Deformations and inversion formulas for formal automorphisms in noncommutative variables

作者：Zhao Wenhua^*

来源：International Journal of Algebra and Computation, 2007, 17(2): 261-288.

DOI：10.1142/S0218196707003676

• noncommutative inversion problem
• deformations of formal maps in noncommutative variables
• the inviscid Burgers-like equations
• noncommutative inversion formulas

Let z = ( z(1), z(2),..., z(n)) be noncommutative free variables and t a formal parameter which commutes with z. Let k be any unital integral domain of any characteristic and F-t( z) = z - H-t(z) with H-t(z) is an element of k[[ t]]<< z >> (x n) and the order o(H-t(z)) >= 2. Note that F-t(z) can be viewed as a deformation of the formal map F(z) := z - H-t=1(z) when it makes sense (for example, when H-t(z)is an element of k[t]<< z >> (x n)). The inverse map G(t)(z) of F-t(z) can always be written as G(t)(z) = z+ M-t(z) with M-t(z)is an element of k[[t]]<< z >> (x n) and o(M-t(z)) = 2. In this paper, we first derive the PDEs satisfied by M-t(z) and u(F-t), u(G(t)) is an element of k[[t]]<< z >> with u(z) is an element of k << z >> in the general case as well as in the special case when H-t(z) = tH(z) for some H(z) is an element of k << z >> (x n). We also show that the elements above are actually characterized by certain Cauchy problems of these PDEs. Secondly, we apply the derived PDEs to prove a recurrent inversion formula for formal maps in noncommutative variables. Finally, for the case char. k = 0, we derive an expansion inversion formula by the planar binary rooted trees.

• 出版日期2007-3