On the algebra of quasi-shuffles

Details On the algebra of quasi-shuffles On the algebra of quasi-shuffles

2005-06-24

arxiv.org/abs/math/0506498v2

Manuscripta Mathematica 123 (2007), no. 1, 79--93

Mathematics

Quantum Algebra

13 pages

Scientific paper

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 $T^q(R)$. We show that if $R$ is the polynomial algebra, then $T^q(R)$ 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 $(Com, As, Lie)$.

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