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

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.

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 = ab + 2aab + 2abb + 4 aabb + 2abab ;

ab ↑ ba = aba + bab + abab + 2abba + 2baab + baba .

The infiltration product is also commutative and associative.

See also

• Hopf algebra of permutations
• Zinbiel algebra

External links

• Shuffle product symbol

• Lothaire (1997) pp.101,126
• Lothaire (1997) p.126
• Lothaire (1997) p.128