Rota-Baxter代数导论

Preface

Leitfaden

Part I The Operator Aspect of Rota-Baxter Algebra

1 Spitzer's Identity

1.1 Basic definitions, examples and properties

1.2 Complete Rota-Baxter algebras

1.3 Formulations of Spitzer's identity, old and new

1.4 Atkinson's multiplicative decomposition and the algebraic Bogoliubov map

1.5 Some related literature and problems

2 Connected Hopf Algebras and Rota-Baxter Algebras

2.1 Algebras and modules revisited

2.2 Coalgebras and bialgebras

2.3 Connected bialgebra and Hopf algebra

2.4 Algebraic Birkhoff decomposition

2.5 Some related literature and problems

Part II The Structure Aspect of Rota-Baxter Algebra

3 Free Commutative Rota-Baxter Algebras and Shuffle

Products

3.1 Shuffle product and generalizations

3.2 Free commutative Rota-Baxter algebras

3.3 Rota-Baxter algebras, Stirling numbers and partitions

3.4 Some related literature and problems

4 Free Noncommutative Rota-Baxter Algebras and Rooted Trees

4.1 The Rota-Baxter algebra of trees

4.2 Free Rota-Baxter algebra on a set

4.3 Unitarization of Rota-Baxter algebras

4.4 Free Rota-Baxter algebra on an algebra

4.5 Some related literature and problems

Part III The Operad Aspect of Rota-Baxter Algebra

5 Rota-Baxter Algebras and Dendriform Algebras

5.1 Concepts and basic properties

5.2 Free objects in the commutative case and shuffle products

5.3 Universal enveloping algebras of tridendriform algebras

5.4 Free dendriform, tridendriform algebras and free

Rota-Baxter algebras

5.5 Some related literature and problems

6 Rota-Baxter Operators on Operads and Manin Products

6.1 Nonsymmetric operads

6.2 Products of binary quadratic NS operads

6.3 Binary quadratic ns operads from linear operators

6.4 Some related literature and problems

Bibliography

Index

Rota-Baxter代数由一个结合代数和一个线性算子组成，该算子满足微积分的分部积分公式中的等式。Rota-Baxter代数20世纪60年代起源于随机理论。本世纪以来，Rota-Baxter代数不仅在理论法方面得到了突飞猛进的发展，并且在数学物理、数论、组合等方面得到了广泛的应用。尽管过去的几十年有很多有关于Rota-Baxter代数的文章，但是还缺乏一本系统介绍Rota-Baxter代数的专著。《Rota-Baxter代数导论（英文版）》就是第一本介绍该领域的著作，通过大量的例子以及各种应用之间的联系，详细介绍了Rota-Baxter代数，包括它的三个重要方面。
本书可作为代数、组合、数论和数学物理领域的研究生教材或参考书，也可供相关的研究人员参考。
作者郭锂为美国Rutgers大学教授，是Rota-Baxter代数及相关结构近年来进展的主要贡献者之一。郭锂教授在Notices of the American Mathematical Society发表的文章WHAT IS a Rota-Baxter Algebra将这个新的研究领域带入了数学公众的视野。