Combinatorics of the double shuffle Lie algebra

In this article we give two combinatorial properties of elements satisfying the stuffle relations; one showing that double shuffle elements are determined by less than the full set of stuffle relations, and the other a cyclic property of their coefficients. Although simple, the properties have some useful applications, of which we give two. The first is a generalization of a theorem of Ihara on the abelianizations of elements of the Grothendieck-Teichmüller Lie algebra grt to elements of the double shuffle Lie algebra in a much larger quotient of the polynomial algebra than the abelianization, namely the trace quotient introduced by Alekseev and Torossian. The second application is a proof that the Grothendieck-Teichmüller Lie algebra grt injects into the double shuffle Lie algebra ds, based on the recent proof by H. Furusho of this theorem in the pro-unipotent situation, but in which the combinatorial properties provide a significant simplification. §1. The cyclic property Write Y for the alphabet {y1, y2, y3, . . .} and U for the alphabet {u1, u2, u3, . . .}, where yi and ui are given the weight i, and these two alphabets are related by the expression (1) u1+u2+· · · = log(1+y1+y2+· · · ) = (y1+y2+· · · )− 2(y1+y2+· · · ) 2+· · · , Received Month Day, Year. Revised Month Day, Year. 2000 Mathematics Subject Classification. Primary 17B40, 17B65, 17B70, 12Y05, 05E99.