nLab
division algebra

Contents

Contents

Definition

A division algebra is a possibly non-associative algebra AA, typically over a field kk, which has the property that for any nonzero element aa and any element bb, the equations ax=ba x=b and xa=bx a=b each have a unique solution for xx 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 kk and that for any aa and bb in the algebra, ab=0a b = 0 implies at least one of aa or bb is already 00. 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 𝕆\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 (X)\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 (nonassociative) algebra AA such that for each nonzero aAa \in A there exists a 1Aa^{-1} \in A with aa 1=a 1a=1a a^{-1} = a^{-1} a = 1, but there are nonzero zero divisors.

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

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

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
AA\phantom{AA}spin groupnormed division algebra\,\, brane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})A\phantom{A} \mathbb{R} the real numberssuper 1-brane in 3d
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})A\phantom{A} \mathbb{C} the complex numberssuper 2-brane in 4d
6=5+16 = 5+1Spin(5,1)Spin(5,1) \simeq SL(2,H)A\phantom{A} \mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1)Spin(9,1) {\simeq}SL(2,O)A\phantom{A} 𝕆\mathbb{O} the octonionsheterotic/type II string

See at division algebra and supersymmetry.

References

See also

Last revised on November 9, 2020 at 12:00:26. See the history of this page for a list of all contributions to it.