In mathematics, a shuffle algebra is a
Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...
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 In mathematics, a free Lie algebra over a field ''K'' is a Lie algebra generated by a set ''X'', without any imposed relations other than the defining relations of alternating ''K''-bilinearity and the Jacobi identity.
Definition
The definition ...
on the set.
Over the rational numbers, the shuffle algebra is isomorphic to the
polynomial algebra
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (mathematics), ring formed from the set (mathematics), set of polynomials in one or more indeterminate (variable), indeterminates (traditionally ...
in the
Lyndon word
In mathematics, in the areas of combinatorics and computer science, a Lyndon word is a nonempty string that is strictly smaller in lexicographic order than all of its rotations. Lyndon words are named after mathematician Roger Lyndon, who investi ...
s.
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
In mathematics, specifically commutative algebra, a divided power structure is a way of introducing items with similar properties as expressions of the form x^n / n! have, also when it is not possible to actually divide by n!.
Definition
Let '' ...
, 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
In formal language theory, the empty string, or empty word, is the unique string of length zero.
Formal theory
Formally, a string is a finite, ordered sequence of characters such as letters, digits or spaces. The empty string is the special cas ...
, ''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. Perhaps most familiar as a pr ...
and
associative
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
.
The shuffle product of two words in some alphabet is often denoted by the shuffle product symbol ⧢ (
Unicode
Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Char ...
character U+29E2 , derived from the
Cyrillic
The Cyrillic script ( ) is a writing system used for various languages across Eurasia. It is the designated national script in various Slavic, Turkic, Mongolic, Uralic, Caucasian and Iranic-speaking countries in Southeastern Europe, Ea ...
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 In algebra, the Malvenuto–Poirier–Reutenauer Hopf algebra of permutations or MPR Hopf algebra is a Hopf algebra with a basis of all elements of all the finite symmetric groups ''S'n'', and is a non-commutative analogue of the Hopf algebra o ...
*
Zinbiel algebra In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity:
:(a \circ b) \circ c = a \circ (b \circ c) + a \circ (c \circ b).
Zinbiel algebras were intro ...
References
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External links
Shuffle product symbol
Combinatorics
Hopf algebras