In algebra, the Malvenuto–Poirier–Reutenauer Hopf algebra of permutations or MPR Hopf 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 of all elements of all the finite symmetric groups ''S''_''n'', and is a non-commutative analogue of the

Hopf algebra of symmetric functions In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in ''n'' indeterminates, as ''n'' goes to infinity. This ring serves as universal structure in which ...

. It is both free as an

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

and graded- cofree as a graded

coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagram ...

, so is in some sense as far as possible from being either commutative or cocommutative. It was introduced by and studied by .

Definition

The underlying

free abelian group In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation (mathematics), operation that is associative, commutative, and inverti ...

of the MPR algebra has a basis consisting of the disjoint union of the symmetric groups ''S''_''n'' for ''n'' = 0, 1, 2, .... , which can be thought of as permutations. The identity 1 is the empty permutation, and the counit takes the empty permutation to 1 and the others to 0. The product of two permutations (''a''₁,...,''a''_''m'') and (''b''₁,...,''b''_''n'') in MPR is given by the

shuffle product 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 ...

(''a''₁,...,''a''_''m'') ''ш'' (''m'' + ''b''₁,...,''m'' + ''b''_''n''). The coproduct of a permutation ''a'' on ''m'' points is given by Σ_{''a''=''b''*''c''} st(''b'') ⊗ st(''c''), where the sum is over the ''m'' + 1 ways to write ''a'' (considered as a sequence of ''m'' integers) as a concatenation of two sequences ''b'' and ''c'', and st(''b'') is the standardization of ''b'', where the elements of the sequence ''b'' are reduced to be a set of the form while preserving their order. The antipode has infinite order.

Relation to other algebras

The Hopf algebra of permutations relates the rings of

symmetric function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\ ...

s, quasisymmetric functions, and

noncommutative symmetric function In mathematics, the noncommutative symmetric functions form a Hopf algebra NSymm analogous to the Hopf algebra of symmetric functions. The Hopf algebra NSymm was introduced by Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladim ...

s, (denoted Sym, QSym, and NSym respectively), as depicted in the following commutative diagram. The duality between QSym and NSym is shown in the main diagonal of this diagram.

References

* * *{{citation, mr=1334836 , last1=Poirier, first1= Stéphane, last2= Reutenauer, first2= Christophe , title=Algèbres de Hopf de tableaux , journal=Ann. Sci. Math. Québec, volume= 19 , year=1995, issue= 1, pages= 79–90 Hopf algebras