Highest-weight string for quantum sl2
= Highest-weight string for quantum sl2
{title2=$EF^jv=[j]\frac{\lambda q^{1-j}-\lambda^{-1}q^{j-1}}{q-q^{-1}}F^{j-1}v$}
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 $\lambda=\epsilon q^n$, $\epsilon=\pm1$. Its distinct $K$ <eigenvalues> make the string vectors independent, and the nonzero raising coefficients prove simplicity.