division algebra



A division algebra is a possibly non-associative algebra AA, typically over a field kk, with the property that for any a,bAa,b \in A then ab=0a b = 0 implies either a=0a = 0 or b=0b = 0 (or ab0a b \ne 0 whenever a,b0a, b \ne 0).

If AA is finite-dimensional (over a field kk), this is equivalent to assuming that for any nonzero aAa \in A, the operations of left multiplication and right multiplication by aa are invertible. If furthermore AA is also associative and unital, this is also equivalent to the existence, for each nonzero aAa \in A, of an element a 1Aa^{-1} \in A with aa 1=a 1a=1a a^{-1} = a^{-1} a = 1. However, it is easy to construct nonassociative unital finite-dimensional algebras over \mathbb{R} (the field of real numbers) such that either:

  • AA is not a division algebra but 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.

  • AA is a division algebra but there exists nonzero aAa \in A for which there is no a 1Aa^{-1} \in A with aa 1=a 1a=1a a^{-1} = a^{-1} a = 1.

For applications to physics, the most interesting division algebras are probably the normed division algebras: the real numbers, complex numbers, quaternions and octonions. These have important relations to supersymmetry.

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})\mathbb{R} the real numbers
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})\mathbb{C} the complex numbers
6=5+16 = 5+1Spin(5,1)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})\mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1)"SL(2,𝕆)"Spin(9,1) {\simeq} \text{"}SL(2,\mathbb{O})\text{"}𝕆\mathbb{O} the octonionsheterotic/type II string

See at division algebra and supersymmetry.


Revised on December 14, 2016 15:05:14 by Urs Schreiber (