SIMPLICITY IN THE FAULKNER CONSTRUCTION
JOSÉ FIGUEROA-O’FARRILL
Abstract. We revisit the Faulkner construction of metric 3-Leibniz algebras admitting anembedding Lie (super)algebra. In the case of positive-definite signature, we relate the various notions of simplicity: of the 3-algebra, of the representation and of the embedding Lie(super)algebra. This reduces their classification to the extant classifications of simple Lie (super)algebras.

Contents
1. Introduction
2. The Faulkner construction
2.1. The real Faulkner construction
2.2. The complex Faulkner construction
2.3. The quaternionic Faulkner construction
3. Simplicity
3.1. Lie triple systems
3.2. N=6 triple systems
3.3. Quaternionic anti-Lie triple systems
3.4. Classifications
Acknowledgments
References

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1. Introduction
Starting with the pioneering proposal of Bagger and Lambert [1, 2] and Gustavsson [3]for a nonabelian superconformal field theory of M2-branes, certain kinds of ternary algebras(contained in the class of metric 3-Leibniz algebras, defined for example in [4]) are known to be
associated to three-dimensional superconformal Chern–Simons + matter theories with various
amounts of supersymmetry. The case of maximal (N =8) supersymmetry corresponds to themetric 3-Lie algebras of Filippov [5], whereas for less-than-maximal supersymmetry a numberof algebras have been shown to play a rôle, starting with the work of Bagger and Lambert[6] for the N =6 theories of Aharony, Bergman, Jafferis and Maldacena [7] and Cherkis andSämann [8] for N =2 theories. In [9] a construction was given of all the 3-algebras underlying
superconformal Chern–Simons + matter theories in three dimensions, purely in terms of Liealgebras and representation theory, by specialising a construction due to Faulkner [10]. The
mathematical literature is replete with a bewildering array of triple systems, to which someauthors have linked the 3-algebras of interest [11, 12, 13].
In particular, in a recent paper [13] Palmkvist revisits the relation between 3-algebras of
the N =6 theories and Lie superalgebras established in [9], refining the result by showing that,
Date: 30th May 2009.
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JOSÉ FIGUEROA-O’FARRILL

in the case of positive-definite signature, several notions of simplicity (or irreducibility) correspond. This is phrased—in the opinion of this author, unnecessarily—in the language ofgeneralised Jordan triple systems and the purpose of this note is to show that this refinement
can be recovered easily from the construction in [9]. In fact, as we will see, the proof that thesimplicity of the triple system corresponds to that of the embedding Lie (super)algebra is mosteasily obtained by first showing that both are separately equivalent to the irreducibility of therepresentation in the Faulkner construction. In other words, it is at the end of the day a result
in representation theory, which is precisely the language in which the Faulkner construction
allows us to phrase properties of the 3-algebras. For completeness we also treat the cases of Lieand anti-Lie triple systems, as a special case of the real and quaternionic Faulkner constructions, respectively. The latter case corresponds to the triple systems underlying (at least someof) the N =4, 5 Chern–Simons + matter theories.
This note is organised as follows. In § 2 we briefly recall the main definitions of the Faulknerconstruction of metric 3-Leibniz algebras: paying particular attention to the real orthogonal,
complex unitary and quaternionic unitary representations. In § 3 we consider the special cases of
the above constructions when the triple system admits an embedding into a Lie (super)algebra.
In those cases there are three notions of simplicity (or irreducibility): simplicity of the triple
system, irreducibility of the representation and simplicity of the embedding Lie (super)algebra
and we show how they are related. We show that in the case of positive-signature, the three
notions are essentially the same. The precise statements are contained in Theorem 1 for the
Lie triple systems, Theorem 2 for the N =6 triple systems and Theorem 3 for the anti-Lie triple
systems arising from a quaternionic unitary representation.
2. The Faulkner construction
In this section we review the Faulkner construction of 3-algebras in [9].
Let g be a real finite-dimensional Lie algebra with an ad-invariant symmetric bilinear form(−, −) and let V be a finite-dimensional faithful representation of g with dual representationV ∗ . We will let h−, −i denote the dual pairing between V and V ∗ . Transposing the g-action
defines for all v ∈ V and α ∈ V ∗ an element D(v, α) ∈ g by
(X, D(v, α)) = hX · v, αifor all X ∈ g,(1)

where the · indicates the g-action on V . Extending D bilinearly, defines a g-equivariant map
D : V ⊗ V ∗ → g, which as shown in [9] is surjective because V is faithful. The g-equivarianceof D is equivalent to the fundamental identity[D(v, α), D(w, β)] = D(D(v, α) · w, β) + D(w, D(v, α) · β) ,(2)

for all v, w ∈ V and α, β ∈ V ∗ , where the dual action D(v, α) · β is defined by
hw, D(v, α) · βi = − hD(v, α) · w, βi .

(3)The map D defines in turn a trilinear product
V ×V∗×V →V(v, α, w) 7→ D(v, α) · w .

(4)

A special case of the Faulkner construction is where V is a faithful unitary representationof g. This means that V is a real, complex or quaternionic representation of g possessing ag-invariant real symmetric, complex hermitian or quaternionic hermitian inner product, respectively. This gives rise, respectively, to a real orthogonal, complex unitary or quaternionicSIMPLICITY IN THE FAULKNER CONSTRUCTION

3

unitary representation of g. As shown in [9], the real case corresponds precisely to the metric 3Leibniz algebras constructed by Cherkis and Sämann in [8], whereas the complex case contains
as a special class the N =6 3-algebras of [6].
2.1. The real Faulkner construction. Let us first consider the case of V, h−, −i a real innerproduct space admitting a faithful orthogonal action of a real metric Lie algebra g, (−, −). The
inner product on V sets up an isomorphism [ : V → V ∗ of g-modules, defined by v [ = hv, −i,with inverse ] : V ∗ → V . The map D : V ⊗ V ∗ → g defined by equation (1) induces a map
D : V ⊗ V → g, by D(v, w) = D(v, w[ ). In other words, for all v, w ∈ V and X ∈ g, we have(D(v, w), X) = hX · v, wi .

(5)

It follows from the g-invariance of the inner product that
(D(v, w), X) = hX · v, wi = − hX · w, vi = − (D(w, v), X) ,(6)

whenceD(v, w) = −D(w, v) .
Using D we can define a 3-bracket on V by

(7)[u, v, w] := D(u, v) · w ,(8)

for all u, v, w ∈ V . The resulting 3-Leibniz algebra, which appeared originally in [10] but more
recently in [8] in the context of superconformal Chern–Simons + matter theories, satisfies thefollowing axioms for all x, y, z, v, w ∈ V :(a) the orthogonality condition
h[x, y, z], wi = − hz, [x, y, w]i ;(9)h[x, y, z], wi = h[z, w, x], yi ;(10)(b) the symmetry condition(c) and the fundamental identity[x, y, [v, w, z]] − [v, w, [x, y, z]] = [[x, y, v], w, z] + [v, [x, y, w], z] .

(11)

It follows from the orthogonality and symmetry conditions that [x, y, z] = −[y, x, z] for all
x, y, z ∈ W , which is nothing but equation (7). A special case are the algebras appearing
in the maximally supersymmetric N = 8 theory of Bagger–Lambert [1, 2] and Gustavsson[3], for which the 3-bracket is totally skewsymmetric. Another special case of these 3-Leibniz
algebras corresponds to metric Lie triple systems, for which the 3-bracket obeys [x, y, z] +[y, z, x] + [z, x, y] = 0. Metric Lie triple systems are characterised by the fact that they embed
into g ⊕ V as a real metric Z2 -graded Lie algebra and are in one-to-one correspondence withpseudoriemannian symmetric spaces.
2.2. The complex Faulkner construction. Let V, h be a complex unitary representationof a real metric Lie algebra g, (−, −). In our somewhat odd conventions, the hermitian innerproduct h is complex linear in the left entry and complex antilinear in the right one. The
hermitian inner product h sets up a complex antilinear isomorphism V → V ∗ . Indeed, we may
parametrise V ∗ by elements of V via v 7→ v [ := h(−, v). The Faulkner map (1) now defines asesquilinear map D : V × V → gC to the complexification of g, defined by(D(v, w), X) = h(X · v, w) ,(12)

for all v, w ∈ V and X ∈ gC . We have extended the action of g on V to gC in the naive way:
(X + iY ) · v = X · v + iY · v, and we have extended the inner product on g to gC complex
bilinearly, turning gC into a complex metric Lie algebra acting faithfully on V . As shown in4

JOSÉ FIGUEROA-O’FARRILL[9], this means that D is surjective. Of course, being complex, gC cannot preserve h — instead
it obeys h(X · v, w) = −h(v, X · w). Complex conjugating (12), we see that this means that
D(u, v) = −D(v, u). We now define a sesquibilinear 3-bracket V × V × V → V , complex linear
in the first two entries and complex antilinear in the third by[u, v; w] := D(v, w) · u ,(13)

for all u, v, w ∈ V . The 3-bracket enjoys the following properties:(a) the unitarity condition
h([x, v; w], y) = h(x, [y, w; v]) ;(14)h([x, v; w], y) = h([v, x; y], w) ;(15)(b) the symmetry condition(c) and the fundamental identity[[z, v; w], x; y] − [[z, x; y], v; w] − [z, [v, x; y]; w] + [z, v; [w, y; x]] = 0 .

(16)

The N =6 3-algebra of [6] obeys in addition the skewsymmetry condition[x, y; z] = −[y, x; z] .

(17)

As proved in [9, Theorem 22], this condition can be understood as the 111 component of theJacobi identity of a 3-graded complex Lie superalgebra k = V ⊕ gC ⊕ V ∗ , in degrees −1, 0, 1
respectively. The Lie bracket of k extends the Lie bracket of gC and the actions of gC on V andV ∗ by defining the [V V ∗ ] bracket by [v, w[ ] = D(v, w).
2.3. The quaternionic Faulkner construction. Although one could work quaternionically,
we will think of quaternionic representations as complex representations with a quaternionic
structure map. Explicitly, by a quaternionic unitary representation of a Lie algebra g, we shallmean a complex unitary representation V, h together with a g-invariant complex antilinear map
J : V → V , obeying J 2 = −1 and which is compatible with the hermitian structure h in thesense that h(Ju, Jv) = h(v, u). Using h and J we may define a complex bilinear symplectic
structureω(u, v) = h(u, Jv) .
(18)
This allows us to transpose the g-action and define a bilinear map Dω : V × V → gC by(Dω (u, v), X) = ω(X · u, v) ,(19)

which in turn defines a trilinear 3-bracket[u, v, w] := Dω (u, v) · w .

(20)

for all u, v, w ∈ V . In terms of the sesquilinear map D : V × V → gC defined in (12), we havethat Dω (u, v) = D(u, Jv), whence [u, v, w] = Dω (u, v) · w = D(u, Jv) · w = [w, u; Jv]. As shownin [9], the resulting complex triple system enjoys the following properties:(a) the symplecticity condition[u, v, w] = [v, u, w] ;(21)

ω([x, y, z], w) = ω([z, w, x], y) ,(22)(b) the symmetry condition(c) the fundamental identity[v, x, [w, y, z]] − [w, y, [v, x, z]] − [[v, x, w], y, z] − [w, [v, x, y], z] = 0 ,(23)SIMPLICITY IN THE FAULKNER CONSTRUCTION

5(d) and the quaternionic condition
J[x, y, z] = [Jx, Jy, Jz] .

(24)

A particular case of these 3-algebras are those for which the 3-bracket obeys the cyclicity
condition[x, y, z] + [y, z, x] + [z, x, y] = 0 .
(25)
Such 3-algebras are known as anti-Lie triple systems. Anti-Lie triple systems admit anembedding in a complex Lie superalgebra in such a way that the 3-bracket is given by a nested
Lie bracket. Indeed, let k = gC ⊕ V be 2-graded by declaring gC to have degree 0 and V tohave degree 1. We give k the structure of a Lie superalgebra by extending the Lie bracket on
gC and the action of gC on V as follows[v, w] = Dω (v, w) ∈ gC .

(26)

Because of the symmetry of Dω , we see that this is indeed the Lie bracket of a Lie super algebra.
The only component of the Jacobi identity which is in question is the 111 component, but thisis precisely the cyclicity condition (25).
3. Simplicity
We have seen above that there are special cases of the Faulkner construction where the triple
product is given by nested Lie brackets in an embedding graded Lie (super)algebra. In the realcase, the Lie triple systems embed in a 2-graded Lie algebra; in the complex case, the N =6
triple systems embed in a 3-graded Lie superalgebra; and in the quaternionic case, the anti-Lie
triple systems embed in a Lie superalgebra.
In all these cases we have three different notions of simplicity and it makes sense that theyshould be related. First we have a notion of simplicity of the triple system, by which we mean
the absence of proper ideals. Each triple system has an obvious notion of morphism and ideals
are defined as kernels of morphisms. We also have a notion of simplicity of the representation V ,by which we mean irreducibility or, more precisely, indecomposability, with which it agrees in
the case of positive-definite signature. Finally we have a notion of simplicity of the embedding
Lie (super)algebra, by which we mean again the absence of proper ideals. As we will now show,
in the case where the inner product on V has positive-definite signature, these different notionsof simplicity agree, with a minor exception in the case of the Lie triple systems (see below).
3.1. Lie triple systems. Let V, h−, −i be a real faithful orthogonal representation of the real
metric Lie algebra g, (−, −). Let D : V × V → g be defined by (5) and the 3-bracket by (8). We
will assume that the 3-bracket defines a Lie triple system, so that in addition to the unitarity(9), symmetry (10) and fundamental identity (11), it also obeys the cyclicity condition[x, y, z] + [y, z, x] + [z, x, y] = 0 .

(27)

Let k = g ⊕ V be a 2-graded vector space with g and V having degrees 0 and 1, respectively.
We define on k a 2-graded Lie algebra structure extending the Lie bracket on g by declaring
that, for all x, y ∈ V and X ∈ g,[x, y] = D(x, y)

and[X, z] = X · z .

(28)

The skewsymmetry condition (7) says that this is a Lie algebra and not a Lie superalgebra.
The Jacobi identities break up into the usual four homogeneous components: the 000 component is zero by virtue of g being a Lie algebra, the 001 component by virtue of V being a
representation, the 011 component by virtue of the fundamental identity (11), rephrased as
the g-equivariance of the map D, and the 111 component is precisely the Lie triple system

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condition (27). Furthermore k is a metric 2-graded Lie algebra by declaring g and V to beperpendicular and using the given inner products (−, −) and h−, −i on g and V , respectively.
The ad-invariance of the inner product is built into the hypothesis of the construction.
We will assume that the inner product h−, −i on V is positive-definite. This does notnecessarily imply that the one on g is positive-definite; although since g is reductive, the innerproduct is arbitrary on the centre but a multiple (not necessarily of a definite sign) of theKilling forms of each of the simple ideals.
Let V, W be two Lie triple systems and let ϕ : V → W be a morphism. This means that for
all x, y, z ∈ V , [ϕx, ϕy, ϕz] = ϕ[x, y, z]. Then U = ker ϕ is an ideal of the Lie triple system,
which means that[U V V ] ⊂ U
and[V V U ] ⊂ U ,(29)
in the obvious notation.
Theorem 1. Let V be a positive-definite metric Lie triple system, let g be the image of theFaulkner map D and let k = g ⊕ V denote its embedding 2-graded Lie algebra. The following
are equivalent:(1) V is irreducible as a g-module,(2) V is simple (or one-dimensional) as a Lie triple system,(3) k is a simple (or one-dimensional) 2-graded Lie algebra.
Proof. The second and third conditions can be paraphrased as saying that V has no proper
ideals and k has no proper homogeneous ideals, since this is what it means to be simple as a
2-graded Lie algebra. This means that they are simple or one-dimensional. We will prove thecontrapositive of each of the statements except the last, which shall be proved directly.
(1) =⇒ (2). Suppose that U C V is a proper ideal of the Lie triple system. Then the second
equation in (29) says that U is a proper g-submodule of V .
(3) =⇒ (1). Suppose that W < V is a proper g-submodule of V , then I = [W W ] ⊕ W
is a proper ideal of k. Indeed, we first notice that by the positive-definiteness assumption
V = W ⊕ W ⊥ , for W ⊥ the perpendicular complement. We then observe that [W W ⊥ ] = 0,
since for all w ∈ W , x ∈ W ⊥ and X ∈ g, ([wx], X) = hX · w, xi, but X · w ∈ W since it
is a submodule and hence the inner product with x is zero. This says that [W ⊥ I] = 0. By
construction [W I] ⊂ I and because W is a g-submodule, [gI] ⊂ I.
(2) =⇒ (1). Suppose again that W < V is a proper g-submodule of V . Then W C V is aproper ideal of V as a Lie triple system. For this we show that [V V W ] ⊂ W and [W V V ] ⊂ W .
The former is precisely the fact that W < V is a g-submodule. For the latter, notice that
[W W ⊥ V ] = 0 because [W W ⊥ ] = 0, which we proved above. Therefore [W V V ] = [W W V ].
Now by cyclicity, [W W V ] ⊂ [V W W ] ⊂ W , since W < V is a g-submodule.
(1) =⇒ (3). Let I = I0 ⊕ I1 C k be a homogeneous ideal. Then [gI1 ] ⊂ I1 says that I1 ⊂ V
is an g-submodule. Since V is irreducible, I1 = 0 or I1 = V . In the former case, I = I0 and[I0 V ] = 0, but g acts faithfully on V , whence I0 = 0 and hence I = 0. In the latter case, I1 = V
then [V V ] ⊂ I0 , but [V V ] = g, whence I0 = g and I = k. So k has no proper homogeneous
ideals.

The theorem can be strengthened by substituting condition (3) in the statement of thetheorem with
(3)0 k is a simple (or one-dimensional) Lie algebra or else V ∼= g (as g-modules) and g is a
simple (or one-dimensional) Lie algebra.
The distinction is that in the case where V ∼= g, the diagonal embedding g ,→ g ⊕ g defines a
non-homogeneous proper ideal of k. The embedding Lie algebra g is then rendered isomorphic

SIMPLICITY IN THE FAULKNER CONSTRUCTION

7

to g ⊕ g as a Lie algebra without 2-grading, with the ideal corresponding to one of the twocopies of g.
Proof that (1) =⇒ (3)0 . We notice that by the theorem it is enough to show that an ideal ishomogeneous, to conclude that it cannot be proper. This will be used implicitly in order tosimplify the exposition. Let I C k be an ideal, not necessarily homogeneous. Let W = I ∩ V .
It is a g-submodule of V , whence it is either 0 or V , since V is irreducible. If W = V , thenI = I0 ⊕ V is homogeneous. If W = 0 then I is transversal to V . Let I¯ denote the projection of
I onto V . The projection is g-equivariant, hence I¯ < V is a g-submodule. Again irreducibility
says that either I¯ = 0, in which case I ⊂ g is homogeneous, or else I¯ = V . In this latter case,
I = I0 ⊕ Γ, where Γ is the graph of a linear map ϕ : V → g. Since I is an ideal, in particular[gI] ⊂ I, whence in particular [gI0 ] ⊂ I0 , and hence I0 is an ideal of g. Since g is reductive,
there’s a complementary ideal I0⊥ and by suitably redefining ϕ, we can assume that ϕ : V → I0⊥ .
Again since I is an ideal, in particular [gΓ] ⊂ Γ, which is equivalent to the g-equivariance of ϕ.
This means that ker ϕ < V is a g-submodule. Again irreducibility says that either ker ϕ = V ,in which case ϕ = 0 and hence Γ = 0, whence I = I0 is homogeneous, or else ker ϕ = 0 and ϕis one-to-one. Its image is thus a g-submodule of I0⊥ ; that is, an ideal of g. Since g is reductive,
we have I0⊥ = im ϕ ⊕ J0 , for some ideal J0 C g. In summary, I = I0 ⊕ Γ and g = I0 ⊕ im ϕ ⊕ J0 .
Now, since I is an ideal, [V I] ⊂ I, which in particular implies that [V I0 ] ⊂ I ∩ V = 0. Since V
is a faithful g-module, this means I0 = 0. Similarly, [V I] ⊂ Γ, which unpacks to the following:that for all v, w ∈ V ,[v, w + ϕw] = [v, w] + [v, ϕ(w)] ∈ Γ =⇒ [v, w] = ϕ[v, ϕw] ,which says that g = im ϕ, or that J0 = 0. In other words, ϕ : V → g is an isomorphismof g-modules. Since V is irreducible, g is irreducible as an adjoint module, whence g has noproper ideals: it is thus simple or one-dimensional. In summary, k = g0 ⊕ g1 , as a vector
space, where both g0 and g1 are isomorphic to g, with subscripts indicating the degrees. The
Lie bracket is 2-graded. Explicitly, if we let Xa , for a = 0, 1, denote the image of X ∈ g
in ga , then [Xa , Yb ] = [X, Y ]a+b , where subscript addition is taken modulo 2. The ideal I ={X0 + X1 |X ∈ g} is the diagonal embedding of g. Ignoring the 2-grading, k = I ⊕J, where both
I and J = {X0 − X1 |X ∈ g} are now commuting ideals isomorphic to g as Lie algebras.

Of course these results are classical. They follow from the classification of irreducible riemannian symmetric spaces, which is equivalent to the classification of simple, positive-definite
Lie triple systems. We include them here because of completeness and in order to illustrate ina simpler case the less familiar complex and quaternionic cases below.
3.2. N=6 triple systems. Let V, h be a faithful complex unitary representation of a real
metric Lie algebra g, (−, −) giving rise to a sesquilinear Faulkner map D : V × V → gC as
in (12) and hence to 3-bracket [x, y; z] := D(y, z) · x as in (13). Assume furthermore that inaddition to the unitarity (14), symmetry (15) and fundamental (16) identities, it also satisfies
the skewsymmetry condition (17). We call such 3-algebras an N=6 triple system. Similarly
to the case of Lie triple systems, an N=6 triple system embeds in a 3-graded Lie superalgebrain such a way that the 3-bracket is giving by nesting the Lie bracket. Indeed, on the complex
vector space k = V ⊕ gC ⊕ V ∗ the Lie bracket on gC together with[u, v [ ] = D(u, v),[X, u] = X · u

and[X, v [ ] = X · v

[

(30)

define the structure of a 3-graded Lie superalgebra, with V, gC , V ∗ having degrees −1, 0, 1
respectively. The Jacobi identity breaks up into 5 homogeneous components: −1 − 11, −100,−101, 000, 001, −111, all other components being zero automatically because k has no elements

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JOSÉ FIGUEROA-O’FARRILL

of degree d with |d| > 1. The 000 component is the Jacobi identity for gC , the 001 and −100
components vanish by virtue of V ∗ and V being gC -modules and the −101 component vanishes
because D is gC -equivariant, which is the content of the fundamental identity. The remaining
components vanish by the skewsymmetry condition (17), whose sign is responsible for thecorrect Jacobi identity for a Lie super algebra, as opposed to a Lie algebra. Furthermore kadmits an ad-invariant complex bilinear inner product extending the one on gC by declaring gC
perpendicular to V ⊕ V ∗ and defining(u, v [ ) = h(u, v) = −(v [ , u) .

(31)

The ad-invariance of this inner product is built into the Faulkner construction, as explained in[9]. We will assume in what follows that h is positive-definite.
A morphism ϕ : V → W of N=6 triple systems is a complex linear map obeying [ϕu, ϕv; ϕw] =ϕ[u, v; w] and h(ϕu, ϕv) = h(u, v); although this latter identity plays no rôle in the following
definition. The kernel of ϕ defines the notion of an ideal for an N=6 triple system. In otherwords, an ideal is a complex subspace U < V such that[U V V ] ⊂ U

and[V V U ] ⊂ U .

(32)

The first condition says that U is a gC -submodule of V , whereas in positive-definite signature,
the second condition is a consequence of the first as we will see below.
Analogously to Theorem 1, we have the following result, a version of which is stated in [13].
Theorem 2. Let V be a positive-definite N=6 triple system, let g be the Faulkner Lie algebra
and let k = V ⊕ gC ⊕ V ∗ denote its embedding 3-graded Lie superalgebra. The following are
equivalent:(1) V is irreducible as a g-module,(2) V is simple (or one-dimensional) as an N=6 triple system,(3) k is a simple (or one-dimensional) 3-graded Lie superalgebra.
Proof. Again, the second and third conditions can be paraphrased as saying that V has noproper ideals and k has no proper homogeneous ideals, since this is what it means to be simple
as a 3-graded Lie superalgebra. This means that they are simple or one-dimensional. We will
prove the contrapositive of each of the statements except the last.
(2) =⇒ (1). Suppose that V is reducible as a g-module and let U < V be a proper
g-submodule. Its perpendicular complement
U ⊥ = {v ∈ V |h(u, v) = 0 ∀u ∈ U }relative to the positive-definite hermitian inner product is also a submodule and V = U ⊕ U ⊥ .
Then this is also a decomposition of the triple system V into as a direct sum of perpendicular
ideals. The first observation is that D(U, U ⊥ ) = 0, in the obvious notation. Indeed, let u ∈ U
and v ∈ U ⊥ , then for any X ∈ gC , we have (D(u, v), X) = h(X · u, v). But, since U is asubmodule, X · u ∈ U and hence is perpendicular to v. This means that [V U U ⊥ ] = 0 andalso by complex conjugation, [V U ⊥ U ] = 0. Using the skewsymmetry condition (17), it alsomeans that [U ⊥ V U ] = 0 and [U V U ⊥ ] = 0. Hence the only nonzero 3-brackets are [U U U ] and[U ⊥ U ⊥ U ⊥ ]. Furthermore [U U U ] ⊂ U , since U is a submodule, and similarly for U ⊥ .
(1) =⇒ (2). Let V = U ⊕ U ⊥ be a decomposition of the triple system V into perpendicularproper ideals. This means that the only 3-brackets which are nonzero are [U U U ] ⊂ U and[U ⊥ U ⊥ U ⊥ ] ⊂ U ⊥ . In particular, [U V V ] ⊂ U , which says that U < V is a proper submodule.
(3) =⇒ (1). As above let V = U ⊕ U ⊥ be a reducible g-module with proper submodules U
and U ⊥ . Define
U ∗ = {u[ | u ∈ U } < V ∗ ,SIMPLICITY IN THE FAULKNER CONSTRUCTION

9which obeys (U ∗ )⊥ = (U ⊥ )∗ , where this latter space is defined in a similar way. In terms of theembedding Lie superalgebra, the vanishing of the 3-brackets above mean that [U (U ⊥ )∗ ] = 0 and[U ⊥ U ∗ ] = 0 and also that [[U U ∗ ]U ⊥ ], [[U U ∗ ](U ⊥ )∗ ], [[U ⊥ (U ⊥ )∗ ]U ] and [[U ⊥ (U ⊥ )∗ ]U ∗ ] all vanish.
This then implies that U ⊕ [U U ∗ ] ⊕ U ∗ is a homogeneous ideal of k. In fact, its perpendicularcomplement in k is U ⊥ ⊕ [U ⊥ (U ⊥ )∗ ] ⊕ (U ⊥ )∗ .
(1) =⇒ (3). Let I = I−1 ⊕ I0 ⊕ I1 C k be a homogeneous ideal. Then in particular [gC I] ⊂ I,
whence I±1 are submodules of V and V ∗ , respectively. Since V and V ∗ are irreducible, thismeans that either I1 = 0 or I1 = V ∗ and similarly that either I−1 = 0 or I−1 = V . Thus there
are four cases to consider:(i) I = I0 . Then [V I] = 0 since I has no piece of degree −1, but since V is a faithfulrepresentation, I = 0.
(ii) I = I0 ⊕ V ∗ . Again [V I0 ] = 0 and by faithfulness I0 = 0, so that I = V ∗ , but this is not
an ideal because [V V ∗] 6= 0.
(iii) I = V ⊕ I0 . Now [V ∗ I0 ] = 0 since I has no piece of degree 1. Since gC acts faithfully on
V ∗ , this means that I0 = 0, whence I = V , which is not an ideal.
(iv) I = V ⊕ I0 ⊕ V ∗ . [V V ∗] ⊂ I0 since I is an ideal, but also [V V ∗] = gC , whence I0 = gC
and I = k.
In summary, k has no proper homogeneous ideals.

As in the case of Lie triple systems, we may strengthen the theorem by substituting condition(3) in the statement of the theorem with
(3)0 k is a simple (or one-dimensional) Lie superalgebra.
In other words, in this case k has no proper ideals of any kind.
Proof that (1) =⇒ (3)0 . Again, the theorem allows us to discard any ideals which are homogeneous, since they cannot be proper. Let I = I0 ⊕ J be an ideal of k, with I0 ⊂ gC andJ ⊂ V ⊕ V ∗ . Then I0 C gC is an ideal and J ⊂ V ⊕ V ∗ is an gC -submodule, whence so are J ∩ V
and J ∩ V ∗ . Since V (and hence V ∗ ) are irreducible, these cannot be proper submodules. Thismeans that we have again four cases to treat:(i) J ∩ V = V and J ∩ V ∗ = V ∗ . Then J = V ⊕ V ∗ , so that I is homogeneous.
(ii) J ∩ V = V but J ∩ V ∗ = 0. Then J = V and again I is homogeneous.
(iii) J ∩ V = 0 but J ∩ V ∗ = V ∗ . Then J = V ∗ and again I is homogeneous.
(iv) J ∩ V = 0 and J ∩ V ∗ = 0. This means that J is transversal to both V and V ∗ . Let
π : J → V denote the projection of J onto V along V ∗ . Since π is gC -equivariant, theimage π(J) ⊂ V is a submodule. Hence it is either 0 or V . If π(J) = 0, then J ⊂ V ∗ ,but since J ∩ V ∗ = 0, this shows that J = 0 and I is again homogeneous. Finally, if
π(J) = V , then J, being a gC -submodule is the graph
J = {v + ϕv|v ∈ V }of a gC -equivariant linear map ϕ : V → V ∗ . Since I is an ideal, in particular [V ∗ I] ⊂ I,
whence [V ∗ J] ⊂ I0 , but [V ∗ J] = [V ∗ V ] = gC , whence I0 = gC . But then the ideal
condition implies that [V ⊕ V ∗ , I0 ] ⊂ J, which is absurd since [V ⊕ V ∗ , gC ] = V ⊕ V ∗ .
Therefore there is no such ideal.

3.3. Quaternionic anti-Lie triple systems. Let V, h, J be a quaternionic anti-Lie triple
system. Recall that we think of it as a complex anti-Lie triple system with a compatible
quaternionic structure. This means that the complex trilinear 3-bracket on V satisfies theconditions (21), (22), (23), (24) and (25), where ω is the complex symplectic structure defined

10

JOSÉ FIGUEROA-O’FARRILL

by h and J by (18). As explained in Section 2.3, we may embed the anti-Lie triple system intoa metric complex Lie superalgebra k = gC ⊕ V , where gC is the complexification of the Faulkner
Lie algebra g. In this section we will assume that the hermitian structure h is positive-definite.
Let ϕ : V → W be a morphism of quaternionic anti-Lie triple systems. This means that ϕ is acomplex linear map, commuting with J, and preserving the 3-bracket: ϕ[u, v, w] = [ϕu, ϕv, ϕw].
Then the kernel of ϕ is an ideal and all ideals are of this form. This means that an ideal U ⊂ V
is a complex subspace, stable under J, and obeying[V V U ] ⊂ U

and[U V V ] ⊂ U .

(33)

The first of the above equations says that U < V is a (quaternionic) g-submodule of V .
Theorem 3. Let V be a positive-definite quaternionic anti-Lie triple system, let g be the corresponding Faulkner Lie algebra and let k = gC ⊕ V denote its embedding Lie superalgebra. The
following are equivalent:(1) V is irreducible as a g-module,(2) V is simple as a triple system,(3) k is a simple Lie superalgebra.
Let us remark that V , being quaternionic, cannot be one-dimensional, hence neither is k.
This means that the absence of proper ideals does imply simplicity.
Proof. We will prove the contrapositive of each of the statements, except the last.
(3) =⇒ (1). Suppose that V = W ⊕ W ⊥ is reducible as g-module, where W is a proper
submodule and W ⊥ is its perpendicular complement. It does not matter whether we define
perpendicularity using h or ω, since W is quaternionic by assumption and hence stable underJ, hence the two notions agree. This implies that Dω (W, W ⊥ ) = 0, in the obvious notation.
Indeed, if w ∈ W and v ∈ W ⊥ , then for all X ∈ gC , (Dω (w, v), X) = ω(X · w, v). Now, since
W is a submodule, X · w ∈ W , which is then perpendicular to v ∈ W ⊥ . In the embedding
Lie superalgebra, this means that [W W ⊥ ] = 0, which in turn implies that [W W ] ⊕ W is aproper ideal of k. Indeed, since W is a submodule, [gC W ] ⊂ W and hence [gC [W W ]] ⊂ [W W ].
Similarly, [V W ] ⊂ [W W ], since [W ⊥ W ] = 0. Finally [V [W W ]] ⊂ [[V W ]W ] ⊂ W , since W is asubmodule.
(2) =⇒ (1). Again let V = W ⊕ W ⊥ , with W and W ⊥ as above. Then [V V W ] ⊂ W ,since W is a submodule. Now notice that Dω (W, W ⊥ ) = 0 implies that [W W ⊥ V ] = 0, whence[W V V ] = [W W V ]. Now by cyclicity, [W W V ] ⊂ [V W W ] ⊂ W . Therefore W is a proper idealof V as an anti-Lie triple system.
(1) =⇒ (2). Let V = W ⊕ W ⊥ be a decomposition of V as an anti-Lie triple system
into perpendicular proper ideals. Then in particular, [V V W ] ⊂ W and hence W is a proper
submodule.
(1) =⇒ (3). Let V be an irreducible module and suppose that I = I0 ⊕ I1 is an ideal of the
Lie superalgebra k. Then [gC I1 ] ⊂ I1 says that I1 < V is a submodule. Since V is irreducible,
then I1 = 0 or else I1 = V . If I1 = 0, then I = I0 . But then [V I0 ] = 0, but since gC actsfaithfully on V , I0 = 0, whence I = 0. Alternatively, if I1 = V , then I = I0 ⊕ V . But [V V ] ⊂ I0
says that I0 = gC , whence I = k. In summary, k has no proper ideals.

3.4. Classifications. The above theorems reduce the classification of positive-definite simple
Lie, N=6, and quaternionic anti-Lie triple systems to the well-known classification of simple
Lie (super)algebras. None of these classifications are new, of course. In the positive-definite
case, the classification of simple Lie triple systems is precisely the classification of irreducibleriemannian symmetric spaces, which goes back to Élie Cartan and is described, for example, inSIMPLICITY IN THE FAULKNER CONSTRUCTION

11[14]. The classification of simple Lie triple systems, without any metricity assumption, is due
to Lister [15]. The classification of simple positive-definite N=6 triple systems follows from theclassification of simple Lie superalgebras, which is sketched in [16] and perhaps more accessibly
also in [17]. As recapitulated in [13] and mentioned already in [9], the corresponding simple Liesuperalgebras are A(n, m) and C(n + 1), agreeing with the classification in [18] and remarksin [19, 20]. Finally, the simple positive-definite quaternionic anti-Lie triple systems are in
one-to-one correspondence with the simple Lie superalgebras whose fermionic subspace admits
a quaternionic structure. These are given by the basic classical Lie superalgebras A(m, n),C(n + 1), B(m, n), D(m, n), F (4), G(3) and D(2, 1; α), in agreement with the results of [21]obtained via conformal limits of gauged three-dimensional supergravities.
Acknowledgments
This work was supported by World Premier International Research Center Initiative (WPI
Initiative), MEXT, Japan. It was done while on sabbatical at the Institute for the Physics andMathematics of the Universe (IPMU) at the University of Tokyo, whose hospitality and support
is hereby acknowledged. It is a pleasure to thank IPMU’s director Hitoshi Murayama who madethis visit possible as well as the Leverhulme Trust for the award of a Research Fellowship freeing
me from my teaching and administrative duties at the University of Edinburgh. Last, but notleast, I would like to thank my collaborators Paul de Medeiros and Elena Méndez-Escobar forthe continuing 3-algebraic discussions.
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Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa,
Chiba 277-8586, Japan
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of
Edinburgh, Edinburgh EH9 3JZ, UK
E-mail address: Jose.Figueroa@ipmu.jp