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Hopf algebra of permutations

Index 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 Sn, and is a non-commutative analogue of the Hopf algebra of symmetric functions. [1]

11 relations: Algebra over a field, Coalgebra, Cofree coalgebra, Free abelian group, Free algebra, Hopf algebra, Noncommutative symmetric function, Quasisymmetric function, Ring of symmetric functions, Shuffle algebra, Symmetric function.

Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.

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Coalgebra

In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras.

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Cofree coalgebra

In algebra, the cofree coalgebra of a vector space or module is a coalgebra analog of the free algebra of a vector space.

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Free abelian group

In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis.

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Free algebra

In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables.

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Hopf algebra

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.

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Noncommutative symmetric function

In mathematics, the noncommutative symmetric functions form a Hopf algebra NSymm analogous to the Hopf algebra of symmetric functions.

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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.

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Ring 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.

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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 X ⧢ Y of two words X, Y: the sum of all ways of interlacing them.

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Symmetric function

In mathematics, a symmetric function of n variables is one whose value given n arguments is the same no matter the order of the arguments.

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Algebra of permutations, MPR Hopf algebra, MPR algebra, MR Hopf algebra, MR algebra, Malvenuto-Reutenauer Hopf algebra, Malvenuto-Reutenauer algebra, Malvenuto–Poirier–Reutenauer Hopf algebra, Malvenuto–Poirier–Reutenauer algebra, Malvenuto–Reutenauer Hopf algebra, Malvenuto–Reutenauer algebra.

References

[1] https://en.wikipedia.org/wiki/Hopf_algebra_of_permutations

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