Question

Keep the notations above and define f, g ∈ E by g(ei) = ei+1, f(ei) = ei−1 (with the convention that e0 = 0). Let R be the k-subalgebra of E generated by f and g. Show that R is isomorphic to S : =k <x, y> /(xy − 1), with a k - isomorphism matching f with x and g with y.