# 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, associative algebras, 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.

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 |