Generalized Shuffles Related to Nijenhuis and TD-Algebras

Abstract

Shuffle type products are well known in mathematics and physics. They are intimately related to Loday's dendriform algebras and were extensively used to give explicit constructions of free 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, 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. Our construction is related to Loday's unital commutative tridendriform algebra, 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.

Key Words:

• Bialgebra
• Dendriform algebra
• Nijenhuis algebra
• Quasi-shuffle product
• Rota–Baxter algebra
• Shuffle product
• Spitzer identity

2000 Mathematics Subject Classification:

• 16A06
• 16W30
• 17A50

ACKNOWLEDGMENTS

The first author acknowledges greatly the support by the European Post-Doctoral Institute and Institut des Hautes Études Scientifiques (I.H.É.S.). L. P. would like to acknowledge the warm hospitality he experienced during his stay at I.H.É.S. when major parts of this article were written. We thank W. Schmitt and D. Manchon for useful discussions and comments.

Notes

Referring to the physicists C. N. Yang from China and the Australian R. Baxter.

Communicated by L. Small.