Tannakian theory for Lie algebras

Solution 1:

Yes, the statement is true for general $G$ (not just reductive).

Write $A$ for the ring of regular functions on $G$, which is a commutative Hopf algebra. For each representation $V$, there is a comodule structure $\Delta_V:F(V)\to F(V)\otimes A$. The multiplication $m:A\otimes A\to A$ is compatible with the tensor product of representations in the sense that the following two maps are equal: $$ F(V)\otimes F(W)\xrightarrow{\Delta_V\otimes\Delta_W} (F(V)\otimes A)\otimes (F(W)\otimes A)\to F(V)\otimes F(W)\otimes A\otimes A\xrightarrow{\mathrm{Id}\otimes m}F(V)\otimes F(W)\otimes A, $$ $$ F(V)\otimes F(W)=F(V\otimes W)\xrightarrow{\Delta_{V\otimes W}} F(V\otimes W)\otimes A=F(V)\otimes F(W)\otimes A. $$

There is a bijection between the functorial endomorphisms of $F$ (ignoring the tensor structure) and the linear maps $\varphi:A\to \mathbb{C}$: the endomorphism corresponding to $\varphi$ is $$ F(V)\xrightarrow{\Delta_V} F(V)\otimes A\xrightarrow{\mathrm{Id}\otimes\varphi} F(V). $$

Write $\epsilon:A\to\mathbb{C}$ for the counit, which corresponds to the identity endomorphism ($\epsilon$ takes a regular function to its value at the identity element of $G$). The condition that $\varphi$ corresponds to an endomorphism satisfying your condition is that the map $$ F(V)\otimes F(W)\xrightarrow{\Delta_V\otimes\Delta_W} (F(V)\otimes A)\otimes (F(W)\otimes A)\to F(V)\otimes F(W)\otimes A\otimes A\xrightarrow{\mathrm{Id}\otimes(\varphi\otimes \epsilon+\epsilon\otimes\varphi)}F(V)\otimes F(W) $$ coincides with the map $$ F(V)\otimes F(W)\xrightarrow{\Delta_{V\otimes W}}F(V\otimes W)\otimes A\xrightarrow{\sim}F(V)\otimes F(W)\otimes A\xrightarrow{\mathrm{Id}\otimes\varphi}F(V)\otimes F(W). $$ Combining this with the compatibility of the multiplication on $A$ and the tensor product, this condition is equivalent to the condition that $\varphi\circ m=\varphi\otimes \epsilon+\epsilon\otimes\varphi$. This is the same as $$ \varphi(xy)=\varphi(x)\epsilon(y)+\epsilon(x)\varphi(y), $$ i.e., that $\varphi$ is a derivation at the identity on $G$. The Lie algebra of a group is identified with the derivations at the identity, so we conclude that endomorphisms $\varphi$ satisfying your condition correspond to elements of the Lie algebra of $G$.