Generic \(8\)-dimensional algebras with mixed basis-graph
Deformation theory is the appropriate tool for describing the irreducible components of the scheme \(\text{Alg}_ n\) which parametrizes the structures of \(n\)-dimensional associative algebras with unit. Each component is ``dominated'' 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 which are used for that classification are the basis-graph, both weighted and unweighted, of an associative algebra. In this paper the 8-dimensional algebras with mixed basis-graph are classified and lower bounds for the numbers of irreducible components of the scheme \(\text{Alg}_ 8\), reduced and non reduced, are given.
scientific article; zbMATH DE number 597577