In his very influential book Infinite dimensional Lie algebras, which is still the main reference for Kac-Moody algebras, in section 0.4 of the introduction, Victor Kac discuss the problem of concrete realization of Kac-Moody algebras.
They are defined via generators and relations using a generalized Cartan matrix $M$, and hence it makes sense to look for a Lie algebra tha we know concretelly and show it is isomorphic the Kac-Moody Lie algebra of $M$.
As a silly example, which nevertheless makes the point clear, we can consider the simple Lie algebras $\mathfrak{sl}_n$. They are can be presented via Serre generators and relations, but this would not make it clear many aspects of this algebra, such as its representation theory. Because of that, its realization as the algebra of traceless $n +1\times n + 1$ matrices is so important.
All finite dimensional Lie algebras have well known concrete realizations, and so have the affine Lie algebras. In the end of $0.4$, Kac says:
"Unfortunately, no simple realization has been found up to now for any nonaffine infinite dimension Kac-Moody algebra.This question appears to be one of the most important open problems of the theory"
It has been years since I moved from Lie algebras to ring theory, so I personally have no knowledge about the status of this problem. A few weeks ago I had a long chat with my former advisor, which is an expert in Lie theory, and he said to my that despite the decades since Kac posed this problem, no advances were made.
I fully believe that the realizationf of all indefinite Kac-Moody algebras is far off limits, but I have some doubts that, not even for a very specific case, such realization has not been achieved.
So, me question is: what is the current status of Kac's problem of concrete realizations of indefinite affine Kac-Moody algebras?
