Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 2 / 2 / a

Fix the quantum enveloping algebra of sl2 conventionThe printed pair of matrices contains an error. On its first basis vector, the printed matrices give $(EF-FE)v_0=ϵ[2{]}^2{v}_0$, whereas the defining commutator gives $ϵ[2]v_0$. Since $q$ is not a root of unity, $[2]\ne 0,1$: $[2]=0$ would give $q^2=-1$, and $[2]=1$ would give $q^2-q+1=0$. Thus the two printed matrices cannot represent this algebra. Keeping the printed $F$ matrix requires exchanging the two nonzero entries of the printed $E$ matrix. We derive that corrected pair rather than assume a classification theorem.

As $q$ is not a root of unity, successive applications of $E$ multiply $K$ eigenvalues by $q^2$. Finiteness of the spectrum therefore makes $E$ a nilpotent operator. Start from a $K$ eigenvector and apply $E$ until the last nonzero vector $v$; then $Ev=0$ and $Kv=\lambda v$, with $\lambda \ne 0$. The defining commutator gives the highest-weight string for quantum sl2For completeness, at $j=1$ this is the commutator applied to $v$. To advance from $j$ to $j+1$, write $E{F}^{j+1}v=F(E{F}^{j}v)+(K-K^{-1})F^{j}v/(q-q^{-1})$, use $K{F}^{j}v=\lambda q^{-2j}{F}^{j}v$, and combine the two coefficients; their sum is the displayed coefficient at $j+1$.

Let $N$ be the first index with $F^{N}v=0$. Such an index exists because nonzero $F^{j}v$ have distinct $K$ eigenvalues $\lambda q^{-2j}$. The linear span of $v,Fv,\dots ,F^{N-1}v$ is stable under $K,E,F$, so simplicity makes it all of $V$. The distinct eigenvalues make these vectors independent; hence $N=3$. Applying the string formula to $F^{3}v=0$ givesAs $[3]\ne 0$ and $F^{2}v\ne 0$, ${\lambda }^2=q^4$, so $\lambda =ϵq^2$ with $ϵ=\pm 1$.

Choose the basis $v_0=v$, $v_1=Fv/[2]$, $v_2=F^{2}v/[2]$. Then $F{v}_0=[2]v_1$, $F{v}_1=v_2$, $E{v}_1=ϵv_0$ and $E{v}_2=ϵ[2]v_1$. The correct matrices in this basis areAlternatively, preserving the printed $E$ instead requires $F_{21}=1$, $F_{32}=[2]$. Both corrected pairs are related by a diagonal change of basis.