2.1 The Fano plane

The quaternions, , are a 4-dimensional algebra with basis . To describe the product we could give a multiplication table, but it is easier to remember that:

- is the multiplicative identity,
- and are square roots of -1,
- we have , , and all identities obtained from these by cyclic permutations of .

When we multiply two elements going clockwise around the circle we get
the next one: for example, . But when we multiply two
going around counterclockwise, we get *minus* the next one:
for example, .

We can use the same sort of picture to remember how to multiply octonions:

This is the **Fano plane**, a little gadget
with 7 points and 7 lines. The `lines' are the sides of the triangle,
its altitudes, and the circle containing all the midpoints of the sides.
Each pair of distinct points lies on a unique line. Each line contains
three points, and each of these triples has has a cyclic ordering
shown by the arrows. If and are cyclically ordered
in this way then

Together with these rules:

- is the multiplicative identity,
- are square roots of -1,

This is certainly a neat mnemonic, but is there anything deeper lurking behind it? Yes! The Fano plane is the projective plane over the 2-element field . In other words, it consists of lines through the origin in the vector space . Since every such line contains a single nonzero element, we can also think of the Fano plane as consisting of the seven nonzero elements of . If we think of the origin in as corresponding to , we get the following picture of the octonions:

Note that planes through the origin of this 3-dimensional vector space give subalgebras of isomorphic to the quaternions, lines through the origin give subalgebras isomorphic to the complex numbers, and the origin itself gives a subalgebra isomorphic to the real numbers.

What we really have here is a description of the octonions as a
`twisted group algebra'. Given any group , the group algebra
consists of all finite formal linear combinations of elements
of with real coefficients. This is an associative algebra with
the product coming from that of . We can use any function

to `twist' this product, defining a new product

by:

where . One can figure out an equation involving that guarantees this new product will be associative. In this case we call a `2-cocycle'. If satisfies a certain extra equation, the product will also be commutative, and we call a `stable 2-cocycle'. For example, the group algebra is isomorphic to a product of 2 copies of , but we can twist it by a stable 2-cocyle to obtain the complex numbers. The group algebra is isomorphic to a product of 4 copies of , but we can twist it by a 2-cocycle to obtain the quaternions. Similarly, the group algebra is a product of 8 copies of , and what we have really done in this section is describe a function that allows us to twist this group algebra to obtain the octonions. Since the octonions are nonassociative, this function is not a 2-cocycle. However, its coboundary is a `stable 3-cocycle', which allows one to define a new associator and braiding for the category of -graded vector spaces, making it into a symmetric monoidal category [3]. In this symmetric monoidal category, the octonions are a commutative monoid object. In less technical terms: this category provides a context in which the octonions

© 2001 John Baez