I am thinking of the following situation:
The lie algebra $g = sl_2(\mathbb(C))$, and $V(1)$ is the unique dimension 2 irreducible representation (the defining representation). Let $U$ be the universal enveloping algebra of $g$.
I pick a basis $v_1, v_2$ for $V(1)$, where $H v_1 = v_1$ and $Hv_2 = -v_2$, and $v^1, v^2$ is the dual basis. (So $Ev_1 = 0$, $Fv_2 = 0$, $E v_2= v_1$, $Fv_1 = v_2$.)
$c^i_j(u) = v^i( u. v_j)$, for $u \in U$, is a matrix coefficient for the Hopf-algebra $H$.
I need to verify that $c^1_1 c^2_2 - c^1_2 c^2_1 = 1$. (In order to show that the algebra of matrix coefficients is isomorphic to the coordinate ring of $SL_2(\mathbb{C})$.)
Here is my attempt:
$c^1_1 c^2_2(u) = m \circ (v^1 \otimes v^2) [ \Delta(u) (v_1 \otimes v_2)] = m \circ (v^1 \otimes v^2)[ ( u \otimes 1 + 1 \otimes u). (v_1 \otimes v_2)] = v^1(uv_1) v^2(v_2) + v^1(v_1) v^2 (u.v_2) = v^1( u. v_1) + v^2 (u. v_2)$
Plugging in $u = E$, $F$ or $H$ from the Lie algebra gives $0$, given my choice of basis above. Plugging in $1$ gives $2$, so $c^1_1 c^2_2 = 2$. (By which I mean, 2 times the unit of $U^*$.)
$c^1_1 c^2_2(u) = m \circ (v^1 \otimes v^2) [ \Delta(u) (v_2 \otimes v_1)] = m \circ (v^1 \otimes v^2)[ ( u \otimes 1 + 1 \otimes u). (v_2 \otimes v_1)] = v^1(uv_2) v^2(v_1) + v^1(v_2) v^2 (u.v_2) = 0$
(since $v^i(v_j) = \delta^i_j$).
Clearly, the difference is not $1$! Please tell me what I am doing wrong.
Thank you for reading!