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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 . It is noncommutative but cocommutative graded Hopf algebra. It has the Hopf algebra of symmetric functions as a quotient, and is a subalgebra of the Hopf algebra of permutations, and is the graded dual of the Hopf algebra of quasisymmetric function. Over the rational numbers it is isomorphic as a Hopf algebra to the universal enveloping algebra of the free Lie algebra on countably many variables. ==Definition== The underlying algebra of the Hopf algebra of noncommutative symmetric functions is the free ring Z⟨''Z''1, ''Z''2,...⟩ generated by non-commuting variables ''Z''1, ''Z''2, ... The coproduct takes ''Z''''n'' to Σ ''Z''''i'' ⊗ ''Z''''n''–''i'', where ''Z''0 = 1 is the identity. The counit takes ''Z''''i'' to 0 for ''i'' > 0 and takes ''Z''0 = 1 to 1. 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「Noncommutative symmetric function」の詳細全文を読む スポンサード リンク
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