Highest-weight string for quantum sl2 ($E{F}^{j}v=[j]\frac{\lambda q^{1-j}-{\lambda }^{-1}{q}^{j-1}}{q-q^{-1}}{F}^{j-1}v$)

If $Ev=0$ and $Kv=\lambda v$, induction using the defining commutator gives the displayed identity. When $q$ is not a root of unity, a finite string of length $n+1$ in a simple module has $\lambda =ϵq^n$, $ϵ=\pm 1$. Its distinct $K$ eigenvalues make the string vectors independent, and the nonzero raising coefficients prove simplicity.