Prelude

Notes on the associator
by Markus Rost (Notes, April 2024/August 2025, 38 pages)
- April 2024: We discuss the 5-term relation for associators, its
relation with the associahedron and ask questions.

- May 2024: Added references to alternative algebras.
- December 2024: Section 6 has an essentially complete discussion
of the associahedral chain complex and its acyclicity. Section 7
has beautiful diagrams!
- August 2025: There are no essential changes since December
2024. Only some corrections, a few further comments and an
addition of diagrams.
Full text (August 21, 2025):
[pdf]
The associahedral chain complex and the cubical associahedron
by Markus Rost (Notes, August/September 2025, 32 pages)
- The first part is a detailed introduction to multi-magmas
(partially parenthesized words).
- The associahedral chain complex is constructed algebraically and
its acyclicity is proven. All details are given.
(It seems that this part is new. Please tell me if you think
otherwise.)
- The cubical associahedron is constructed, also with all details.
(This part is perhaps new as well.)
- The sign game relating the definition of A(∞)-algebras is
detailed.
- Possible expansions are indicated.
Full text (September 28, 2025):
[pdf]
Associahedron drawings
Cubical associahedron drawings
The 2-dimensional associahedron
The pentagon as rectangle with an edge subdivided:

The 3-dimensional associahedron
Main page: The 3-dimensional
associahedron unfolded
As cuboid with two faces subdivided:

The 4-dimensional associahedron
Main page: The 4-dimensional
cubical associahedron
The first image shows the 1-skeleton with all 42 points and 84
edges. Further the two basic 3-dimensional subassociahedra
•(...), (...)• (the shoebox cuboids) and the cells
parallel to them.

[high
resolution image]
The next image includes orientations of the edges. The black (as
opposed to blue) arrows depict for each point the "canonical
directed path" (Mac Lane 1963 MR 170925 p. 34) to the base
point (the red one on the bottom cuboid).

[high
resolution image]
Here are labels for
- the path with 4 directions (starting in the
gray left bottom corner of the upper cuboid)
- the maximal path with
no direction change (the rightmost green line with the 5 red
points)
The parenthesis pairs are enumerated for identification along the
path. See
also cube-4-84-parens-large.png.

[high resolution
image]
[black and white]
[TeX source]
Notes on associator identities
by Markus Rost (Notes, May 2024, 10 pages)
The text contains notes on some (well-known) identities in
non-associative algebras.
Among other things, we establish the 5-term relation for
associators and this identity in alternative algebras:
![An identity in alternative algebras establishing the alternating 4-form (x,yz,t) - (x,z,t) y - z (x,y,t) = (x,z,[y,t]) + ([x,z],y,t)](images/assoc4.png)
Full text (June 18, 2024):
[pdf]
Notes on free alternative algebras
by Markus Rost (Notes, June 2024, 12 pages)
We compute the free alternative algebra up to degree 4.

Full text (June 22, 2024):
[pdf]
TeX sources of displayed formulas:
A1A2A3A4.tex
· assoc1.tex
· assoc4.tex
· assoc5.tex
TeX sources of the drawings can be found on the main pages for
the 3-dimensional
and 4-dimensional
associahedron.
Go to: Publications and Preprints
· Markus Rost's Web Page