The geometric classification of nilpotent algebras

Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

“…Indeed Flanigan has shown in [55] that the variety of 3-dimensional nilpotent associative algebras has an irreducible component which does not contain any rigid algebras-it is instead defined by the closure of a union of a one-parameter family of algebras. We encounter similar situations in [78] and in [80].…”

“…The notion of length for nonassociative algebras has been recently introduced in [66], generalizing the corresponding notion for associative algebras. Using the above result, we show in [80,Cor. 39] that the length of an arbitrary (i.e.…”

“…Informally, although Theorem 7.5 shows that there is no single generic 6-dimensional nilpotent anticommutative algebra, one can see the family A 82 (α) given below as the generic family in the variety. In [80] we completely solve the geometric classification problem for nilpotent, commutative nilpotent and anticommutative nilpotent algebras of arbitrary dimension. We prove that the corresponding geometric varieties are irreducible, find their dimensions and describe explicit generic families of algebras which define each of these varieties.…”

“…We show that the family S n given in [80,Def. 15] is generic in the variety of ndimensional commutative nilpotent algebras and inductively give an algorithmic procedure to obtain any n-dimensional nilpotent commutative algebra as a degeneration from S n .…”