1 How many algebras are there?
Unlike categories of discrete objects, such as simple graphs with vertices, (see article on enumerating graphs (http://planetmath.org/EnumeratingGraphs)) such a question is a little malposed as the quantity can be infinite. However the spirit of the question can be addressed by appealing to algebraic varieties and considering their dimension.
From every basis for , the addition of the algebra is completely understood as all -dimensional -vector spaces are isomorphic. Thus we must consider only the multiplication. For this the structure constants of the algebra are considered. That is:
for . These structure constants completely define the algebra .
Hence we find
for all . Likewise the Jacobi identity/associativity/commutative conditions each imply their particular relations. If one replaces the structure constants with variables we find that each algebra of a given type (Lie/Associative/Commutative/etc.) is a solution to the polynomial equations given by the relations of the algebra. Thus the algebras themselves are parameterized by the algebraic variety, in -dimensional affine space, of these equations.
Theorem 1 (Neretin, 1987).
The dimension of the algebraic variety for -dimensional Lie algebras, associative algebras, and commutative algebras is respectively
As with the related problems for -groups, it is also expected that the true upper bound has error term [Neretin,Sims].
Neretin, Yu. A., An estimate for the number of parameters defining an
-dimensional algebra, Izv. Akad. Nauk SSSR Ser. Mat., vol. 51,1987, no. 2, pp. 306–318, 447.
Mann, Avinoam, Some questions about -groups, J. Austral. Math. Soc. Ser. A, vol. 67, 1999, no. 3, pp. 356–379.
|Date of creation||2013-03-22 15:50:48|
|Last modified on||2013-03-22 15:50:48|
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