n-ary algebras: a review with applications

Abstract

Abstract. This paper reviews the properties and applications of certain n-ary generalizations of Lie algebras in a self-contained and unified way. These generalizations are algebraic structures in which the two entries Lie bracket has been replaced by a bracket with n entries. Each type of n-ary bracket satisfies a specific characteristic identity which plays the rôle of the Jacobi identity for Lie algebras. Particular attention will be paid to generalized Lie algebras, which are defined by even multibrackets obtained by antisymmetrizing the associative products of its n components and that satisfy the generalized Jacobi identity, and to Filippov (or n-Lie) algebras, which are defined by fully antisymmetric n-brackets that satisfy the Filippov identity. Three-Lie algebras have surfaced recently in multi-brane theory in the context of the Bagger-LambertGustavsson model. Because of this, Filippov algebras will be discussed at length, including the cohomology complexes that govern their central extensions and their deformations (it turns out that Whitehead's lemma extends to all semisimple n-Lie algebras). When the skewsymmetry of the Lie or n-Lie algebra bracket is relaxed, one is led to a more general type of n-algebras, the n-Leibniz algebras. These will be discussed as well, since they underlie the cohomological properties of n-Lie algebras.The standard Poisson structure may also be extended to the n-ary case. We shall review here the even generalized Poisson structures, whose generalized Jacobi identity reproduces the pattern of the generalized Lie algebras, and the Nambu-Poisson structures, which satisfy the Filippov identity and determine Filippov algebras. Finally, the recent work of Bagger-Lambert and Gustavsson on superconformal Chern-Simons theory will be briefly discussed. Emphasis will be made on the appearance of the 3-Lie algebra structure and on why the A 4 model may be formulated in terms of an ordinary Lie algebra, and on its Nambu bracket generalization.

Higher Spins from Nambu–Chern–Simons Theory

“…Let us mention here that we will not actually use 3-algebra notation, and will instead follow [15] in writing down expressions adapted to SDiff(M n ). Another paper which treats SDiff gauge theory is the review [26], where 3-algebra expressions can be found.…”

Commutator algebras arising from splicing operations

“…Later there has been an increasing interest of developing the various -ary generalization of Lie algebra and their applications in -body mechanics, quantum physics, geometry; mathematical genetics and etc. A detailed survey on -ary generalizations of Lie algebras and their applications in physics have been given by de Azcárraga and Izquierdo [1].…”

“…the class of algebras are satisfied of all identities of A. For a commutator and a Jordan (symmetrized) product, we use the following standard notation [ ] = − • = + Let A be an arbitrary -ary algebra over F with -ary multiplication 1 ∈ A. We will denote by…”

Brackets

“…induces an action of the multiplicative R. Remark 6.4. As the Grassmann algebra A(E * ) can be understood as the algebra of smooth functions on the graded manifold E [1] (an N-manifold of degree 1 in the terminology ofŠevera and Roytenberg [86,87]), following [97,98] we can view the de Rham derivative d Π as a vector field of degree 1 on E [1]. This vector field is homological, (d Π ) 2 = 0, if and only if we are actually dealing with a Lie algebroid.…”