= Quantum enveloping algebra action on the quantum plane
{title2=$E(x^ry^s)=[s]_qx^{r+1}y^{s-1}$}
For $yx=qxy$, the <quantum enveloping algebra of sl2> acts by $K(x^ry^s)=q^{r-s}x^ry^s$, $E(x^ry^s)=[s]_qx^{r+1}y^{s-1}$ and $F(x^ry^s)=[r]_qx^{r-1}y^{s+1}$. Negative exponents here have zero coefficients and are omitted. The <quantum integer> identity $[r][s+1]-[s][r+1]=[r-s]$ verifies the commutator. These operations preserve total degree and make the <quantum plane> a <module algebra>.
Back to article page
