Volterra Series and Permutation Groups

shuffle product of the permutations u E E k and v' E E k, we mean the following element from ~+~, /,, (i), i < k , vmv'=: Z o'o(~,| where (v~v') ( i ) = ~ . v , ( i _ k ) + l e ' i ' > k . a(~sk+n'cn} Since the permutations form an additive basis of the space ~ , the shuffle multiplication is uniquely extended "by linearity" to any pair of elements from ~ and defines in $ = ~0$n a structure of graded associative algebra. Moreover, the shuffle multiplication can be extended uniquely "by continuity" to an associative multiplication in the completion ~ of the graded space ~. As usually, by the completion of a graded space we mean the space of formal series of the form X an, anE~n, n ~ O while multiplication is performed according to Cauchy: an Ul bn = a a m b n _ k . n~0 \k=0 / The algebra 6 , just as other algebra of formal series, is considered with the standard topology of termwise convergence. The algebra ~ is indeed the completion of ~ in this topology, while the multiplication "according to Cauchy" :is the extension by continuity of the shuffle multiplication in 6. Everywhere in the sequel, u.nless otherwise mentioned, by an associative algebra we mean an associative algebra with identity over R. For each associative algebra ~t, by [ ~] we denote its associated Lie algebra with Lie multiplication [a, b] = a b ba, b E ~. 3. We recall the nonstationary vector field h t in which we were interested in the first section. The mapping h , : v ~ s n I " ' l h'v'"' . . . . . h,v, t , d t l . . , d tn , vET.n, n > 0 , A n can be extended in a unique manner to a continuous homomorphism h of the algebra ~ with shuffle multiplication into the algebra of chronological series. For n = 1, 2 . . . . we denote by 1 n the identity element in the group I~ n, 10 = 1 E R. We have h , In i d + ~ s a . . . f t n . . . . . h , , d t I . . . d r n a ~ l Art 2 the asymptotic expansion of the right chronological exponential. We introduce the notation exp (e )= enl n, n m O elements ex--p(e) are invertible in ~ as any formal series with invertible constant terms. Def'~ition. The group T , generated by the elements exp(t), t ~ R, with the multiplication operation called an abstract variational group. eEl~ . The