Rota-Baxter 代數(shù)導(dǎo)論

內(nèi)容概要

　　Rota-Baxter代數(shù)由一個結(jié)合代數(shù)和一個線性算子組成，該算子滿足微積分的分部積分公式中的等式。Rota-Baxter代數(shù)20世紀(jì)60年代起源于隨機理論。本世紀(jì)以來，Rota-Baxter代數(shù)不僅在理論法方面得到了突飛猛進的發(fā)展，并且在數(shù)學(xué)物理、數(shù)論、組合等方面得到了廣泛的應(yīng)用。盡管過去的幾十年有很多有關(guān)于Rota-Baxter代數(shù)的文章，但是還缺乏一本系統(tǒng)介紹Rota-Baxter代數(shù)的專著?！禦ota-Baxter代數(shù)導(dǎo)論（英文版）》就是第一本介紹該領(lǐng)域的著作，通過大量的例子以及各種應(yīng)用之間的聯(lián)系，詳細(xì)介紹了Rota-Baxter代數(shù)，包括它的三個重要方面。
　　本書可作為代數(shù)、組合、數(shù)論和數(shù)學(xué)物理領(lǐng)域的研究生教材或參考書，也可供相關(guān)的研究人員參考。
　　作者郭鋰為美國Rutgers大學(xué)教授，是Rota-Baxter代數(shù)及相關(guān)結(jié)構(gòu)近年來進展的主要貢獻者之一。郭鋰教授在Notices
of the American Mathematical Society發(fā)表的文章WHAT IS a Rota-Baxter
Algebra將這個新的研究領(lǐng)域帶入了數(shù)學(xué)公眾的視野。

書籍目錄

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

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