# 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 $e_{1},\dots,e_{n}$ 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:

 $e_{i}e_{j}=\sum_{k=1}^{n}c^{k}_{ij}e_{k}$

for $c^{k}_{ij}\in k$. 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

 $c^{k}_{ii}=0$

for all $1\leq i\leq n$. 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, associative algebras, and commutative algebras is respectively

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

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.

Title enumerating algebras EnumeratingAlgebras 2013-03-22 15:50:48 2013-03-22 15:50:48 Algeboy (12884) Algeboy (12884) 9 Algeboy (12884) Theorem msc 08B99 msc 05A16 EnumeratingGroups