Algebraic geography: Varieties of structure constants

Straightening-out and semirigidity in associative algebras

“…See Theorem 4 below provides an answer to the first question, describing the "size and shape" of the radical A^ as an S-bimodule. In Theorem 5 we see that the "pre-matrix" part of nt can essentially be given by polynomials in /; such theorems (see also [1]) assure us that the explicit deformations we construct by trial and error, adjusting various powers of t, are, in the sense of equivalence, all. And in Theorem 6 we give sufficient conditions for Type II deformability, reducing the problem to that of finding a nilpotent deformation within the radical.…”

“…In accord with that example, it is entirely straightforward to [April show: for each ordered pair of distinct a,ße<[p~), there exists an a/J-block Naß0 such that, writing Up = ®Naß0 with distinct a, ß e </?>, the off-diagonal blocks of Sp are filled by (UP)K. Also Up is an S<p>-bimodule and the ic-algebra PP = UP ® S<p> deforms via w, into 2P. Thus Pp is a pre-matrix algebra in the language of [1], that is, a fc-algebra with unit which deforms into the algebra of all Ä-matrices of a given rank, while maintaining the same unit.…”

“…First, since Pv is a pre-matrix algebra, we know from [1] that there is a ¿-basis for P'v (cf. the Reduction Principle of (ii) above) of the forrn zaB0 with a, ß e </?> (here zaa0=ea) wherein irt is equivalent to the multiplication Ut(zaB0, zBy0) = tmzay0;…”

“…By the Reduction Principle and Theorem 4, therefore, we need only consider the multiplications TTÁZaeo, Zßyd with a, ß e </?>, y e (q} and /= 1. Just as in [1], the key observation is that such a product is a scalar (in K) multiple of a single basis element; that is, nfeaBo, zßri)= &ccyi with ¿¡=$aByoi an element of k [[t]]. We factor |=imK where m = maBy0l and K = KaBy0i is a power series in t with nonzero constant term.…”

The moduli space of commutative algebras of finite rank

“…Various analogues of B n have been studied in the literature before, at least over a base field. The paper [Fla68] defines the moduli space for based associative (but not necessarily commutative or unital) algebras, defines a GL n -action as we do below, and studies GL n -orbits in its irreducible components. The moduli space of based associative algebras with 1, along with the GL n -action, is studied in [Gab74,Maz79,Maz82,LBR99].…”