Noncommutative symmetric functions and the inversion problem

作者：Zhao Wenhua^*

来源：International Journal of Algebra and Computation, 2008, 18(5): 869-899.

DOI：10.1142/S0218196708004615

• NCS systems
• noncommutative symmetric functions
• formal automorphisms in commutative or noncommutative variables
• D-log's
• the formal flows
• the Jacobian conjecture

Let K be any unital commutative Q-algebra and z = (z(1), z(2), ... , zn) commutative or noncommutative variables. Let t be a formal central parameter and K[[t]]<< Z >> the formal power series algebra of z over K[[t]]. In [29], for each automorphism F-t(z) = z - H-t(z) of K[[t]]<< z >> with H-t (= 0)(z) = 0 and o(H(z)) >= 1, a NCS (noncommutative symmetric) system [28] Omega(Ft) has been constructed. Consequently, we get a Hopf algebra homomorphism S-Ft : NSym -> D << z >> from the Hopf algebra NSym [9] of NCSFs (noncommutative symmetric functions). In this paper, we first give a list for the identities between any two sequences of differential operators in the NCS system Omega(Ft) by using some identities of NCSFs derived in [9] and the homomorphism S-Ft. Secondly, we apply these identities to derive some formulas in terms of differential operator in the system Omega(Ft) for the Taylor series expansions of u(F-t) and u(F-t(-1)) (u(z) is an element of K[[t]]<< z >>); the D-Log and the formal flow of F-t and inversion formulas for the inverse map of F-t. Finally, we discuss a connection of the well-known Jacobian conjecture with NCSFs.

• 出版日期2008-8