NCS Systems over Differential Operator Algebras and the Grossman-Larson Hopf Algebras of Labeled Rooted Trees · arXivDeskAbstract
Let K be any unital commutative -algebra and W any non-empty subset of ^+. Let z=(z1,...,zn)
be commutative or noncommutative free variables and
a formal central parameter. % Denote uniformly by
and
the formal power series algebras % of
over
and
, respectively. Let
be the unital algebra generated by the differential operators of
which increase the degree in
by at least
and
the group of automorphisms
Ft(z)=z−Ht(z) of
with
o(Ht(z))≥α and
Ht=0(z)=0 . First, we study a connection of the systems
(F_t∈ )
(GTS-I, GTS-II) over the differential operators algebra
and the system
Ω_^W
(GTS-IV) over the Grossman-Larson Hopf algebra
_GL^W
(GL, F1, F2) of
-labeled rooted trees. We construct a Hopf algebra homomorphism
A_F_t: _GL^W →
(F_t∈ )
such that
A_F_t^× 5(Ω_^W) =Ω_F_t
. Secondly, we generalize the tree expansion formulas for the inverse map (BCW, Wr3), the D-Log and the formal flow (WZ) of
in the commutative case to the noncommutative case. Thirdly, we prove the injectivity of the specialization
: NSym → _GL^^+
(GTS-IV) of NCSF's (noncommutative symmetric functions) (G-T). Finally, we show the family of the specializations
_F_t
of NCSF's with all
and the polynomial automorphisms
Ft=z−Ht(z) with
homogeneous and the Jacobian matrix
strictly lower triangular can distinguish any two different NCSF's. The graded dualized versions of the main results above are also discussed.