Algebraic geography: Varieties of structure constants

Let ^ = c ?{n, ) denote the algebraic set of structure constants for ^-dimensional associative algebras, a subset of n *. Here is a universal domain over a prime field F and a point c = (CMJ) with h, i, j = 1, , n is in ^ if and only if the multiplication (x h , Xi) -> XhX% = 2^ ChijXj is associative. The set ^ is readily seen to be F-closed in the Zariski topology on nZ and is in fact a finite union of irreducible closed cones (the components of ^) with the origin as vertex. The natural "change of basis" action of the group G = GL(n, ) on ^ yields a one-one correspondence between orbits G-c on & and ^-dimensional 2-algebras. One studies the globality of these algebras (and of algebras defined over subfields of ) by examining the geography of ^. Thus if S is a semi-simple 2-algebra (more generally, if the Hochschild group H 2 (S, S) = (0)) then its corresponding orbit (denoted G-S) is open and therefore dense in its component ^o of ^. Thus S determines all algebras which live on if 0 . One checks that dim ^0 = n 2n + s, where S = Si 0 & for simple S a . Moreover, in the language of Gerstenhaber and Nijenhuis-Richardson, one may hope to deform the algebras on ^ into S. In commencing a study of the parameter space ^, therefore it seems a natural first question to ask whether every irreducible component of ^ is dominated by such an open orbit or, in the sense of deformation theory, "Does every algebra deform into a rigid algebra?" We show here that the answer is no.

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