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Let 𝒞 = 𝒞(n,Ω) denote the
algebraic set of structure constants for n-dimensional associative algebras, a subset of
Ωn3
. Here Ω is a universal domain over a prime field F and a point c = (chij) with
h,i,j = 1,⋯,n is in 𝒞 if and only if the multiplication (xh,xi) → xhxi = ∑
jchijxj is
associative. The set 𝒞 is readily seen to be F-closed in the Zariski topology on Ωn3
and is in fact a finite union of irreducible closed cones (the components of 𝒞)
with the origin as vertex. The natural “change of basis” action of the group
G = GL(n,Ω) on 𝒞 yields a one-one correspondence between orbits G ⋅ c on 𝒞
and n-dimensional Ω-algebras. One studies the globality of these algebras
(and of algebras defined over subfields of Ω) by examining the geography of
𝒞.
Thus if S is a semi-simple Ω-algebra (more generally, if the Hochschild group
H2(S,S) = (0)) then its corresponding orbit (denoted G ⋅ S) is open and therefore
dense in its component 𝒞0 of 𝒞. Thus S determines all algebras which live on 𝒞0. One
checks that dim𝒞0 = n2 −n + s, where S = S1 ⊕⋯⊕Ss for simple Sα. Moreover, in
the language of Gerstenhaber and Nijenhuis-Richardson, one may hope to deform the
algebras on 𝒞0 into S. In commencing a study of the parameter space 𝒞, therefore it
seems a natural first question to ask whether every irreducible component of 𝒞 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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