Algebraic Formulae for the $q$-inverse in a Free Group

J. Math. Sci. Univ. Tokyo
Vol. 8 (2001), No. 4, Page 721--734.

Habiro, K. ; Kamada, S. ; Matsumoto, Y. ; Yoshikawa, K.
Algebraic Formulae for the $q$-inverse in a Free Group
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Let $F_n$ be a free group with $n$ fixed generators, which we assume are linearly ordered. In a previous paper [\rM], a curious mapping $I: F_n \to F_n$ was introduced pictorially. It is a \lq\lq square root\rq\rq\ of the inner automorphism of $F_n$ induced by the \lq\lq smallest\rq\rq\ generator. In the present paper, two algebraic formulae will be given, by which one can compute the mapping $I$ purely algebraically.

Mathematics Subject Classification (2000): Primary 20F36, 57M05; Secondary 57M25, 57M50
Mathematical Reviews Number: MR1868298

Received: 2001-05-25