nLab alternative algebra

Definitions

Definition

Consider the following equational laws of a binary operation (written multiplicatively):

• left alternativity: $(x x) y = x (x y)$;
• flexibility: $(x y) x = x (y x)$;
• right alternativity: $(y x) x = y (x x)$.

An operation satisfying one of these conditions is left-alternative, flexible, or right-alternative, respectively, and it is alternative if it is both left-alternative and right-alternative. A magma is so if its binary operation is so, and a nonassociative algebra (or nonassociative ring) is so if its multiplication operation is so.

Properties

A commutative operation/magma/algebra must be flexible, and it is left-alternative iff it is right-alternative (and so simply alternative). An associative operation/magma/algebra is both alternative and flexible.

Among algebras (but not magmas), any algebra with two of these three properties must have the third. In particular, an alternative algebra must be flexible. This follows from the characterization in terms of the associator below.

In terms of the associator

Proposition

For a nonassociative algebra $A$, alternativity according to def. is equivalent to the condition that the associator, i.e. the tri-linear map

$$[-,-,-] \;\colon\; A \otimes A \otimes A \longrightarrow A$$

given by

$$[a,b,c] \coloneqq (a b) c - a (b c)$$

is alternating, in that whenever two of the three arguments are equal, the result is zero.

Corollary

Alternativity implies that the associator is skew-symmetric?, in that for any permutation $\sigma$ of three elements then $[a_{\sigma_1}, a_{\sigma_2}, a_{\sigma_3}] = (-1)^{\vert \sigma\vert} [a_1, a_2, a_3]$ for ${\vert\sigma\vert}$ the signature of the permutation. Over a field whose characteristic is different from $2$, or more generally over any commutative ring in which $2$ is invertible or even cancellable, alternativity is equivalent to skew-symmetry of the associator.

Proof

In one direction, left alternativity says directly that the associator is alternating in the first two arguments:

$$[x,x,y] = (x x) y - x (x y) = (x x) y - (x x) y = 0 ,$$

and right alternativity says the same thing in the last two arguments:

$$[x,y,y] = (x y) y - x (y y) = (x y) y - (x y) y = 0 .$$

To be fully alternating, we then argue using multi-linearity of the associator:

$$[x,y,x] = [x,x,x] + [x,y,x] + [y,x,x] + [y,y,x] = [(x+y),(x+y),x] = 0 .$$

Multilinearity also proves that the associator is skew-symmetric, in adjacent arguments:

$$0 = [(x+y),(x+y),z] = [x,y,z] + [y,x,z]$$

$$0 = [z,(x+y),(x+y)] = [z,x,y] + [z,y,x]$$

and hence in all arguments.

In the other direction, skew-symmetry of the associator implies alternativity by

$$\begin{aligned} & [x,x,y] = -[x,x,y] \\ \Leftrightarrow & 2 [x,x,y] = 0 \\ \Leftrightarrow & [x,x,y] = 0 \end{aligned}$$

using the assumption that $2$ is cancellable in $A$, and similarly for $[y,x,x] = 0$.

Proposition

A nonassociative algebra is alternative, def. , prop. , precisely if the subalgebra? generated by any two elements is an associative algebra.

This is due to Emil Artin, see for instance (Schafer 95, p. 18).

Proposition

The only alternative division algebras over the real numbers are the real numbers themselves, the complex numbers, the quaternions and the octonions.

This is due to (Zorn 30).

Examples

Every associative algebra is alternative and flexible. Every Lie algebra or Jordan algebra is flexible.

Every Cayley–Dickson algebra over a commutative ring $R$ is flexible. The first three (corresponding, if we start with the real numbers, to the real numbers, complex numbers, and quaternions) are associative and hence alternative. The next one (corresponding to the octonions) is still alternative despite not being associative (unless $R$ has characteristic $2$). After that (corresponding to the sedenions and above), they are not even alternative (unless $R$ has characteristic $2$).

Related concepts

• Cayley-Dickson construction

• composition algebra

References

• R. D. Schafer, chapter III of Introduction to Non-Associative Algebras Dover, New York, 1995. (web)

• Max Zorn, Theorie der alternativen Ringe, Abhandlungen des Mathematischen Seminars der Universität Hamburg 8 (1930), 123-147

• Wikipedia, Alternative algebra, Flexible algebra