enumerating algebras

1 How many algebras are there?

Unlike categories of discrete objects, such as simple graphsMathworldPlanetmath with n 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 dimensionPlanetmathPlanetmath.

Let A be an non-associative algebra over a field k of dimension n. For example, A could be a Lie algebraMathworldPlanetmath, an associative algebra, or a commutative algebra.

From every basis e1,,en for A, the addition of the algebraMathworldPlanetmathPlanetmathPlanetmath is completely understood as all n-dimensional k-vector spaces are isomorphicPlanetmathPlanetmathPlanetmathPlanetmath. Thus we must consider only the multiplication. For this the structure constants of the algebra are considered. That is:


for cijkk. These structure constants completely define the algebra A.

Due to the axioms of multiplication, the structure constants satisfy certain relationsPlanetmathPlanetmath. For example, if A is a Lie algebra then multiplication is via the associated Lie bracket and we know


Hence we find


for all 1in. Likewise the Jacobi identityMathworldPlanetmath/associativity/commutativePlanetmathPlanetmathPlanetmath conditions each imply their particular relations. If one replaces the structure constants with variables xijk we find that each algebra A 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 n3-dimensional affine space, of these equations.

Theorem 1 (Neretin, 1987).

The dimension of the algebraic variety for n-dimensional Lie algebras, associative algebras, and commutative algebras is respectively

 and 227n3+O(n8/3).

Lower bounds of 227n3+O(n2) (and/or 427+O(n2)) are attainable by exhibiting large families of algebras. For example, class 2 nilpotent Lie algebrasMathworldPlanetmath attain the lower bound.

As with the related problems for p-groups, it is also expected that the true upper bound has error term O(n2) [Neretin,Sims].

Neretin, Yu. A., An estimate for the number of parameters defining an n-dimensional algebra, Izv. Akad. Nauk SSSR Ser. Mat., vol. 51,1987, no. 2, pp. 306–318, 447.

Mann, Avinoam, Some questions about p-groups, J. Austral. Math. Soc. Ser. A, vol. 67, 1999, no. 3, pp. 356–379.

Title enumerating algebras
Canonical name EnumeratingAlgebras
Date of creation 2013-03-22 15:50:48
Last modified on 2013-03-22 15:50:48
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 9
Author Algeboy (12884)
Entry type Theorem
Classification msc 08B99
Classification msc 05A16
Related topic EnumeratingGroups