nLab division algebra

Definition

A division algebra is a possibly non-associative algebra $A$, typically over a field $k$, which has the property that for any nonzero element $a$ and any element $b$, the equations $a x=b$ and $x a=b$ each have a unique solution for $x$ in the algebra.

Usually the algebra is assumed to have a multiplicative identity, in which case this condition implies that each element has a left inverse and a right inverse. These inverses necessarily coincide if the algebra is associative, but this may fail in the absence of associativity.

A sub-topic of interest is the existence of a multiplicative norm for the algebra, see at normed division algebra.

Other definitions

An alternative definition is used in Baez 02: The axioms given there require that the algebra is finite-dimensional over $k$ and that for any $a$ and $b$ in the algebra, $a b = 0$ implies at least one of $a$ or $b$ is already $0$. This coincides with the definition given above for finite dimensional algebras.

Examples

• Perhaps the most famous division algebras are the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, the quaternions $\mathbb{H}$, and the octonions $\mathbb{O}$. According to the Hurwitz theorem, these are the only normed, finite dimensional, real division algebras.

• Any division ring is an associative division algebra over its center and has identity, but it may not be finite dimensional over its center.

• The ring of rational polynomials $\mathbb{R}(X)$ is an infinite dimensional real associative division algebra.

• There exists a division algebra with identity that does not have two-sided inverses for every nonzero element.

Counter-examples

• There exists a reciprocal algebra with nonzero zero divisors.

• The sedenions $\mathbb{S}$ are a finite-dimensional normed real algebra that is not a division algebra (it has zero divisors.)

• The ideal generated by $Y$ in the polynomial ring $\mathbb{R}[Y]$ is an associative, commutative, infinite dimensional real algebra without identity and without zero divisors. The equation $Yx=Y$ does not have a solution for $Y$ in the algebra.

Related concepts

For many applications (also to physics) the most interesting division algebras are the normed division algebras over the real numbers: By the Hurwitz theorem these are the real numbers, complex numbers, quaternions and octonions. These have important relations to supersymmetry.

exceptional spinors and real normed division algebras

Lorentzian spacetime dimension	$\phantom{AA}$spin group	normed division algebra	$\,\,$ brane scan entry
$3 = 2+1$	$Spin(2,1) \simeq SL(2,\mathbb{R})$	$\phantom{A}$ $\mathbb{R}$ the real numbers	super 1-brane in 3d
$4 = 3+1$	$Spin(3,1) \simeq SL(2, \mathbb{C})$	$\phantom{A}$ $\mathbb{C}$ the complex numbers	super 2-brane in 4d
$6 = 5+1$	$Spin(5,1) \simeq$ SL(2,H)	$\phantom{A}$ $\mathbb{H}$ the quaternions	little string
$10 = 9+1$	Spin(9,1) ${\simeq}$ “SL(2,O)”	$\phantom{A}$ $\mathbb{O}$ the octonions	heterotic/type II string

See at division algebra and supersymmetry.

• Hopf invariant one

• division ring

References

See also

• eom, Division algebra

• Wikipedia, Division algebra

• John Baez, Section 1.1 Preliminaries In: The octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205. (doi:10.1090/S0273-0979-01-00934-X, website)