Algorithms for the explicit matrix isomorphism problem over $\mathbb{C}$

Question

Suppose that $A$ is a $d^2$ dimensional algebra over $\mathbb{C}$ and we know the multiplication tensor $c_{ij}^k$ and the unit $u^k$ in some basis. If $A$ is semi-simple and has a single simple representation, then computing an explicit isomorphism $A \cong M_d(\mathbb{C})$ is often called the explicit isomorphism problem. This is equivalent to constructing an explicit $d$ dimensional representation of $A$.

Most of the literature I have been able to find about the explicit isomorphism problem assumes you are working over some number field. I want to know the state of the art for working over $\mathbb{C}$. If $d=2$, Grobner bases work fine, but I don't know what I am doing and this approach takes too long when $d>2$.

Question: What is known about the explicit isomorphism problem over $\mathbb{C}$?

Answer 1

When specifying an algorithmic problem "over $\mathbb{C}$", you should say precisely what the input and output should be.

The isomorphism problem is less studied over $\mathbb{C}$ because it is much easier than over a number field.

Here is the basic idea: if you can find an element $a\in A$ such that right multiplication by $a$ on $A$ has rank $d$ (i.e. the corresponding matrix, which you don't know yet, has rank $1$), then the left ideal $Aa$ is isomorphic to $\mathbb{C}^d$ as a vector space, and left multiplication on $Aa$ induces an isomorphism $A\cong M_d(\mathbb{C})$. If you are able to draw "random" elements in your algebra, then with probablility $1$ a random element $b\in A$ will correspond to a matrix with $d$ distinct eigenvalues; then the characteristic polynomial of right multiplication of $b$ on $A$ is $\prod_{i=1}^d(X-\lambda_i)^d$ and from this factorisation you can find a polynomial $P$ such that $a=P(b)$ corresponds to a matrix of rank $1$.

Some references :

• Eberly, Decomposition of algebras over R or C
• de Graaf, Ivanyos, Finding maximal tori and splitting elements in matrix algebras.