THE PETER–WEYL THEOREM FOR CLASSICAL AND QUANTUM slN

as (G,G)-bimodules, and where the sum is over all irreducible representations of G. Let H be a Hopf algebra. If H is finite dimensional, then H∗ is also a Hopf algebra by dualizing all operations from H. We run into issues if H is infinite-dimensional, but we can find a fix. For a finite-dimensional H-module V , an element v ∈ V , and f ∈ V ∗, define a linear functional on H by cf,v(u) = f(uv). Call these linear functionals matrix coefficients. Proposition 0.2. (1) cf,vcg,w = cg⊗f,v⊗w (2) For a representation V , let {e1, . . . , en} be a basis and {e1, . . . , en} be a dual basis for V ∗. Then ∆(cf,v) = ∑ i cf,ei ⊗ cei,v (3) H0 has an antipode S = S∗. (4) Suppose φ : V →W is a map of H-modules. Then cf,φv = cφ∗f,v. The algebra of matrix coefficients is the subalgebra H0 of H∗ spanned by all matrix coefficients. Recall that H∗ is an H −H-bimodule as follows. For f ∈ H∗, a, b ∈ H, (a⊗ b)f is the function satisfying: (a⊗ b)f(u) = f(S(a)ub). This equips H0 with a bimodule structure by restriction.