Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 24, No. 3, pp. 271-277 (2008)

Cohomology of deformation parameters of diagonal noncommutative nonassociative graded algebras

Luis Alberto Wills-Toro, Thomas Craven and Juan Diego Vélez

American University of Sharjah, University of Hawaii at Manoa and Universidad Nacional de Colombia

Abstract: We study graded algebras with no monomial in the generators having zero divisors and graded over a finite abelian group. As a vector space over the field, the algebra is generated by a set of algebra elements with as many elements as the grading group, and each generator is graded by a different element of the grading group. Their noncommutativity and nonassociativity turns out to be diagonal and governed by structure constants of any (pure grade) generating basis as a vector space over the field. There are functions q and r coding the noncommutativity and nonassociativity of the algebra. We study the cohomology of such q- and r-functions. We discover that the r-function coding nonassociativity has always trivial cohomology. Quaternions and octonions are constructed in this manner and we study their noncommutativity and nonassociativity using cohomological tools.

Keywords: Noncommutative algebras, nonassociative algebras, cohomology of deformation parameters, perfect algebra.

Classification (MSC2000): 17A99; 17D99, 13D03, 20J06

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