What is $\mathfrak{gl}(\infty)$

Question

As title says, I know what is $\mathfrak{gl}(n,\mathbb{C})$, but what is $\mathfrak{gl}(\infty)$? Where can I find good reference for this?

Answer 1

This is the finitary Lie algebra $$ \mathfrak{gl}(\infty)=\bigcup_{n\in \mathbb{N}}\mathfrak{gl}(n). $$ The simple ones have been classified:

Theorem(Baranov): There are only three nonisomorphic finitary simple Lie algebras of countable dimension: $\mathfrak{sl}(\infty)$, $\mathfrak{so}(\infty)$, and $\mathfrak{sp}(\infty)$.

The paper of Baranov here contains a number of good references: http://www2.le.ac.uk/departments/mathematics/extranet/staff-material/staff-profiles/ab155/research/sdiag.pdf. Finitary Lie algebras have been studied by many authors (e.g., Penkov, Bahturin, Strade, Zalesskii, and many others).