Prelude

((A₁⊗A₂)⊗A₃)⊗A₄ = A₁⊗(A₂⊗(A₃⊗A₄))

Notes on the associator

by Markus Rost (Notes, April 2024/August 2025, 38 pages)

Full text (August 21, 2025): [pdf]

The associahedral chain complex and the cubical associahedron

by Markus Rost (Notes, August/September 2025, 32 pages)

Full text (September 28, 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

Main page: The 3-dimensional associahedron unfolded

As cuboid with two faces subdivided:

A 3-dimensional associahedron

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.

A 4-dimensional associahedron

[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).

A 4-dimensional associahedron (with arrows)

[high resolution image]

Here are labels for

The parenthesis pairs are enumerated for identification along the path. See also cube-4-84-parens-large.png.

Some labels for the 4-dimensional associahedron

[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:

(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]


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.


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