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de Azcarraga, J. A., & Izquierdo, J. M. (2010). n-ary algebras: a review with applications. J. Phys. A, 43(29), 293001–117pp.
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 These generalizations are algebraic structures in which the two-entry Lie bracket has been replaced by a bracket with n entries.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 role 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. 3-Lie algebras have surfaced recently in multi-brane theory in the context of the Bagger-Lambert-Gustavsson model. As a result, 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.
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de Azcarraga, J. A., & Izquierdo, J. M. (2011). On a class of n-Leibniz deformations of the simple Filippov algebras. J. Math. Phys., 52(2), 023521–13pp.
Abstract: We study the problem of infinitesimal deformations of all real, simple, finite-dimensional Filippov (or n-Lie) algebras, considered as a class of n-Leibniz algebras characterized by having an n-bracket skewsymmetric in its n-1 first arguments. We prove that all n > 3 simple finite-dimensional Filippov algebras (FAs) are rigid as n-Leibniz algebras of this class. This rigidity also holds for the Leibniz deformations of the semisimple n = 2 Filippov (i.e., Lie) algebras. The n = 3 simple FAs, however, admit a nontrivial one-parameter infinitesimal 3-Leibniz algebra deformation. We also show that the n >= 3 simple Filippov algebras do not admit nontrivial central extensions as n-Leibniz algebras of the above class.
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de Azcarraga, J. A., Kamimura, K., & Lukierski, J. (2011). Generalized cosmological term from Maxwell symmetries. Phys. Rev. D, 83(12), 124036–8pp.
Abstract: By gauging the Maxwell spacetime algebra, the standard geometric framework of Einstein gravity with cosmological constant term is extended by adding six four-vector fields A(mu)(ab)(x) associated with the six Abelian tensorial charges in the Maxwell algebra. In the simplest Maxwell extension of Einstein gravity this leads to a generalized cosmological term that includes a contribution from these vector fields. We also consider going beyond the basic gravitational model by means of bilinear actions for the new Abelian gauge fields. Finally, an analogy with the supersymmetric generalization of gravity is indicated. In an appendix, we propose an equivalent description of the model in terms of a shift of the standard spin connection by the A(mu)(ab)(x) fields.
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de Azcarraga, J. A., & Izquierdo, J. M. (2012). D=3 (p, q)-Poincare supergravities from Lie algebra expansions. Nucl. Phys. B, 854(1), 276–291.
Abstract: We use the expansion of superalgebras procedure (summarized in the text) to derive Chem-Simons (CS) actions for the (p, q)-Poincare supergravities in three-dimensional spacetimes. After deriving the action for the (p, 0)-Poincare supergravity as a CS theory for the expansion osp(p vertical bar 2: R)(2, 1) of osp(p vertical bar 2: R), we find the general (p, q)-Poincare superalgebras and their associated D = 3 supergravity actions as CS gauge theories from an expansion of the simple osp(p + q vertical bar 2, R) superalgebras, namely osp(p + q vertical bar 2, R)(2, 1, 2).
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de Azcarraga, J. A., Izquierdo, J. M., Lukierski, J., & Woronowicz, M. (2013). Generalizations of Maxwell (super)algebras by the expansion method. Nucl. Phys. B, 869(2), 303–314.
Abstract: The Lie algebras expansion method is used to show that the four-dimensional spacetime Maxwell (super)algebras and some of their generalizations can be derived in a simple way as particular expansions of o(3,2) and osp(N vertical bar 4).
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