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Part Ⅰ The Operator Aspect of Rota-Baxter Algebra 3

1 Spitzer's Identity 3

1.1 Basic definitions,examples and properties 4

1.2 Complete Rota-Baxter algebras 19

1.3 Formulations of Spitzer's identity,old and new 22

1.4 Atkinson's multiplicative decomposition and the algebraic Bogoliubov map 39

1.5 Some related literature and problems 43

2 Connected Hopf Algebras and Rota-Baxter Algebras 47

2.1 Algebras and modules revisited 48

2.2 Coalgebras and bialgebras 50

2.3 Connected bialgebra and Hopf algebra 54

2.4 Algebraic Birkhoff decomposition 62

2.5 Some related literature and problems 69

Part Ⅱ The Structure Aspect of Rota-Baxter Algebra 73

3 Free Commutative Rota-Baxter Algebras and Shuffle Products 73

3.1 Shuffle product and generalizations 74

3.2 Free commutative Rota-Baxter algebras 92

3.3 Rota-Baxter algebras,Stirling numbers and partitions 107

3.4 Some related literature and problems 116

4 Free Noncommutative Rota-Baxter Algebras and Rooted Trees 119

4.1 The Rota-Baxter algebra of trees 120

4.2 Free Rota-Baxter algebra on a set 128

4.3 Unitarization of Rota-Baxter algebras 147

4.4 Free Rota-Baxter algebra on an algebra 151

4.5 Some related literature and problems 163

Part Ⅲ The Operad Aspect of Rota-Baxter Algebra 167

5 Rota-Baxter Algebras and Dendriform Algebras 167

5.1 Concepts and basic properties 168

5.2 Free objects in the commutative case and shuffle products 171

5.3 Universal enveloping algebras of tridendriform algebras 177

5.4 Free dendriform,tridendriform algebras and free Rota-Baxter algebras 180

5.5 Some related literature and problems 190

6 Rota-Baxter Operators on Operads and Manin Products 191

6.1 Nonsymmetric operads 192

6.2 Products of binary quadratic NS operads 196

6.3 Binary quadratic ns operads from linear operators 204

6.4 Some related literature and problems 209

Bibliography 211

Index 223