Josi M. Izquierdo

n-ary algebras: a review with applications

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 These generalizations are algebraic structures in which the two entries Lie bracket has been replaced by a bracket with n entries.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 r\^ole 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 (GJI), and to Filippov (or n-Lie) algebras, which are defined by fully antisymmetric n-brackets that satisfy the Filippov identity (FI). Three-Lie algebras have surfaced recently in multi-brane theory in the context of the Bagger-Lambert-Gustavsson model. Because of this, Filippov algebras will be discussed at length, including the cohomology complexes that govern their central extensions and their deformations (Whiteheads lemma extends to all semisimple n-Lie algebras). When the skewsymmetry of the n-Lie algebra is relaxed, one is led 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 GJI reproduces the pattern of the generalized Lie algebras, and the Nambu-Poisson structures, which satisfy the FI 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.

Classical anomalies of supersymmetric extended objects

Abstract The hamiltonian form of the action for a p -extended supersymmetric object is presented, and used to deduce both the algebra generated by the constraints, in agreement with previous results for p =1,2, and the algebra of the supersymmetry charges. The “anomalous” contributions in each algebra (for given p ) are shown to be related, and the origin of their different properties is exhibited. In particular, it is shown why only in the charge algebra are the “anomalous” contributions always topological and the commutators of the translations always zero.

Generalizations of Maxwell (super)algebras by the expansion method

This paper has been supported by research grants from the Spanish MINECO (FIS2008-01980, FIS2009-09002, CONSOLIDER CPAN-CSD2007-00042), from the Polish Ministry of Science and Education (202332139) and from the Polish National Science Center (project 2011/01/B/ST2/0335).

The Z2-graded Schouten–Nijenhuis bracket and generalized super-Poisson structures

The super or Z2-graded Schouten–Nijenhuis bracket is introduced. Using it, new generalized super-Poisson structures are found which are given in terms of certain graded-skew-symmetric contravariant tensors Λ of even order. The corresponding super “Jacobi identities” are expressed by stating that these tensors have a zero super Schouten–Nijenhuis bracket with themselves [Λ,Λ]=0. As a particular case, we provide the linear generalized super-Poisson structures which can be constructed on the dual spaces of simple superalgebras with a non-degenerate Killing metric. The su(3,1) superalgebra is given as a representative example.

Two-twistor particle models and free massive higher spin fields

A bstractWe present D = 3 and D = 4 world-line models for massive particles moving in a new type of enlarged spacetime, with D−1 additional vector coordinates, which after quantization lead to towers of massive higher spin (HS) free fields. Two classically equivalent formulations are presented: one with a hybrid spacetime/bispinor variables and a second described by a free two-twistor dynamics with constraints. After first quantization in the D = 3 and D = 4 cases, the wave functions satisfying a massive version of Vasiliev’s free unfolded equations are given as functions on the SL(2, ℝ) and SL(2, ℂ) group manifolds respectively, which describe arbitrary on-shell momenta and spin degrees of freedom. Further we comment on the D = 6 case, and possible supersymmetric extensions are mentioned as well. Finally, the description of interactions and the AdS/CFT duality are briefly considered for massive HS fields.

A supergroup based supersigma model

The supergroup UOSp(1,2) is used to construct geometrically a twice graded supersymmetric supersigma model. In an appropriate contraction limit, the model reduces to the conventional supersymmetric sigma model.

Models with Hopf terms

The geometrical structure of the Hopf term and its relation to the spin of a skyrmion is studied. An ansatz describing the most general one‐skyrmion field configuration in three dimensions is introduced and various properties of the physical Hopf term for it are exhibited. The extension to seven dimensions is also made. Some comments are also made about the quantization of the coefficient of the Hopf term in the action.

Anomalies from the Wess-Zumino-Witten action

We show that the two most frequent expressions for the anomalous commutators can be both derived from quantities associated with the WZW model.

k -Leibniz algebras from lower order ones: From Lie triple to Lie ℓ-ple systems

Two types of higher order Lie l-ple systems are introduced in this paper. They are defined by brackets with l > 3 arguments satisfying certain conditions, and generalize the well known Lie triple systems. One of the generalizations uses a construction that allows us to associate a (2n − 3)-Leibniz algebra L with a metric n-Leibniz algebra ˜ L by using a 2(n − 1)-linear Kasymov trace form for ˜ L. Some specific types of k-Leibniz algebras, relevant in the construction, are introduced as well. Both higher order Lie l-ple generalizations reduce to the standard Lie triple systems for l = 3.

Poincaré supergravities from Lie algebra expansions

Abstract We use the expansion of superalgebras procedure (summarized in the text) to derive Chern–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 | 2 ; R ) ( 2 , 1 ) of osp ( p | 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 | 2 , R ) superalgebras, namely osp ( p + q | 2 , R ) ( 2 , 1 , 2 ) .