Shuffle algebra

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Let be a set (alphabet) and consider the free associative algebra on over the integers, provided with the Hopf algebra structure given by , , . As an Abelian group, is free and graded. Its graded dual is again a Hopf algebra, sometimes called the shuffle-cut Hopf algebra or merge-cut Hopf algebra. Its underlying algebra is the shuffle algebra . As an Abelian group, has as basis the elements of the free monoid (see Free semi-group) of all words in the alphabet . The product of two such words , is the sum of all words of length that are permutations of such that both and appear in their original order. E.g.,

This is the shuffle product. It derives its name from the familiar rifle shuffle of decks of playing cards.

As an algebra over , is a free commutative algebra with as free commutative generators the Lyndon words in , see Lyndon word. I.e.,

[a1]. It is not true that is free over .


[a1] C. Reutenauer, "Free Lie algebras" , Oxford Univ. Press (1993)
How to Cite This Entry:
Shuffle algebra. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article