by Markus Rost (Notes, April 2024/July 2025, 33 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!
February 2025: I realized that the cubical presentation of the associahedron is known as "Tamari polytope". See the updated preface.
March 2025: Section 8 has a TeX version of the original Tamari diagram. A variant (April 2025) shows the homotopy, another variant (June 2025) the dual.
Full text (July 8, 2025): [pdf]
The 2-dimensional associahedron (the pentagon as rectangle with an edge subdivided):
The 3-dimensional associahedron (as cuboid with two faces subdivided): [main page: The 3-dimensional associahedron unfolded]
The 4-dimensional cubical associahedron (showing at least its 1-skeleton with 42 points and 84 edges):
The 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 cube).
Here are labels for the maximal path with no direction change (the straight line with the 5 red vertices) and for the path with 4 directions (starting in the gray opposite corner of the upper cube):
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]
