If *G* is a group, its **abelianization** is the abelian group *A* and the map *G* → *A* such that any map *G* → *B* with *B* abelian factors through *A*. Abelianization is a functor, and in general a very lossy operation. The map *G* → *A* is always a surjection/quotient, because we can construct *A* by dividing *G* by the minimal normal subgroup that contains all conjugations *ghg-1h-1* for *g,h*∈*G*.


If *V* is a finite-dimensional (super)vector space over a field *K*, then the abelianization of GL(*V*) is isomorphic to the multiplicative group *K*\* of non-zero numbers in *K*. Indeed, the determinant exhibits the desired isomorphism.


Here are two questions I'm curious about:


1. What can be said about the abelianizations of other (finite-dimensional) Lie groups?
2. If *V* is an infinite-dimensional vector space, what can be said about the abelianization of GL(*V*)? Most infinite-dimensional vector spaces have some analytic structure, e.g. topological vector spaces, and so it's reasonable to ask that the operators in GL(*V*) should preserve that structure; you are welcome to take your favorite type of infinite-dimensional vector space and your favorite type of GL(*V*), if you want.



Abelianization of Lie groups
I don't have anything to say about specific examples, but here are some general remarks. A way to construct the abelianization of any compact group is to consider its image under the product of all its 1-dimensional unitary representations. This is because a compact abelian group is characterized by its set of characters by Pontrjagin duality. More generally, you can construct the double Pontrjagin dual of a locally compact group to get its locally compact abelianization as a subgroup of a space of maps to U(1) with the compact-open topology.


