扭双代数与色Fock函子上的压缩与抽取
Contractions and extractions on twisted bialgebras and coloured Fock functors
摘要：
本文在扭双代数上引入了抽缩余积的概念，即在线性种范畴内的扭双代数上。如果$P$是一个扭曲双代数，压缩抽取余积将$P[X]$发送到$P[X/\sim]\otimes P[X]$，对于任何有限集$X$和$X$上的任何等价关系$\sim$，具有与$P$的积和余积的协结合约束和相容性。证明了若$P$是一个具有抽取-压缩余积的扭曲双代数，则$P\circ Com$是余代数种范畴中的双代数，即余代数范畴中的种，并引入了玻色Fock函子的着色形式.这就导出了一个双函子，它与任何双代数$(V,\cdot,δ_V)$和任何扭曲双代数$P$相关联，并具有一个抽取-压缩余积一个余模双代数$F_V[P]$：这个对象继承了一个积$m$和两个余积$Δ$和$δ$，使得$(F_V[P],m,Δ)$是右$(F_V[P],m,δ)$ -余模范畴中的双代数。作为一个例子，这被应用于图的扭曲双代数。有色福克函子，然后允许延长建设的双重双代数的图形双代数的图形，其顶点是装饰的元素的任何双代数$V$。其他例子（关于混合图，超图，非交叉分区.）将在一系列即将发表的论文中发表。
英文摘要：
We introduce a notion of extraction-contraction coproduct on twisted bialgebras, that is to say bialgebras in the category of linear species. If $P$ is a twisted bialgebra, a contraction-extraction coproduct sends $P[X]$ to $P[X/\sim]\otimes P[X]$ for any finite set $X$ and any equivalence relation $\sim$ on $X$, with a coassociativity constraint and compatibilities with the product and coproduct of $P$. We prove that if $P$ is a twisted bialgebra with an extraction-contraction coproduct, then $P\circ Com$ is a bialgebra in the category of coalgebraic species, that is to say species in the category of coalgebras.We then introduce a coloured version of the bosonic Fock functor. This induces a bifunctor which associates to any bialgebra $(V,\cdot,δ_V)$ and to any twisted bialgebra $P$ with an extraction-contraction coproduct a comodule-bialgebra $F_V[P]$: this object inherits a product $m$ and two coproducts $Δ$ and $δ$, such that $(F_V[P],m,Δ)$ is a bialgebra in the category of right $(F_V[P],m,δ)$-comodules.As an example, this is applied to the twisted bialgebra of graphs. The coloured Fock functors then allow to extend the construction of the double bialgebra of graphs to double bialgebras of graphs which vertices are decorated by elements of any bialgebra $V$. Other examples (on mixed graphs, hypergraphs, noncrossing partitions...) will be given in a series of forthcoming papers.
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