enumerating algebras

1 How many algebras are there?

Unlike categories of discrete objects, such as simple graphs 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 dimension.

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

From every basis $e_1,\mathrm{\dots },e_n$ for $A$, the addition of the algebra is completely understood as all $n$-dimensional $k$-vector spaces are isomorphic. Thus we must consider only the multiplication. For this the structure constants of the algebra are considered. That is:

$$e_i{e}_j=\sum _{k=1}^n{c}_{ij}^k{e}_k$$

for $c_{ij}^k\in k$. These structure constants completely define the algebra $A$.

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

$$[e_i,e_i]=0$$

Hence we find

$$c_{ii}^k=0$$

for all $1\le i\le n$. Likewise the Jacobi identity/associativity/commutative conditions each imply their particular relations. If one replaces the structure constants with variables $x_{ijk}$ 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 $n^3$-dimensional affine space, of these equations.

Theorem 1 (Neretin, 1987).

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

$$\frac{2}{27}{n}^3+O(n^{8/3}),\frac{4}{27}{n}^3+O(n^{8/3}),$$

$$\text{ and }\frac{2}{27}{n}^3+O(n^{8/3}).$$

Lower bounds of $\frac{2}{27}{n}^3+O(n^2)$ (and/or $\frac{4}{27}+O(n^2)$) are attainable by exhibiting large families of algebras. For example, class 2 nilpotent Lie algebras attain the lower bound.

As with the related problems for $p$-groups, it is also expected that the true upper bound has error term $O(n^2)$ [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.