Compilation of relations for the antisymmetric tensors defined by the Lie algebra cocycles of su(n)su(n)

This paper attempts to provide a comprehensive compilation of results, many new here, involving the invariant totally antisymmetric tensors (Omega tensors) which define the Lie algebra cohomology cocycles of $su(n)$, and that play an essential role in the optimal definition of Racah-Casimir operators of $su(n)$. Since the Omega tensors occur naturally within the algebra of totally antisymmetrised products of $\lambda$-matrices of $su(n)$, relations within this algebra are studied in detail, and then employed to provide a powerful means of deriving important Omega tensor/cocycle identities. The results include formulas for the squares of all the Omega tensors of $su(n)$. Various key derivations are given to illustrate the methods employed.Comment: Latex file (run thrice). Misprints corrected, Refs. updated. Published in IJMPA 16, 1377-1405 (2001

Superalgebra cohomology, the geometry of extended superspaces and superbranes

We present here a cohomological analysis of the new spacetime superalgebras that arise in the context of superbrane theory. They lead to enlarged superspaces that allow us to write D-brane actions in terms of fields associated with the additional superspace variables. This suggests that there is an extended superspace/worldvolume fields democracy for superbranes.Comment: 12 pages, LATEX. Invited lecture delivered at the XXXVII Karpacz Winter School on "New Developments in Fundamental Interaction Theories" (6-15 February, 2001, Karpacz, Poland). To be published in the Proceeding

Topics on n-ary algebras

We describe the basic properties of two n-ary algebras, the Generalized Lie Algebras (GLAs) and, particularly, the Filippov (or n-Lie) algebras (FAs), and comment on their n-ary Poisson counterparts, the Generalized Poisson (GP) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends the familiar Whitehead's lemma to all $n\geq 2$ FAs, n=2 being the standard Lie algebra case. When the n-bracket of the FAs is no longer required to be fully skewsymmetric one is led to the n-Leibniz (or Loday's) algebra structure. Using that FAs are a particular case of n-Leibniz algebras, those with an anticommutative n-bracket, we study the class of n-Leibniz deformations of simple FAs that retain the skewsymmetry for the first n-1 entires of the n-Leibniz bracket.Comment: 11 page

The Ortographic Irregularities in the Manuscript M1 of the Library of the Universidad Complutense de Madrid

In this paper, I study some ortographic irregularities of the text of the manuscript M1 of the Library of the Universidad Complutense de Madrid. These are the cases of extraordinary points, suspended letters, inverted nuns, and large and small letters. I also analyze the masoroth of these cases, and compare them with those of the oldest manuscripts: Aleppo, Cairo, and Leningrad.En este artículo he estudiado algunas de las grafías extraordinarias en el texto del manuscrito M1 de la Universidad Complutense de Madrid. Estas son: las letras o palabras puntuadas, las letras suspendidas, los nûnîn invertidos y las letras de mayor y menor tamaño que las de su contexto. Así mismo he analizado las masoras de estos casos y he comparado el texto y las masoras de M1 con las de los más antiguos manuscritos bíblicos: Alepo, Cairo y Leningrado

Braided structure of fractional Z3Z_3-supersymmetry

It is shown that fractional $Z_3$-superspace is isomorphic to the $q\to\exp(2\pi i/3)$ limit of the braided line. $Z_3$-supersymmetry is identified as translational invariance along this line. The fractional translation generator and its associated covariant derivative emerge as the $q\to\exp(2\pi i/3)$ limits of the left and right derivatives from the calculus on the braided lineComment: 8 pages, LaTeX, submitted to Proceedings of the 5th Colloquium `Quantum groups and integrable systems', Prague, June 1996 (to appear in Czech. J. Phys.

The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures

Newly introduced generalized Poisson structures based on suitable skew-symmetric contravariant tensors of even order are discussed in terms of the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are expressed as conditions on these tensors, the cohomological contents of which is given. In particular, we determine the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.Comment: 29 pages. Plain TeX. Phyzzx needed. An example and some references added. To appear in J. Phys.