Shuffle algebra

In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product XшY of two words X, Y: the sum of all ways of interlacing them. The shuffle product was introduced by Eilenberg & Mac Lane (1953). The name "shuffle product" refers to the fact that the product can be thought of as a sum over all ways of riffle shuffling two words together.

The shuffle algebra on a finite set is the graded dual of the universal enveloping algebra of the free Lie algebra on the set.

Over the rational numbers, the shuffle algebra is isomorphic to the polynomial algebra in the Lyndon words.

The shuffle product of two words in some alphabet is often denoted by the shuffle product symbol ш (a cyrillic sha, or the unicode character SHUFFLE PRODUCT (U+29E2)).

Definition

The shuffle product of words of lengths m and n is a sum over the (m+n)!/m!n! ways of interleaving the two words, as shown in the following examples:

ab ш xy = abxy + axby + xaby + axyb + xayb + xyab

aaa ш aa = 10aaaaa

References

Eilenberg, Samuel; Mac Lane, Saunders (1953), "On the groups of H(Π,n). I", Annals of Mathematics. Second Series 58: 55–106, ISSN 0003-486X, JSTOR 1969820, MR 0056295
Green, J. A. (1995), Shuffle algebras, Lie algebras and quantum groups, Textos de Matemática. Série B, 9, Coimbra: Universidade de Coimbra Departamento de Matemática, MR 1399082, http://books.google.com/books?id=jxvvAAAAMAAJ
Hazewinkel, M. (2001), "Shuffle algebra", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=S/s110110
Reutenauer, Christophe (1993), Free Lie algebras, London Mathematical Society Monographs. New Series, 7, The Clarendon Press Oxford University Press, ISBN 978-0-19-853679-6, MR 1231799, http://books.google.com/books?id=cBvvAAAAMAAJ

External links

Shuffle product symbol