In algebra, the Malvenuto–Poirier–Reutenauer Hopf algebra of permutations or MPR Hopf 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), Swedis ...

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 one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

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 diagrams ...

, 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 basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...

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 i ...

(''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\l ...

quasisymmetric function In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables. This ring ge ...

s, 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 the following commutative diagram. The duality between QSym and NSym is shown in the main diagonal of this diagram.

References

* * *{{citation, MR=1334836 , last=Poirier, first= 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