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 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.
Shuffle product
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 .
It may be defined inductively by[1]
- ua ш vb = (u ш vb)a + (ua ш v)b .
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 product is commutative and associative.[2]
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)).
Infiltration product
The closely related infiltration product was introduced by Chen, Fox & Lyndon (1958). It is defined inductively on words over an alphabet A by
- fa ↑ ga = (f↑ga)a + (fa↑g)a + (f↑g)a ;
- fa ↑ gb = (f↑gb)a + (fa↑g)b .
For example:
- ab ↑ ab = abb + 2aab + 2abb + 4 aabb + 2abab ;
- ab ↑ ba = aba + bab + abab + 2abba + 2baab + baba .
The infiltration product is also commutative and associative.[3]
References
- Chen, Kuo-Tsai; Fox, Ralph H.; Lyndon, Roger C. (1958), "Free differential calculus. IV. The quotient groups of the lower central series", Annals of Mathematics. Second Series 68 (1): 81–95, doi:10.2307/1970044, ISSN 0003-486X, JSTOR 1970044, MR 0102539, Zbl 0142.22304 Unknown parameter
|not-used-author1-link=ignored (help) - 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, Zbl 0050.39304
- 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
- Hazewinkel, M. (2001), "S/s110110", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Lothaire, M. (1997), Combinatorics on words, Encyclopedia of Mathematics and Its Applications 17, Perrin, D.; Reutenauer, C.; Berstel, J.; Pin, J. E.; Pirillo, G.; Foata, D.; Sakarovitch, J.; Simon, I.; Schützenberger, M. P.; Choffrut, C.; Cori, R.; Lyndon, Roger; Rota, Gian-Carlo. Foreword by Roger Lyndon (2nd ed.), Cambridge University Press, ISBN 0-521-59924-5, Zbl 0874.20040
- 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, Zbl 0798.17001