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Algebra and Applications 2
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1
Algebraic Background for Numerical Methods, Control Theory and Renormalization
1.1. Introduction
1.2. Hopf algebras: general properties
1.2.1.
Algebras
1.2.2.
Coalgebras
1.2.3.
Convolution product
1.2.4.
Bialgebras and Hopf algebras
1.2.5.
Some simple examples of Hopf algebras
1.2.5.1.
The Hopf algebra of a group
1.2.5.2.
Tensor algebras
1.2.5.3.
Enveloping algebras
1.2.6.
Some basic properties of Hopf algebras
1.3. Connected Hopf algebras
1.3.1.
Connected graded bialgebras
1.3.2.
An example: the Hopf algebra of decorated rooted trees
1.3.3.
Connected filtered bialgebras
1.3.4.
The convolution product
1.3.5.
Characters
1.3.6.
Group schemes and the Cartier-Milnor-Moore-Quillen theorem
1.3.7.
Renormalization in connected filtered Hopf algebras
1.4. Pre-Lie algebras
1.4.1.
Definition and general properties
1.4.2.
The group of formal flows
1.4.3.
The pre-Lie Poincaré–Birkhoff–Witt theorem
1.5. Algebraic operads
1.5.1.
Manipulating algebraic operations
1.5.2.
The operad of multi-linear operations
1.5.3.
A definition for linear operads
1.5.4.
A few examples of operads
1.5.4.1.
The operad
ASSOC
1.5.4.2.
The operad
COM
1.5.4.3.
Associative algebras
1.6. Pre-Lie algebras (continued) 1.6.1.
Pre-Lie algebras and augmented operads
1.6.1.1.
General construction
1.6.1.2.
The pre-Lie operad
1.6.2.
A pedestrian approach to free pre-Lie algebra
1.6.3.
Right-sided commutative Hopf algebras and the Loday-Ronco theorem
1.6.4.
Pre-Lie algebras of vector fields
1.6.4.1.
Flat torsion-free connections
1.6.4.2.
Relating two pre-Lie structures
1.6.5.
B-series, composition and substitution
1.7. Other related algebraic structures 1.7.1.
NAP algebras
1.7.1.1.
Definition and general properties
1.7.1.2.
Free NAP algebras
1.7.1.3.
NAP algebras of vector fields
1.7.2.
Novikov algebras
1.7.3.
Assosymmetric algebras
1.7.4.
Dendriform algebras
1.7.5.
Post-Lie algebras
1.8. References
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