On the structure of k-Lie algebras

We show that the structure constants of k-Lie algebras, k > 3, with a positive definite metric are the sum of the volume forms of orthogonal k-planes. This generalizes the result for k = 3 in arXiv:0804.2662 and arXiv:0804.3078, and confirms a conjecture in math/0211170. Metric k-Lie algebras, k > 2, have emerged in the investigation of maximally supersymmetric supergravity solutions in [1]. In particular, one finds that the Killing spinor equations of IIB supergravity require that the 5-form field strength of maximally supersymmetric backgrounds to obey the Jacobi identity of the structure constants of a 10-dimensional 4-Lie algebra. So the classification of such backgrounds relies on the understanding of the solutions of the associated Jacobi identity. This was achieved in [2], after a lengthy computation, for the particular case that applies in IIB supergravity and some other related cases. In the same paper, it was realized that k-Lie algebras, k > 2, are highly constrained. In particular for all cases investigated it was found that the structure constants are the sum of the volume forms of orthogonal planes. So it was conjectured that this is likely to be the case for all metric k-Lie algebras, k > 2, with Euclidean signature metrics. More recently, 3-Lie algebras have appeared in an attempt to construct a multiple M2-brane theory [3, 4, 5]. This followed earlier attempts to construct superconformal N = 8 Chern-Simons [6] and multiple M2-brane theories [7]. Some other aspects have been examined in [12, 13, 14, 15, 16, 17]. Consistency requires that one should be able to relate such a M2-brane theory to the U(N) maximally supersymmetric gauge theory in 3-dimensions which describes N D2-branes [8, 9, 10, 11]. This relation also implies that there must be 3-Lie algebras which contain the Lie algebra u(N) of the gauge group of D2-branes. This again raises the issue of the solution of the Jacobi identities for metric 3-Lie algebras and requires that there must be more general solutions than those suggested in [2]. However, it was shown in [19] that one cannot embed most semi-simple Lie algebras in metric 3-Lie algebras. Moreover it was confirmed in [19, 20] that the only solutions of the Jacobi identity of metric 3-Lie algebras with a positive definite metric are those stated in the conjecture of [2]. In this paper, we shall confirm the conjecture of [2] for a k-Lie algebra, a[k], with a positive definite metric. In particular, we shall show that the structure constants of such an algebra are of the form