|Rota-Baxter Algebras III: Splitting an RBA of type I (Outline)|
|1) Recall on
RB-operator (type 1) and algebras; notation
- Proposition RBO projector <=> direct sum of algebras
- In the filtered (graded) case, this is the generic case!
- Proposition (new?): Graded connected RBA => i) R projector, ii) R(1)=0 or 1. proof: ...
- Corollary (new?):
i) Graded connected RBA <=> direct sum of graded connected algebras,
ii) classification of RBOs on GC-RBAs: R=R+ iff R1=1, R=R- iff R1=0.
- The filtered RBA should be thought of as a topological RBA (degree topology), separated if connected.
- The graded RBA may be thought of as a geometric RBA (Lie group/Hilbert space sense).
2) The \chi map from the Article (constructive):
- "Derived products" and powers notation,
- There is a unique map \chi s.t. \chi(a)=a-BCH(R(\chi(a)),R'(\chi(a))).
- Derived products and "propagators";
- Symmetrization of the derived product and "star product" of a RBA \star (deformations): -\star=\cdot - \cup
- Short note on BCH-formula: transfer of structure, deformation, Maurer-Cartan initial value problem, formal Moyal's formula (on graphs)
- Comparing + & o-plus: A) the star-convolution,
B) Theorem: (i) \chi is the inverse of R\star R', ii) \Lambda as an infinitesimal "defect" (transition function)
C) Conjecture: R\chi transfers to an RB-group structure on G=1+A1