# Is there any Lie algebra structure on the sheaf of sections of adjoint bundle

Let $$X$$ be an irreducible smooth projective variety over $$\mathbb{C}$$. Let $$G$$ be an affine algebraic group over $$\mathbb{C}$$. Let $$p : E_G \longrightarrow X$$ be a holomorphic principal $$G$$-bundle on $$X$$. Let $$ad(E_G) = E_G \times^G \mathfrak{g}$$ be the adjoint vector bundle of $$E_G$$ associated to the adjoint representation $$ad : G \longrightarrow End(\mathfrak{g})$$ of $$G$$ on its Lie algebra $$\mathfrak{g}$$. The fibers of $$ad(E_G)$$ are $$\mathbb{C}$$-linearly isomorphic to $$\mathfrak{g}$$. Consider $$ad(E_G)$$ as a sheaf of $$\mathcal{O}_X$$-modules on $$X$$.

Question: Is there any $$\mathcal{O}_X$$-bilinear homomorphism $$[,] : ad(E_G)\times ad(E_G) \to ad(E_G)$$ giving a Lie algebra structure on the sheaf $$ad(E_G)$$?

A principal $$G$$-bundle gives a monoidal functor from the category of representations of $$G$$ to the category of vector bundles. In particular, it takes the morphism $$[-,-] \colon \mathfrak{g} \otimes \mathfrak{g} \to \mathfrak{g}$$ of $$G$$-representations (for the adjoint action) to a morphism of vector bundles $$[-,-] \colon ad(E_G) \otimes ad(E_G) \to ad(E_G).$$ By functoriality, it is skew-symmetric and satisfies the Jacobi identity, hence provides the sheaf $$ad(E_G)$$ with a Lie algebra structure.
• @VítTuček: The Jacobian identity says that the sum of three maps $ad(E_G) \otimes ad(E_G) \otimes ad(E_G) \to ad(E_G)$ vanishes. These maps come from three maps $\mathfrak{g} \otimes \mathfrak{g} \otimes \mathfrak{g} \to \mathfrak{g}$. The sum of the latter maps is zero, hence so is the sum of the former maps. – Sasha Dec 10 '18 at 16:41
Yes. It boils down to natural isomorphism $$ad(E_G) \otimes ad(E_G) \simeq E_G \times^G (\mathfrak{g}\otimes \mathfrak{g})$$ which allows you to compose tensor product of sections with the bracket on $$\mathfrak{g}\otimes \mathfrak{g}$$.