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Metric Lie n-algebras and double extensions

We prove a structure theorem for Lie n-algebras possessing an invariant inner product. We define the notion of a double extension of a metric Lie n-algebra by another Lie n-algebra and prove that all metric Lie n-algebras are obtained from the simple and one-dimensional ones by iterating the operations of orthogonal direct sum and double extension.


Introduction
A (finite-dimensional, real) Lie n-algebra consists of a finite-dimensional real vector space V together with a linear map Φ : Λ n V → V , denoted simply as an n-bracket, obeying a generalisation of the Jacobi identity. To define it, let us recall that an endomorphism D ∈ End V is said to be a derivation if for all x i ∈ V . Then (V, Φ) defines a Lie n-algebra if the endomorphisms ad x 1 ...x n−1 ∈ End V , defined by ad x 1 ...x n−1 y = [x 1 . . . x n−1 y], are derivations. When n = 2 this clearly agrees with the Jacobi identity of a Lie algebra. For n > 2 we will call it the n-Jacobi identity. The vector space of derivations is a Lie subalgebra of gl(V ) denoted Der V . The derivations ad x 1 ...x n−1 ∈ Der V span the ideal ad V ⊳Der V consisting of inner derivations.
From now on, whenever we write Lie n-algebra, we will assume that n > 2 unless otherwise stated. In this paper we will only work with finite-dimensional real Lie nalgebras.
Lie n-algebras were introduced by Filippov [1] and have been studied further by a number of people. We mention here only two outstanding works beyond Filippov's original paper: the pioneering work of Kasymov [2] and the PhD thesis of Ling [3]. Kasymov studied the various notions of solvability and nilpotency for Lie n-algebras, introduced the notion of representation of a Lie n-algebra and proved an Engel-type theorem and a Cartan-like criterion for solvability. Ling classified simple Lie n-algebras and proved a very useful Levitype decomposition. It is perhaps remarkable that most structural results in the theory of Lie n-algebras are actually consequences of similar results for the Lie algebra of derivations. In this sense it is to be expected that results for Lie algebras should have their analogue in the theory of Lie n-algebras; although it seems that Lie n-algebras become more and more rare as n increases, due perhaps to the fact that as n increases, the n-Jacobi identity imposes more and more conditions.
For example, over the complex numbers there is up to isomorphism a unique simple Lie n-algebra for every n > 2, of dimension n + 1 and whose n-bracket is given relative to a basis (e i ) by where a hat denotes omission. Over the reals, they are all given by attaching a sign ε i to each e i on the right-hand side of the bracket.
A class of Lie n-algebras which have appeared naturally in mathematical physics are those which possess a nondegenerate inner product which is invariant under the inner derivations. We call them metric Lie n-algebras. They seem to have arisen for the first time in work of Papadopoulos and the author [4] in the classification of maximally supersymmetric type IIB supergravity backgrounds [5], and more recently, for the case of n = 3, in the work of Bagger and Lambert [6,7] and Gustavsson [8] on a superconformal field theory for multiple M2-branes. It is this latter work which has revived the interest of part of the mathematical physics community on metric Lie n-algebras.
Metric Lie algebras are not as well understood as the simple Lie algebras; although, shy of a classification, a number of structural results are known. It is a classic result that Lie algebras possessing a positive-definite invariant inner product are reductive, whence isomorphic to an orthogonal direct sum of simple and one-dimensional Lie algebras. In lorentzian signature (i.e., index 1) there is a classification due to Medina [9]. The indecomposable lorentzian Lie algebras are constructed out of the one-dimensional Lie algebra by iterating two constructions: orthogonal direct sum and double extension. This was later extended by Medina and Revoy [10] (see also work of Stanciu and the author [11]), who showed that indecomposable metric Lie algebras are constructed by again iterating the operations of direct sum and the (generalised) double extension, using again as ingredients the simple and one-dimensional Lie algebras. This was used in [9] to construct all possible indecomposable metric Lie algebras of index 2 (i.e., signature (2, p)). Contrary to the lorentzian case, there is a certain ambiguity in this construction, which prompted Kath and Olbrich [12] to approach the classification problem for metric Lie algebras from a cohomological perspective. In particular they classified indecomposable metric Lie algebras with index 2, a result which had been announced in [13]. For more indefinite signatures, the classification problem is still largely open.
Much less is known about metric Lie n-algebras. There is a classification for euclidean [14] (see also [15]) and lorentzian [16] metric Lie n-algebras and also a classification of index-2 metric Lie 3-algebras [17]. In that paper there is also a structure theorem for metric Lie 3-algebras and a definition of double extension. In this note we will extend these results to n > 3. We prove a structure theorem for metric Lie n-algebras and in particular introduce the notion of a double extension of a metric Lie n-algebra by another Lie n-algebra.

Acknowledgments
It is a pleasure to thank Paul de Medeiros and Elena Méndez-Escobar for many entertaining and illuminating n-algebraic discussions.

Metric Lie n-algebras
We recall that a metric Lie n-algebra is a triple (V, Φ, b) consisting of a finite-dimensional real vector space V , a linear map Φ : Λ n V → V , denoted simply by an n-bracket, and a nondegenerate symmetric bilinear form b : S 2 V → R, denoted simply by −, − , subject to the n-Jacobi identity and the invariance condition of the inner product for all x i , y i ∈ V . Given two metric Lie n-algebras for all x i ∈ V i and all y i ∈ V 1 ⊕ V 2 . The resulting object is again a metric Lie n-algebra. A metric Lie n-algebra is said to be indecomposable if it is not isomorphic to an orthogonal direct sum of metric Lie n-algebras In order to classify the metric Lie n-algebras, it is clearly enough to classify the indecomposable ones. In Section 3 we will prove a structure theorem for indecomposable Lie n-algebras.

2.1.
Basic facts about Lie n-algebras. From now on let (V, Φ) be a Lie n-algebra.
We will use freely the notions of subalgebra, ideal and homomorphisms as reviewed in [16]. In particular a subalgebra for all x i ∈ V 1 . An isomorphism is a bijective homomorphism. There is a one-to-one correspondence between ideals and homomorphisms and all the standard theorems hold. In particular, intersection and sums of ideals are ideals. An ideal I ⊳ V is said to be minimal if any other ideal J ⊳ V contained in I is either 0 or I. Dually, an ideal I ⊳ V is said to be maximal if any other ideal J ⊳ V containing I is either V and I. If I ⊳ V is any ideal, we define the centraliser Z(I) of I to be the subalgebra defined by [Z(I)IV . . . V ] = 0. Taking V as an ideal of itself, we define the centre Z(V ) by the condition [Z(V )V . . . V ] = 0. A Lie n-algebra is said to be simple if it has no proper ideals and dim V > 1.
Simple Lie n-algebras have been classified.
. A simple real Lie n-algebra is isomorphic to one of the (n + 1)dimensional Lie n-algebras defined, relative to a basis e i , by where a hat denotes omission and where the ε i are signs.
It is plain to see that simple real Lie n-algebras admit invariant inner products of any signature. Indeed, the Lie n-algebra in (3) leaves invariant the diagonal inner product with entries (ε 1 , . . . , ε n+1 ).
Complementary to the notion of semisimplicity is that of solvability. As shown by Kasymov [2], there is a whole spectrum of notions of solvability for Lie n-algebras. However we will use here the original notion introduced by Filippov [1]. Let I ⊳ V be an ideal. We define inductively a sequence of ideals We say that I is solvable if I (s) = 0 for some s, and we say that V is solvable if it is solvable as an ideal of itself. If I, J ⊳ V are solvable ideals, so is their sum I + J, leading to the notion of a maximal solvable ideal Rad V , known as the radical of V . A Lie n-algebra V is said to be semisimple if Rad V = 0. Ling [3] showed that a semisimple Lie n-algebra is isomorphic to the direct sum of its simple ideals. The following result is due to Filippov [1] and can be paraphrased as saying that the radical is a characteristic ideal.
We say that a subalgebra L < V is a Levi subalgebra if V = L ⊕ Rad V as vector spaces. Ling showed that, as in the theory of Lie algebras, Lie n-algebras admit a Levi decomposition. A further result of Ling's which we shall need is the following. Let us say that a Lie n-algebra is reductive if its radical coincides with its centre: Rad V = Z(V ).
Theorem 5 ([3, Theorem 2.10]). Let V be a Lie n-algebra. Then V is reductive if and only if the Lie algebra ad V of inner derivations is semisimple. If in addition Der V = ad V , V is semisimple.
In turn this allows us to prove the following useful result. Proposition 6. Let 0 → A → B → C → 0 be an exact sequence of Lie n-algebras. If A and C are semisimple, then so is B.
Proof. Since A is semisimple, Theorem 5 says that ad A is semisimple. B is a representation of ad A, hence fully reducible. Since A is an ad A-submodule of B, we have B = A ⊕ C, where C is a complementary ad A-submodule. Since A ⊳ B is an ideal (being the kernel of a homomorphism), ad A(C) = 0, whence [A . . . AC] = 0.
The subspace C is actually a subalgebra, since the component [C . . . C] A of [C . . . C] along A is ad A-invariant by the n-Jacobi identity and the fact that C is ad A-invariant. This means that [C . . . C] A is central in A, but A is semisimple, whence it must vanish. Hence, [C . . . C] ⊂ C. Since the projection B → C maps C isomorphically to C, we see that this isomorphism is one of Lie n-algebras, hence C < B is semisimple and indeed [C . . . C] = C.
Next we show that [AC . . . C] = 0. Indeed, for c 1 , . . . , c n−1 ∈ C, the map a → [c 1 . . . c n−1 a] is a derivation of A. Since A is semisimple, it is an inner derivation. However since ad A acts trivially on C, this derivation is ad A-invariant, which means that it is central. Since ad A has trivial centre, we see that it must be zero. This shows that B = A ⊕ C is also a direct sum of ad C-modules, with A being a trivial ad C-module.
. We have seen that W 0 = A, W 1 = 0 = W n−1 and W n = C. We claim that W 1<k<n−1 = 0 as well. Indeed, W k is the image of Λ n−k A⊗Λ k C → A (since A is an ideal) under the bracket. Since the bracket is ad V -equivariant, it is in particular ad C-equivariant, but now A is a trivial ad C-module and C, being semisimple, decomposes into nontrivial irreducible ad C-modules. Therefore the only ad C-equivariant map Λ n−k A ⊗ Λ k C → A, for k ≥ 1, is the zero map.
In other words, [ACB . . . B] = 0, whence B = A ⊕ C is the direct sum of the two commuting ideals A and C. Since A and C are themselves direct sum of simple ideals, so is B.
A useful notion that we will need is that of a representation of a Lie n-algebra. A representation of Lie n-algebra V on a vector space W is a Lie n-algebra structure on the direct sum V ⊕ W satisfying the following three properties: We will often say that W is a V -module, although this is slightly misleading in the absence of a notion of a "universal enveloping algebra" for a Lie n-algebra. The second of the above conditions says that if W is a representation of V , we have a map ad V → End W from inner derivations of V to linear transformations on W . The n-Jacobi identity for V ⊕ W says that this map is a representation of the Lie algebra ad V . Viceversa, any representation ad V → End W defines a Lie n-algebra structure on V ⊕ W extending the Lie n-algebra structure of V and demanding that [V, . . . , V, W, W ] = 0. Taking W = V gives rise to the adjoint representation, whereas taking W = V * gives rise to the coadjoint representation, where if α ∈ V * then 2.2. Basic notions about metric Lie n-algebras. Let us now introduce an inner product, so that (V, Φ, b) is a metric Lie n-algebra.
If W ⊂ V is any subspace, we define (1) I ⊥ ⊳ V is also an ideal; (2) I ⊥ ⊳ Z(I); and (3) if I is minimal then I ⊥ is maximal.

Structure of metric Lie n-algebras
We now investigate the structure of metric Lie n-algebras. If a Lie n-algebra V is not simple or one-dimensional, then it has a proper ideal and hence a minimal ideal. Let I ⊳ V be a minimal ideal of a metric Lie n-algebra. Then I ∩ I ⊥ , being an ideal contained in I, is either 0 or I. In other words, minimal ideals are either nondegenerate or isotropic. If nondegenerate, V = I ⊕ I ⊥ is decomposable. Therefore if V is indecomposable, I is isotropic. Moreover, by Proposition 9 (2), I is abelian and furthermore, because I is isotropic, [IIV . . . V ] = 0.
It follows that if V is euclidean and indecomposable, it is either one-dimensional or simple, whence of the form (3) with all ε i = 1. This result, originally due to Nagy [14] (see also [15]), was conjectured in [4], albeit in the guise of a conjectural generalisation of the classical Plücker identity.
Let V be an indecomposable metric Lie n-algebra. Then V is either simple, onedimensional (provided the index of the inner product is < 2) or possesses an isotropic proper minimal ideal I which obeys [IIV . . . V ] = 0. The perpendicular ideal I ⊥ is maximal and hence by Lemma 1, U := V /I ⊥ is simple or one-dimensional, whereas by Lemma 7, W := I ⊥ /I is a metric Lie n-algebra. The inner product on V induces a nondegenerate pairing g : U ⊗ I → R. Indeed, let [u] = u + I ⊥ ∈ U and v ∈ I. Then we define g ([u], v) = u, v , which is clearly independent of the coset representative for [u]. In particular, I ∼ = U * is either one-or (n + 1)-dimensional. If the signature of the metric of W is (p, q), that of V is (p + r, q + r) where r = dim I = dim U . There are two possibilities for U: either it is one-dimensional or else it is simple. We will treat both cases separately.
3.1. U is one-dimensional. If the quotient Lie n-algebra U = V /I ⊥ is one-dimensional, so is the minimal ideal I. Let u ∈ V be such that u ∈ I ⊥ , whence its image in U generates it. Because I ∼ = U * is induced by the inner product, there is v ∈ I such that u, v = 1.
The subspace spanned by u and v is therefore nondegenerate, and hence as a vector space we have an orthogonal decomposition V = R(u, v) ⊕ W , where W is the perpendicular complement of R(u, v). It is clear that W ⊂ I ⊥ , and that I ⊥ = I ⊕ W as a vector space. Indeed, the projection , for all x i ∈ W , whence v is central. Metricity then implies that the only nonzero n-brackets take the form which defines [x 1 . . . x n−1 ] and [x 1 . . . x n ] W and where x i ∈ W . The n-Jacobi identity is equivalent to the following two conditions: (1) [x 1 . . . x n−1 ] defines a Lie (n − 1)-algebra structure on W , which leaves the inner product invariant due to the skewsymmetry of [x 1 . . . x n−1 ], x n ; and (2) [x 1 . . . x n ] W defines a metric Lie n-algebra structure on W which is invariant under the (n − 1)-algebra structure.
As we will see below, this makes V into the double extension of the metric Lie n-algebra W by the one-dimensional Lie n-algebra U.
3.2. U is simple. Consider I ⊥ as a Lie n-algebra in its own right and let R = Rad I ⊥ denote its radical. By Theorem 4, I ⊥ admits a Levi subalgebra L < I ⊥ . Since I ⊥ ⊳ V and R ⊳ I ⊥ is a characteristic ideal, R ⊳ V . Indeed, for all x i ∈ V , ad x 1 ...x n−1 is a derivation of I ⊥ (since I ⊥ ⊳ V ) and by Theorem 3, it preserves R. Let M = V /R. Notice that U = V /I ⊥ ∼ = (V /R)/(I ⊥ /R) = M/L , by the standard homomorphism theorems. Since L and U are semisimple, Proposition 6 says that so is M and moreover that M ∼ = L ⊕ U. This means that R is also the radical of V , whence M is a Levi factor of V . This discussion is summarised by the following commutative diagram with exact rows and columns: The map M → U admits a section, so that M has a subalgebra U isomorphic to U and such that M = U ⊕L. Then the vertical map V → M also admits a section, whence there is a subalgebra U < V isomorphic to U such that V = I ⊥ ⊕U (as vector space). Furthermore, the inner product on V pairs I and U nondegenerately, whence I ⊕ U is a nondegenerate subspace. Let W denote its perpendicular complement, whence V = W ⊕ I ⊕ U. Clearly I ⊥ = W ⊕ I, whence the canonical projection I ⊥ → W maps W isomorphically onto W .
Let us now write the possible n-brackets for V = W ⊕ I ⊕ U.
for 0 < k < n − 1. We notice that whence the component of V k along I agrees up to a sign with the component of V k+1 along W . In principle all such brackets occur and the only conditions apart from the metricity come from the Jacobi identity of V .
Similarly to the case when U is one-dimensional, we will interpret V as the double extension of the metric Lie n-algebra W by the simple Lie n-algebra U.
3.3. Double extensions and the structure theorem. More generally we have the following definition.
Definition 10. Let W be a metric Lie n-algebra and let U be a Lie n-algebra. Then by the double extension of W by U we mean the metric Lie n-algebra on the vector space W ⊕ U ⊕ U * with the following nonzero n-brackets subject to the Jacobi identity for V : • ].
The resulting Lie n-algebra is metric, with inner product which extends the one on W by the dual pairing between U and U * . One can also add any invariant symmetric bilinear form on U, even if degenerate.
For n = 2 this construction is due to Medina and Revoy [10], whereas for n = 3 it is due to the authors of [17].
In summary we have proved the following Theorem 11. Every indecomposable metric Lie n-algebra is either one-dimensional, simple or else it is the double extension of a metric Lie n-algebra of smaller dimension by a one-dimensional or a simple Lie n-algebra.
An easy induction argument and the fact that metric Lie n-algebras are orthogonal direct sums of their indecomposable components yields the following Corollary 12. The class of metric Lie n-algebras is generated by the simple and onedimensional Lie n-algebras under the operations of orthogonal direct sum and double extension.
It is clear that the subclass of euclidean metric Lie n-algebras is generated by the simple and one-dimensional euclidean Lie 3-algebras under orthogonal direct sum, since double extension always incurs in indefinite signature. Therefore an indecomposable euclidean metric Lie n-algebra is either one-dimensional or simple [4,14,15]. The lorentzian indecomposables admit at most one double extension by a one-dimensional Lie n-algebra and are easy to classify [18,16].