Prelude

((A₁⊗A₂)⊗A₃)⊗A₄ = A₁⊗(A₂⊗(A₃⊗A₄))

Notes on the associator

by Markus Rost (Notes, April 2024/July 2025, 34 pages)

April 2024: We discuss the 5-term relation for associators, its relation with the associahedron and ask questions.

(x,y,z)t + (x,yz,t) + x(y,z,t) = (x,y,zt) + (xy,z,t)

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 noticed the cubical presentation of the associahedron known as "Tamari polytope". See Cubical drawings from other sources. The relation with the one discussed in Section 6 is not yet clear to me.

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 25, 2025): [pdf]

Associahedron drawings

Cubical associahedron drawings

The 2-dimensional associahedron (the pentagon as rectangle with an edge subdivided):

The pentagon and its contraction

The 3-dimensional associahedron (as cuboid with two faces subdivided): [main page: The 3-dimensional associahedron unfolded]

A 3-dimensional associahedron

The 4-dimensional cubical associahedron: [main page: The 4-dimensional cubical associahedron]

The image shows the 1-skeleton with 42 points and 84 oriented 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 cube).

A 4-dimensional associahedron

[high resolution image]

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):

Some labels for the 4-dimensional associahedron

[high resolution image]

[colored]

[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:

(x,yz,t) - (x,z,t) y - z (x,y,t) = (x,z,[y,t]) + ([x,z],y,t)

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]


Go to: Publications and Preprints · Markus Rost's Web Page