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Zeitschrift für verallgemeinerte Lügentheorie und -anwendungen

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Volumen 3, Ausgabe 2 (2009)

Forschungsartikel

Cohomology and duality for coalgebras over a quadratic operad

Anita MAJUMDAR a and Donald YAU

The cohomology of a finite-dimensional coalgebra over a finitely generated quadratic operad, with coefficients in itself, is defined and is shown to have the structure of a graded Lie algebra. The cohomology of such a coalgebra is isomorphic to the cohomology of its linear dual as graded Lie algebras.

Forschungsartikel

Color Lie algebras and Lie algebras of order F

R.CAMPOAMOR-STURSBERG and M.RAUSCH DETRAUBENBERG

The notion of color algebras is generalized to the class of F-ary algebras, and corresponding decoloration theorems are established. This is used to give a construction of colored structures by means of tensor products with Cli ord-like algebras. It is, moreover, shown that color algebras admit realizations as q = 0 quon algebras.

Forschungsartikel

Deforming K(1) superalgebra modules of symbols

Faouzi AMMAR and Kaouthar KAMOUN

We study nontrivial deformations of the natural action of the Lie superalgebra K(1) of contact vector elds on the (1; 1)-dimensional superspace R1j1 on the space of symbols e Sn  = Ln k=0 F􀀀k2 . We calculate obstructions for integrability of in nitesimal multiparameter deformations and determine the complete local commutative algebra corresponding to the miniversal deformation.

Rezensionsartikel

Algebras with ternary law of composition and their realization by cubic matrices

V. ABRAMOV, R. KERNER, O. LIIVAPUU, and S.SHITOV

We study partially and totally associative ternary algebras of first and second kind. Assuming the vector space underlying a ternary algebra to be a topological space and a triple product to be continuous mapping, we consider the trivial vector bundle over a ternary algebra and show that a triple product induces a structure of binary algebra in each fiber of this vector bundle. We find the sufficient and necessary condition for a ternary multiplication to induce a structure of associative binary algebra in each fiber of this vector bundle. Given two modules over the algebras with involutions, we construct a ternary algebra which is used as a building block for a Lie algebra. We construct ternary algebras of cubic matrices and find four different totally associative ternary multiplications of second kind of cubic matrices. It is proved that these are the only totally associative ternary multiplications of second kind in the case of cubic matrices. We describe a ternary analog of Lie algebra of cubic matrices of second order which is based on a notion of j-commutator and find all commutation relations of generators of this algebra.

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