Quasi-shuffle products

Details Quasi-shuffle products Quasi-shuffle products

1999-07-27

arxiv.org/abs/math/9907173v1

J. Algebraic Combin. 11 (2000), 49-68.

Mathematics

Quantum Algebra

18 pages. See also http://www.nadn.navy.mil/Users/math/meh/research.html

Scientific paper

Given a locally finite graded set A and a commutative, associative operation on A that adds degrees, we construct a commutative multiplication * on the set of noncommutative polynomials in A which we call a quasi-shuffle product; it can be viewed as a generalization of the shuffle product. The resulting commutative algebra can be given the structure of a Hopf algebra (_A_,*,Delta). In the case where A is the set of positive integers and the operation on A is addition, (_A_,*,Delta) is the Hopf algebra of quasi-symmetric functions. If rational coefficients are allowed, there is a Hopf algebra isomorphism exp from the shuffle Hopf algebra on A onto (_A_,*,Delta). We discuss the dual of (_A_,*,Delta), and define a deformation *_q of * that coincides with * when q = 1 and is isomorphic to the concatenation product when q is not a root of unity. Finally, we discuss various examples of this construction.

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