by Markus Rost (Notes, April 2024/June 2025, 32 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 (June 21, 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 (at least its 1-skeleton with 42 points and 84 edges):
The arrows show 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).
[variant with different scale of the 4th dimension]
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]
