Braided Bialgebras of Hecke-type

The paper is devoted to prove a version of Milnor-Moore Theorem for connected braided bialgebras that are infinitesimally cocommutative. Namely in characteristic different from $2$, we prove that, for a given connected braided bialgebra $(A,\mathfrak{c}_A)$ which is infinitesimally $\lambda $-cocommutative for some element $\lambda \neq 0$ that is not a root of one in the base field, then the infinitesimal braiding of $A$ is of Hecke-type of mark $\lambda $ and $A$ is isomorphic as a braided bialgebra to the symmetric algebra of the braided subspace of its primitive elements.