division algebra




A division algebra is a possibly non-associative algebra AA, typically over a field kk, which is also a division ring, hence 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 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

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)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{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.


Last revised on December 2, 2018 at 10:15:06. See the history of this page for a list of all contributions to it.