Topics on Noncummutative Algebra

topics_in_noncommutative_algebra.txt

# Video:

https://youtu.be/MAL6rz3FyVw

In this video I talk about the Book 
"Topics in Noncommutative Algebra - The Theorems of Campell, Baker, Hausdorff and Dynkin" 
by Andrea Bonfilio and Roberta Fulci.
I tease some of my motivation with the topic by starting out ranting about differential equation and exponential growth, 
such as seen in baceria growth or virus outbreaks.

# Topics in Noncommutative Algebra 

Book layout:

540 pages

########## Part 1 Algebraic Proofs (pp. 49-392)
## Capter 1, Historical Overview (pp. 1 - 45)
## Preface (ten pages)
* Motivation/History
* Overview
* Etc.

## Capter 2, Background Algebra (pp. 49-113)
Starts with 50 pages algebra review/recalls.

## Capter 3, Main Proof (pp. 114-172)
## Capter 4, Some "Short" Proofs of the CBHD Theorem (pp. 172-264)
## Capter 5, Convergence of the CBHD Series and Associativity of the CBHD Operation (pp. 265-370)
## Capter 6, Relationship Between the CBHD Theorem, the PBW Theorems and the Free Lie algebra (pp. 371-392)

########## Part 2 Proofs of algebraic Prequesists (pp. 393-522)

## Capter 7, Proofs of the Algebraic Prerequesits.


########## Notes on Chapter 1

Prehistory/origin:
	* Lie (Transformation groups)

Two periods of work on CBHD: 
	* 1890's - 1950'
	* rest
People (p. 2):
	* Schur (1890)
	* Campbell (1897)
	* Poincare (1899)
	* Baker (1901)
	* Pascal (1902)
	* Hausdorff (1906)
	* Yosida (1837)
	* Dynkin (1947)
(The dates are the date of the first cited notable paper on the subject.)

	* Bourbaki
	* ...

### The origin of the Problem (pp. 2-8)

Emphasis in the book that Lie's theory doesn't cover Lie groups or algebras in our modern sense, but instead R^n transformation groups and both their (algebraically expressed geometric) properties. 

A bunch of discussion are on concrete vs. symbolic approaches.

Poincare is praised for symbolic and universal approach (recall Universal enveloping algebra).
Criticism of Hausdorff and Bourbaki for Schur's and Poincare's work is recollected:
	* Comprehensiveness and completeness is criticized

Book tries to sketch (on a few page) how modern theories corresponds to Lie's results about his object of study. E.g. Lie's third theorem

(pp. 9-18) Discussion/Sketch of contributions of: 
	Schur, Poincare, Pascal
(pp. 19-36) Discussion/Sketch of contributions of: 
	Campbell, Baker, Hausdorff, Dynkin
(pp. 37-43) Discussion/Sketch of contributions in "Modern Era" of CBHD Theorem
(p. 44-45) Naming discussions