Differential operator specializations of noncommutative symmetric functions

Abstract

Let K be any unital commutative Q-algebra and z = (z 1 , ..., z n ) commutative or noncommutative free variables. Let t be a formal parameter which commutes with z and elements of K. We denote uniformly by K z and K[[t]] z the formal power series algebras of z over K and K[[t]], respectively. For any α ≥ 1, let D [α] z be the unital algebra generated by the differential operators of K z which increase the degree in z by at least α − 1 and Az , we introduce five sequences of differential operators of K z and show that their generating functions form a NCS (noncommutative symmetric) system ([Z4]) over the differential algebra D [α] z . Consequently, by the universal property of the NCS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [GKLLRT], we obtain a family of Hopf algebra homomorphisms S Ft :, which are also grading-preserving when F t satisfies certain conditions. Note that, the homomorphisms S Ft above can also be viewed as specializations of NCSFs by the differential operators of K z . Secondly, we show that, in both commutative and noncommutative cases, this family S Ft (with all n ≥ 1 and F t ∈ A[α] t z ) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions ([Ge], [MR], [S]) are also discussed.

Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra

“…In the reverse direction, Wenhua Zhao (see [33], [34], [35]) has for his part rediscovered the constructions of coarborification and then obtained in effect the mechanisms of arborification by dualizing and going to CK. Notably, Zhao's results concern in fact both plain and contracting arborification.…”

“…Proof The proof is based on the following result of Grossman and Larson (see [33], [34]): Let τ the map from GL H to [x, ∂ x ] defined by…”

Noncommutative symmetric systems over associative algebras

“…In the second part, Sections 3 and 4, we review some of the main results that will appear in the sequels [34,36,37]. The main reasons for our including Sections 3 and 4 in this paper are as follows.…”

“…First, we think it is better to provide some concrete examples for the general discussions of NCS systems given in Section 2, so the paper can be read as a more complete introduction to the newly defined NCS systems. Secondly, considering the length of the whole series of papers, we hope that discussions in Sections 3 and 4 can also serve as a shorter survey or review of some of the main results obtained in the following papers [34,36,37] (precisely speaking, they should be read as an announcement since these sequel papers for the time being are still under submission).…”

“…In Section 3, we discuss the NCS systems that will be constructed in [34] over differential operator algebras in commutative or noncommutative free variables. Certain properties of the resulting specialization of NCSFs by differential operators, which will be proved in [34] and [37], are also discussed.…”