Séminaire Lotharingien de Combinatoire, B74f (2018), 22 pp.
Van Chien Bui, Gérard H.E. Duchamp,
Quoc Huan Ngô, Vincel Hoang Ngoc Minh
and Christophe Tollu
(Pure) Transcendence Bases in φ-Deformed Shuffle Bialgebras
Abstract.
Computations with integro-differential operators are often carried out
in an associative algebra with unit,
and they are essentially non-commutative computations. By adjoining a
cocommutative co-product,
one can have those operators
act on a bialgebra isomorphic to an enveloping algebra.
This gives an adequate framework
for a computer-algebra implementation via monoidal factorization,
(pure) transcendence bases and Poincaré-Birkhoff-Witt bases.
In this paper, we systematically study these deformations, obtaining
necessary and sufficient conditions for the operators to exist, and we
give the most general cocommutative deformations of the shuffle
co-product
and an effective construction of pairs of bases in duality. The paper
ends by the combinatorial setting of local systems of coordinates on
the group of group-like series.
Received: July 3, 2015.
Revised: May 11, 2017.
Accepted: January 21, 2018.
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