Vol. 27, No. 1, 1968

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Algebraic geography: Varieties of structure constants

Francis James Flanigan

Vol. 27 (1968), No. 1, 71–79

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.

Mathematical Subject Classification
Primary: 16.90
Secondary: 14.00
Received: 12 January 1968
Published: 1 October 1968
Francis James Flanigan