Abstract
Deformation theory is the appropriate tool for describing the irreducible components of the scheme AlgM which parametrizes the structures of /?-dimensiona l associative algebras with unit. Each component is by one generic or quasi-generic algebra or family of algebras (genericity means that the algebra or the family has only trivial infinitesimal deformations, and quasi-genericity means that the algebra or the family has non trivial infinitesimal deformations, but no algebraic deformation). The components dominated by a generic algebra (or family) are reduced, while the components dominated by a quasi-generic family are non reduced. The invariants we use for that classification are the basis-graph, both weighted and unweighted, of an associative algebra. In this paper, we classify the 8-dimensional algebras with mixed basis-graph and give lower bounds for the numbers of irreducible components of the scheme Algg, reduced and non reduced.
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