Cohomology of deformation parameters of diagonal noncommutative nonassociative graded algebras

Wills-Toro, Luis Alberto and Craven, Thomas and Vélez, Juan Diego (2008) Cohomology of deformation parameters of diagonal noncommutative nonassociative graded algebras. ACTA MATHEMATICA ACADEMIAE PAEDAGOGICAE NYÍREGYHÁZIENSIS, 24 (3). pp. 271-277. ISSN 0866-0174

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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.

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