On the algebra of quasi-shuffles

By Jean-Louis Loday. 2005

68 citations recorded by the source corpus.

For any commutative algebra R the shuffle product on the tensor module T(R) can be deformed to a new product. It is called the quasi-shuffle algebra, or stuffle algebra, and denoted Tq(R). We show that if R is the polynomial algebra, then Tq(R) is free for some algebraic structure called Commutative TriDendriform (CTD-algebras). This result is part of a structure theorem for CTD-bialgebras which are associative as coalgebras and whose primitive part is commutative. In other words, there is a good triple of operads (As, CTD, Com) analogous to (Com, As, Lie). In the last part we give a similar interpretation of the quasi-shuffle algebra in the noncommutative setting.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f40772811044574de1dd6ce0601d1f107b578fe8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Generalized bialgebras and triples of operads (2006, background): “(4)The relation x · (y ≺ z) = (x ≺ z) · y is missing in the axioms of a CTD-algebra given in [49].”
FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES (2005, background): “Furthermore, the mixable shuffle product in the construction of [36] appeared also in the work of Goncharov [30] to study motivic shuffle relations and the work of Hazewinkle [40] on overlapping shuffles.”
Renormalization of multiple zeta values (2006, background): “It is shown [16] to be the same as the mixable shuffle product [26, 27] which is also called overlapping shuffles [28] and generalized shuffles [23], and can be interpreted in terms of Delannoy paths [1, 21, 35 ].”
Combinatorial Hopf algebras (2008, background): “Then, the 7 relations come down to the following 4 relations: (x ≺ y) ≺ z = x ≺ (y ≺ z + z ≺ y + y · z), (x ≺ y) · z = x · (z ≺ y), (x · y) ≺ z = x · (y ≺ z), (x · y) · z = x · (y · z), The second relation has been, unfortunately, mistakenly omitted in [37].”