Combinatorial Hopf Algebras

这个项目将有助于建立Hopf代数组合作为一个主要组成部分，在更广泛的背景下代数组合，通过开发新的结果，统一和推广组合中的重要结构。构成这些构造基础的霍普夫代数也与其他数学和物理领域有着深刻的联系，如运算理论、重整化理论和表示论。因此，本项目还将处理结合破缺理论中的抽象结构和相应类型的代数，量子电动力学树的Hopf代数的研究，以及与对称函数代数密切相关的各种对象的研究，如准对称和非交换对称函数。组合问题研究各种计数方法。在这个建议中，代数方法将用于对这些计数问题的一些新的见解。此外，PI将向研究生介绍这一研究领域，并在拉丁美洲和欧洲进行国际联系和合作。

This project will contribute to establish Hopf algebraic combinatorics as a major component within the wider context of algebraic combinatorics, by developing new results that unify and generalize important constructions in combinatorics. The same Hopf algebras that are at the basis of these constructions also have deep connections with other areas of mathematics and physics such as the theory of operads, renormalization theory, and representation theory. Thus, this project will also deal with abstract constructions in the theory of associativity breaking and the corresponding types of algebras, the investigation of the Hopf algebras of trees of quantum electrodynamics, and the investigation of various objects closely related to the algebra of symmetric functions, such as quasi-symmetric and non-commutative symmetric functions.Combinatorial questions study various ways of counting. In this proposal methods from algebra will be used to about new insight into some of these counting questions. In addition the PI will introduce graduate students to this area of research and make international contacts and collaborations in Latin America and Europe.