Natural endomorphisms of quasi-shuffle Hopf algebras
Natural endomorphisms of quasi-shuffle Hopf algebras
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DOI：
10.24033/bsmf.2644
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发表时间：
2011-01
期刊：
影响因子：
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通讯作者：
J. Novelli;F. Patras;J. Thibon
中科院分区：
文献类型：
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作者：
J. Novelli;F. Patras;J. Thibon
词拟对称函数的Hopf代数（$\WQSym$）是拟对称函数的Hopf代数的非交换推广，它可以被赋予一个内积，该内积与$\WQSym$上的其他运算具有若干相容性。这扩展了自由李代数、非交换对称函数及其各种应用领域理论中熟悉和核心的构造，并允许将$\WQSym$解释为拟洗牌代数的线性自同态的卷积代数。然后，我们使用这种解释来研究准洗牌代数（MZVs，自由Rota-Baxter代数.）的精细结构。特别是，我们计算他们的亚当斯操作，并证明了广义欧拉幂等元的存在性，也就是说，一个典型的左逆自然满射映射到他们的不可分解，允许这些代数的自由多项式生成器的组合建设。
The Hopf algebra of word-quasi-symmetric functions ($\WQSym$), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on $\WQSym$. This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret $\WQSym$ as a convolution algebra of linear endomorphisms of quasi-shuffle algebras. We then use this interpretation to study the fine structure of quasi-shuffle algebras (MZVs, free Rota-Baxter algebras...). In particular, we compute their Adams operations and prove the existence of generalized Eulerian idempotents, that is, of a canonical left-inverse to the natural surjection map to their indecomposables, allowing for the combinatorial construction of free polynomial generators for these algebras.