Superbranes, D=11 CJS supergravity and enlarged superspace coordinates/fields correspondence

We discuss the r\^ole of enlarged superspaces in two seemingly different contexts, the structure of the $p$-brane actions and that of the Cremmer-Julia-Scherk eleven-dimensional supergravity. Both provide examples of a common principle: the existence of an {\it enlarged superspaces coordinates/fields correspondence} by which all the (worldvolume or spacetime) fields of the theory are associated to coordinates of enlarged superspaces. In the context of $p$-branes, enlarged superspaces may be used to construct manifestly supersymmetry-invariant Wess-Zumino terms and as a way of expressing the Born-Infeld worldvolume fields of D-branes and the worldvolume M5-brane two-form in terms of fields associated to the coordinates of these enlarged superspaces. This is tantamount to saying that the Born-Infeld fields have a superspace origin, as do the other worldvolume fields, and that they have a composite structure. In $D$=11 supergravity theory enlarged superspaces arise when its underlying gauge structure is investigated and, as a result, the composite nature of the $A_3$ field is revealed: there is a full one-parametric family of enlarged superspace groups that solve the problem of expressing $A_3$ in terms of spacetime fields associated to their coordinates. The corresponding enlarged supersymmetry algebras turn out to be deformations of an {\it expansion} of the $osp(1|32)$ algebra. The unifying mathematical structure underlying all these facts is the cohomology of the supersymmetry algebras involved.Comment: plain latex, 29 pages, no figures. To appear in the Am. Inst. of Phys. Proc. Serie

Central extensions, classical non-equivariant maps and residual symmetries

The arising of central extensions is discussed in two contexts. At first classical counterparts of quantum anomalies (deserving being named as "classical anomalies") are associated with a peculiar subclass of the non-equivariant maps. Further, the notion of "residual symmetry" for theories formulated in given non-vanishing EM backgrounds is introduced. It is pointed out that this is a Lie-algebraic, model-independent, concept.Comment: 8 pages, LaTex. Talk given at the International Conference "Renormalization Group and Anomalies in Gravitation and Cosmology", Ouro Preto, Brazil, March 2003. To appear in the Proceeding

Minimal D=4 supergravity from the superMaxwell algebra

We show that the first-order D=4, N=1 pure supergravity lagrangian four-form can be obtained geometrically as a quadratic expression in the curvatures of the Maxwell superalgebra. This is achieved by noticing that the relative coefficient between the two terms of the Lagrangian that makes the action locally supersymmetric also determines trivial field equations for the gauge fields associated with the extra generators of the Maxwell superalgebra. Along the way, a convenient geometric procedure to check the local supersymmetry of a class of lagrangians is developed.Comment: Plain latex, 14 pages. Two misprints corrected, one reference adde

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 (Whitehead's 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.Comment: Invited topical review for JPA Math.Theor. v2: minor changes, references added. 120 pages, 318 reference

Generating Higher-Order Lie Algebras by Expanding Maurer Cartan Forms

By means of a generalization of the Maurer-Cartan expansion method we construct a procedure to obtain expanded higher-order Lie algebras. The expanded higher order Maurer-Cartan equations for the case $\mathcal{G}=V_{0}\oplus V_{1}$ are found. A dual formulation for the S-expansion multialgebra procedure is also considered. The expanded higher order Maurer Cartan equations are recovered from S-expansion formalism by choosing a special semigroup. This dual method could be useful in finding a generalization to the case of a generalized free differential algebra, which may be relevant for physical applications in, e.g., higher-spin gauge theories

Contractions, Hopf algebra extensions and cov. differential calculus

We re-examine all the contractions related with the ${\cal U}_q(su(2))$ deformed algebra and study the consequences that the contraction process has for their structure. We also show using ${\cal U}_q(su(2))\times{\cal U}(u(1))$ as an example that, as in the undeformed case, the contraction may generate Hopf algebra cohomology. We shall show that most of the different Hopf algebra deformations obtained have a bicrossproduct or a cocycle bicrossproduct structure, for which we shall also give their dual `group' versions. The bicovariant differential calculi on the deformed spaces associated with the contracted algebras and the requirements for their existence are examined as well.Comment: TeX file, 25 pages. Macros are include