Graded Lie algebras and q-commutative and r-associative parameters

Resumo

We study graded Lie algebras whose transformation parameters are graded

q-commutativive and r-associative. We study first some graded algebras over a field, with no zero divisors at the level of monomials in their graded algebra generators. These generators are q-commutative and r-associative. We address the cohomology of the q-function and r-functions, in particular we study quaternions and octonions. We then define algebras whose transformation parameters are q-commutative and r-associative. We address a generalization of a theorem by Scheunert on its relation to Lie (super)algebras. We show finally that for the cases studied by Scheunert there is always a real and faithful transformation parameter basis with the required q-commutativity. We use this basis to perform a transformation on the graded Lie algebra that relates it to a plain Lie (super)algebra while respecting the self-adjoint character of generators and preserving the group grading. Keywords: Graded Lie (super)algebras, Color Lie (super)algebras, noncommutative algebras, nonassociative algebras, cohomology of deformation parameters, perfect algebra. AMS-MSC: 17B70, 17B75, 22E60, 17A99, 17D99, 13D03, 20J06.