In mathematics, a shuffle algebra is a
Hopf algebra Hopf is a German surname. Notable people with the surname include:
* Eberhard Hopf (1902–1983), Austrian mathematician
* Hans Hopf (1916–1993), German tenor
* Heinz Hopf (1894–1971), German mathematician
* Heinz Hopf (actor) (1934–2001), Sw ...
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 In the mathematics of permutations and the study of shuffling playing cards, a riffle shuffle permutation is one of the permutations of a set of n items that can be obtained by a single riffle shuffle, in which a sorted deck of n cards is cut into ...
.
The shuffle algebra on a finite set is the graded dual of the
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representa ...
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 In the mathematics of permutations and the study of shuffling playing cards, a riffle shuffle permutation is one of the permutations of a set of n items that can be obtained by a single riffle shuffle, in which a sorted deck of n cards is cut into ...
. 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 ways of interleaving the two words, as shown in the following examples:
:''ab'' ⧢ ''xy'' = ''abxy'' + ''axby'' + ''xaby'' + ''axyb'' + ''xayb'' + ''xyab''
:''aaa'' ⧢ ''aa'' = 10''aaaaa''
It may be defined inductively by
:''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 . 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 In the mathematics of permutations and the study of shuffling playing cards, a riffle shuffle permutation is one of the permutations of a set of n items that can be obtained by a single riffle shuffle, in which a sorted deck of n cards is cut into ...
. The product is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
and
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
.
The shuffle product of two words in some alphabet is often denoted by the shuffle product symbol ⧢ (
Unicode
Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
character U+29E2 , derived from the
Cyrillic letter
sha).
Infiltration product
The closely related infiltration product was introduced by . 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'' + 2''aab'' + 2''abb'' + 4 ''aabb'' + 2''abab''
:''ab'' ↑ ''ba'' = ''aba'' + ''bab'' + ''abab'' + 2''abba'' + 2''baab'' + ''baba''
The infiltration product is also commutative and associative.
See also
*
Hopf algebra of permutations
*
Zinbiel algebra
References
*
*
*
*
*
*
*
{{refend
External links
Shuffle product symbol
Combinatorics
Algebra