Natural linear and coalgebra transformations of tensor algebras are studied. The representations of certain combinatorial groups are given. These representations are connected to natural transformations of tensor algebras and to the groups of the homotopy classes of maps from the James construction to loop spaces. Applications to homotopy theory appear in a sequel [Applications of combinatorial groups to Hopf invariant and the exponent problem, Algebr. Geom. Topol. 6 (2006) 2229-2255].
"Natural transformations of tensor algebras and representations of combinatorial groups." Algebr. Geom. Topol. 6 (5) 2189 - 2228, 2006. https://doi.org/10.2140/agt.2006.6.2189