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 X ⧢ Y of two words X, Y: the sum of all ways of interlacing them. The interlacing is given by the riffle shuffle permutation.
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 occurs in generic settings in non-commutative algebras; this is because it is able to preserve the relative order of factors being multiplied together - the riffle shuffle permutation. This can be held in contrast to the divided power structure, which becomes appropriate when factors are commutative.
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 ⧢ xy = abxy + axby + xaby + axyb + xayb + xyab
- aaa ⧢ aa = 10aaaaa
It may be defined inductively by[1]
- u ⧢ ε = ε ⧢ u = u
- ua ⧢ vb = (u ⧢ vb)a + (ua ⧢ v)b
where ε is the empty word, a and b are single elements, and u and v are arbitrary words.
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: this is the riffle shuffle permutation. The product is commutative and associative.[2]
The shuffle product of two words in some alphabet is often denoted by the shuffle product symbol ⧢ (Unicode character U+29E2 SHUFFLE PRODUCT, derived from the Cyrillic letter ⟨ш⟩ sha).
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 = ab + 2aab + 2abb + 4 aabb + 2abab
- ab ↑ ba = aba + bab + abab + 2abba + 2baab + baba
The infiltration product is also commutative and associative.[3]
References
- Lothaire 1997, p. 101,126
- Lothaire 1997, p. 126
- Lothaire 1997, p. 128
- 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, JSTOR 1970044, MR 0102539, Zbl 0142.22304
- Eilenberg, Samuel; Mac Lane, Saunders (1953), "On the groups of H(Π,n). I", Annals of Mathematics, Second Series, 58 (1): 55–106, doi:10.2307/1969820, 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, vol. 9, Coimbra: Universidade de Coimbra Departamento de Matemática, MR 1399082
- Hazewinkel, M. (2001) [1994], "Shuffle algebra", Encyclopedia of Mathematics, EMS Press
- Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010), Algebras, rings and modules. Lie algebras and Hopf algebras, Mathematical Surveys and Monographs, vol. 168, American Mathematical Society, doi:10.1090/surv/168, ISBN 978-0-8218-5262-0, MR 2724822, Zbl 1211.16023
- Lothaire, M. (1997), Combinatorics on words, Encyclopedia of Mathematics and Its Applications, vol. 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, vol. 7, Oxford University Press, ISBN 978-0-19-853679-6, MR 1231799, Zbl 0798.17001