The groups $U(N)$ and $SU(N)$ are the most important Lie groups in quantum field theory. The most popular are the $U(1),SU(2),SU(3)$ groups (these gauge groups form the Standard model). But is there mentioned a $SU(\infty)$ gauge theory in physics literature?
An example of such a theory could be the following: May be $g \in SU(\infty)$ smooth and for a function $f(x,y)$ with spacetime coordinate $x$ and the new $SU(\infty)$ degree of freedom $y$ it holds $gf(x,y) = \int d^4y (g(x,y,y')f(x,y'))$. Now it is straighforward to define a gauge connection and the gauge field strength.
In more non-theoretical words: Some quantum states have degeneracies and these degeneracies are based on a special symmetry (operator) that exists in a quantum system. If now the degeneracy symmetry operator is unitary and local symmetry one can define a gauge theory. Was this concept used in quantum mechanics or does such a concept makes sense?
Another interesting case is this: One can perform the following switch of coordinates $g(x,y,y') = g(x,x-y,x-y')$ and hence the generators $T_a(y,y')$ defined by $g(x,y,y') = \sum_a g_a(x)T_a(y,y')$ become dependent on the spacetime coordinate. Another question: Is it possible to define spacetime dependent generators of a Lie algebra?