This question is related to the following general question:

Given a variety of (non-associative) algebras $\mathcal V$, a finite field $\mathbb{F}_q$, with $q$ elements, and a positive integer $n$, how many $n$-dimensional $\mathbb F_q$-algebras in $\mathcal V$ are there?

It is well known that if $A$ is an $n$-dimensional algebra over a field $F$, with basis $\{e_1, \dots, e_n\}$ then its algebra structure is uniquely determined by the $n^3$-tuple $(\alpha_{ij}^{(k)})\in F^{n^3}$, defined by $e_i e_j=\sum_{k=1}^n \alpha_{ij}^{(k)} e_k$.

So the general question above can be reformulated as:

How many of these $n^3$-tuples of elements of $\mathbb F_q$ define algebras in $\mathcal V$?

Or in another language: given an arbitrary $n$-dimensional algebra, what is the probability that it lies in $\mathcal V$?

Let us denote such number by $N_{q,n}(\mathcal V)$.

Some examples are simple to compute. For example, if $q$ is odd, one can easily show that if $\mathcal V$ is the variety of anticommutative algebras (i.e., the class of all algebras satisfying the identity $xy+yx=0$), then $N_{q,n}(\mathcal V)=q^{n^2(n-1)/2}$ and if $\mathcal C$ is the variety of commutative algebras, then $N_{q,n}(\mathcal C)=q^{n^2(n+1)/2}$.

But other examples seem to be much more difficult, for example for the varieties of Lie and associative algebras.

So my main questions (for now) are the following:

1. How many Lie algebras of dimension $n$ over a field with $n$ elements are there?
2. How many associative algebras of dimension $n$ over a field with $n$ elements are there?

I'd like to stress that I am not interested in isomorphism classes, but in the number of such algebras only (that is to say this is a problem of combinatorics and not of algebra).

Finally, I would like to remark that I have considered the possibility to write a computer program to compute some cases (example for $q=3$ and $n \leq 6$), so I could have a guess of the general answer, but in a first look I realized that this is would take too much time.

This question is related to the following general question:

Given a variety of (non-associative) algebras $\mathcal V$, a finite field $\mathbb{F}_q$, with $q$ elements, and a positive integer $n$, how many $n$-dimensional $\mathbb F_q$-algebras in $\mathcal V$ are there?

It is well known that if $A$ is an $n$-dimensional algebra over a field $F$, with basis $\{e_1, \dots, e_n\}$ then its algebra structure is uniquely determined by the $n^3$-tuple $(\alpha_{ij}^{(k)})\in F^{n^3}$, defined by $e_i e_j=\sum_{k=1}^n \alpha_{ij}^{(k)} e_k$.

So the general question above can be reformulated as:

How many of these $n^3$-tuples of elements of $\mathbb F_q$ define algebras in $\mathcal V$?

Or in another language: given an arbitrary $n$-dimensional algebra, what is the probability that it lies in $\mathcal V$?

Let us denote such number by $N_{q,n}(\mathcal V)$.

Some examples are simple to compute. For example, if $q$ is odd, one can easily show that if $\mathcal V$ is the variety of anticommutative algebras (i.e., the class of all algebras satisfying the identity $xy+yx=0$), then $N_{q,n}(\mathcal V)=q^{n^2(n-1)/2}$ and if $\mathcal C$ is the variety of commutative algebras, then $N_{q,n}(\mathcal C)=q^{n^2(n+1)/2}$.

But other examples seem to be much more difficult, for example for the varieties of Lie and associative algebras.

So my main questions (for now) are the following:

1. How many Lie algebras of dimension $n$ over a field with $q$ elements are there?
2. How many associative algebras of dimension $n$ over a field with $q$ elements are there?

I'd like to stress that I am not interested in isomorphism classes, but in the number of such algebras only (that is to say this is a problem of combinatorics and not of algebra).

Finally, I would like to remark that I have considered the possibility to write a computer program to compute some cases (example for $q=3$ and $n \leq 6$), so I could have a guess of the general answer, but in a first look I realized that this is would take too much time.

EDIT:

The answer I am expecting is an explicit formula for $N_{q,n}(\mathcal V)$ when $\mathcal V$ is the variety of associative or Lie algebras.