I’m working on some group-theoretic stuff related to n copies of a unital quantum channel and am in search of some isomorphism between the dihedral group and the unitary group U(n).
Now, what’s interesting is that the unitary group U(1) is isomorphic to the circle group, i.e. the geometric circle, while can be thought of as representing the discrete rotations of an n-gon. So one would think that as , would become isomorphic to U(1) since the n-gon geometrically approaches a circle. But is isomorphic to which geometrically is a line infinite in both directions.
Technically, for what I’m interested in, I suspect U(1) would be too restrictive anyway. I’m more interested in finding a direct isomorphism between and U(n) or, at the very least, something broader than U(1) and, preferably, broader than SU(2) as well.
So if you’re a regular reader of this blog and you know of any such isomorphism, post it here!