On the algebra of quasi-shuffles. (English) Zbl 1126.16029
A commutative tri-dendriform algebra (CTD-algebra) is determined by two operations, denoted by \(\prec\) (left) and \(\cdot\) (dot), satisfying certain axioms. The aim of the paper is to show that the tensor module of the (non-constant) polynomial algebra with the quasi-shuffle algebra structure (called the stuffle algebra) is a free unital CTD-algebra. A non-commutative version of this is also proved in the paper. A quasi-shuffle algebra is in fact a CTD-bialgebra. The author proves a structure theorem for connected CTD-bialgebras.
Reviewer: Goutam Mukherjee (Kolkata)
MSC:
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 18D50 | Operads (MSC2010) |
| 17A30 | Nonassociative algebras satisfying other identities |
Keywords:
shuffle products; tensor modules; quasi-shuffle algebras; commutative tri-dendriform algebras; bialgebras; coalgebras; operadsReferences:
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