Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 2 / 4 / b / Solution

Every ordered monomial in the degree-$n$ homogeneous component has the form $x^{n-j}{y}^j$, $0\le j\le n$. The formulas in part (a) showEach nonzero image still has total degree $n$; $K^{-1}$ does too. Since $E,F,K,K^{-1}$ generate the quantum enveloping algebra of sl2, the degree-$n$ component is a submodule, with basis $x^n,x^{n-1}y,\dots ,y^n$ and dimension $n+1$.