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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 of symmetric functions. It is both free as an algebra and graded-cofree as a graded coalgebra, 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 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 (''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hopf algebra of permutations」の詳細全文を読む スポンサード リンク
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