论文标题
准对称函数的重新归一化
Renormalization of quasisymmetric functions
论文作者
论文摘要
作为准对称函数的HOPF代数的自然基础，单次准对称函数是由组成定义的形式功率序列。相同的定义适用于左弱组合物，而其他弱组成的分歧则导致差异。我们适应了Connes和Kreimer框架中的量子场理论中重新归一化的方法，以处理这种分歧。这种方法定义了任何弱组合物作为功率序列的单一准对称函数，同时扩展了通常的准对称函数满足的准避难（sTECTLE）关系。因此，获得的准对称函数的代数被证明是弱组成的准剃须刀代数的同构，使前者具有天然的Hopf代数结构，而后者则是功率系列实现。同构也使自由交换性rota-baxter代数具有功率序列实现，以支持Rota的建议，即Rota rota-baxter代数应为对称函数的概括提供广泛的背景。
As a natural basis of the Hopf algebra of quasisymmetric functions, monomial quasisymmetric functions are formal power series defined from compositions. The same definition applies to left weak compositions, while leads to divergence for other weak compositions. We adapt the method of renormalization in quantum field theory, in the framework of Connes and Kreimer, to deal with such divergency. This approach defines monomial quasisymmetric functions for any weak composition as power series while extending the quasi-shuffle (stuffle) relation satisfied by the usual quasisymmetric functions. The algebra of renormalized quasisymmetric functions thus obtained turns out to be isomorphic to the quasi-shuffle algebra of weak compositions, giving the former a natural Hopf algebra structure and the latter a power series realization. This isomorphism also gives the free commutative Rota-Baxter algebra a power series realization, in support of a suggestion of Rota that Rota-Baxter algebra should provide a broad context for generalizations of symmetric functions.
