论文标题
Bialgebras Overs另一个双齿和quasishuffle double Bialgebras
Bialgebras overs another bialgebras and quasishuffle double bialgebras
论文作者
论文摘要
Quasishuffle Hopf algebras, usually defined on a commutative monoid, can be more generally defined on any associative algebra V. If V is a commutative and cocommutative bialgebra, the associated quasishuffle bialgebra QSh(V) inherits a second coproduct $δ$ of contraction and extraction of words, cointeracting with the deconcatenation coproduct $Δ$,使QSH（v）成为双重双重性。为了概括Qsym的Hopf代数的普遍特性（QSym的一种特定案例（一种特定的quasishuff hopf代数案例），由Aguiar，Bergeron和Sottile所公开，我们介绍了V. bialgebra的第二个bialgebra的概念，而V. bialgebra在V. bialgebra上是v-comod的第二个bialgebra，是v-comoduct of v-comodul的概念。 V. V的双重双重双重bras。我们证明，quasishufle bialgebra qsh（v）是v上的双重双重bialgebra，它满足了普遍的特性：对于v上的任何bialgebra b，对于v和任何字符的$λ$ b，在连接性的条件下，在连接性的条件下，存在一个独特的bialgebbras v v v v v v v v v v v v v v） ϕ =λ$。当V是V上的双重双重双子时，我们从B到QSH（v）上获得了双重双ggebras的独特形态，并表明这种形态$ ϕ_1 $允许Bialgebra的任何形态从B到B到QSH（V）的任何bialgebra的形态，这要归功于角色的单调动作。这种形式主义适用于V型图形的双重双齿。
Quasishuffle Hopf algebras, usually defined on a commutative monoid, can be more generally defined on any associative algebra V. If V is a commutative and cocommutative bialgebra, the associated quasishuffle bialgebra QSh(V) inherits a second coproduct $δ$ of contraction and extraction of words, cointeracting with the deconcatenation coproduct $Δ$, making QSh(V) a double bialgebra. In order to generalize the universal property of the Hopf algebra of quasisymmetric functions QSym (a particular case of quasishuffle Hopf algebra) as exposed by Aguiar, Bergeron and Sottile, we introduce the notion of double bialgebra over V. A bialgebra over V is a bialgebra in the category of right V-comodules and an extra condition is required on the second coproduct for double bialgebras over V. We prove that the quasishuffle bialgebra QSh(V) is a double bialgebra over V , and that it satisfies a universal property: for any bialgebra B over V and for any character $λ$ of B, under a connectedness condition, there exists a unique morphism $ϕ$ of bialgebras over V from B to QSh(V) such that $ε_δ\circ ϕ=λ$. When V is a double bialgebra over V , we obtain a unique morphism of double bialgebras over V from B to QSh(V), and show that this morphism $ϕ_1$ allows to obtain any morphism of bialgebra over V from B to QSh(V) thanks to an action of a monoid of characters. This formalism is applied to a double bialgebra of V-decorated graphs.
