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]
by Markus Rost (Notes, August 2025, 21 pages)
We construct the associahedral chain complex algebraically and prove its acyclicity. All details are given. The methods seems to be new.
Future possible expansions are indicated.
Full text (September 1, 2025): [pdf]
The pentagon as rectangle with an edge subdivided:
Main page: The 3-dimensional associahedron unfolded
As cuboid with two faces subdivided:
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.
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).
Here are labels for
The parenthesis pairs are enumerated for identification along the path. See also cube-4-84-parens-large.png.
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:
![(x,yz,t) - (x,z,t) y - z (x,y,t) = (x,z,[y,t]) + ([x,z],y,t)](images/assoc4.png){kind=small}
Full text (June 18, 2024): [pdf]
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.
