second slot. The spaces Hom_{𝔬𝔰𝔭(1,2)}(V ⊗ V, F) and Hom_{𝔬𝔰𝔭(1,2)}(V ⊗V, 𝔬𝔰𝔭(1, 2)) are 1-dimensional. By identifying V with the tiny Kaplansky superalgebra K, we may describe the generators of these spaces nicely by the maps u ⊗ v ↦ (u | v) and u ⊗ v ↦ [Ł_{u}, Ł_{v}] ∈ 𝔬𝔰𝔭(1, 2) for u, v ∈ K. From this, the multiplication in the Kac superalgebra K_{10} is almost uniquely determined in terms of K.

References

[BDE] P. Benito, C. Draper and A. Elduque, On some algebras related to simple Lie triple systems, J. Algebra 219 (1999), 234–254.

[HK] L. Hogben and V.G. Kac, Erratum: Classification of simple ℤ-graded Lie superalgebras and simple Jordan superalgebras, Comm. Algebra 5 (1977), 1375–1400, Comm. Algebra 11 (1983), 1155–1156.

[K] V.G. Kac, Classification of ℤ-graded Lie superalgebras and simple Jordan superalgebras, Comm. Algebra 5 (1977), 1375–1400.

[Kp] I. Kaplansky, Graded Jordan Algebras I, preprint.

[M] K. McCrimmon, Kaplansky superalgebras, J. Algebra 164 (1994), 656-694.

[MZ] C. Martínez and E. Zelmanov, Simple finite dimensional Jordan superalgebras of prime characteristic, J. Algebra 236 (2001), 575-629.

[RZ] M. Racine and E. Zelmanov, Classification of simple Jordan superalgebras with semisimple even part (to appear).

[S] A.S. Shtern, Representation of an exceptional Jordan superalgebra, Funktsional Anal. i Prilozhen 21 (1987), 93–94 (Russian); English transl. in Functional Anal. Appl. 21 (1987), 253–254.

Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

   E-mail address: benkart@math.wisc.edu

Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain

   E-mail address: elduque@posta.unizar.es