| 词条 | Shuffle algebra |
| 释义 |
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. Shuffle productThe shuffle product of words of lengths m and n is a sum over the {{sfrac|(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] ua ⧢ vb = (u ⧢ vb)a + (ua ⧢ v)b The shuffle product was introduced by {{harvtxt|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+2932 {{sc|shuffle product}}, derived from the Cyrillic letter {{angbr|ш}} sha). Infiltration productThe closely related infiltration product was introduced by {{harvtxt|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] See also
References1. ^Lothaire (1997) pp.101,126 2. ^Lothaire (1997) p.126 3. ^Lothaire (1997) p.128
|last=Hazewinkel|first= Michiel|last2= Gubareni|first2= Nadiya|last3= Kirichenko|first3= V. V. |title=Algebras, rings and modules. Lie algebras and Hopf algebras|series= Mathematical Surveys and Monographs|volume= 168|publisher= American Mathematical Society|place= Providence, RI|year= 2010|ISBN= 978-0-8218-5262-0|doi=10.1090/surv/168}}
External links
2 : Combinatorics|Algebra |
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