Generalized shuffles related to Nijenhuis and TD-algebras

Abstract

Shuffle and quasi-shuffle products are well-known in the mathematics literature. They are intimately related to Loday's dendriform algebras, and were extensively used to give explicit constructions of free commutative Rota-Baxter algebras. In the literature there exist at least two other Rota-Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle product, the so-called left-shift shuffle, can be used to give an explicit construction of the free unital commutative TD-algebra. We explore some basic properties of TD-operators and show that the free unital commutative Nijenhuis algebra is a TD-algebra. We relate our construction to Loday's unital commutative dendriform trialgebras, including the involutive case. The concept of Rota-Baxter, Nijenhuis and TD-bialgebras is introduced at the end and we show that any commutative bialgebra provides such objects.

20 pages, typos corrected, accepted for publication in Communications in Algebra

References in corpus (6)

• Loday-type Algebras and the Rota-Baxter Relation
• Mixable Shuffles, Quasi-shuffles and Hopf Algebras
• Infinitesimal bialgebras, pre-Lie and dendriform algebras
• On Products and Duality of Binary, Quadratic, Regular Operads
• On some remarkable operads constructed from Baxter operators
• A LL-lattice reformulation of arithmetree over planar rooted trees. Part II