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normed division algebra

Normed division algebras

Idea

A normed division algebra is a generalisation of the real numbers, complex numbers, quaternions, and octonions. Amazingly, there are no other examples!

Definition

A normed division algebra is a Banach algebra that is also a division algebra.

Properties

While the norm in a Banach algebra is normally only submultiplicative (xyxy{\|x y\|} \leq {\|x\|} {\|y\|}), the norm in a normed division algebra must be multiplicative (xy=xy{\|x y\|} = {\|x\|} {\|y\|}). Accordingly, this norm is considered to be an absolute value and often written ||{|{-}|} instead of {\|{-}\|}. There is also a converse: if the norm on a Banach algebra is multiplicative (including 1=1{\|1\|} = 1), then it must be a division algebra. While the term ‘normed division algebra’ does not seem to include the completeness condition of a Banach algebra, in fact the only examples have finite dimension and are therefore complete.

The only normed division algebra over the complex numbers is the algebra of complex numbers themselves. Up to isomorphism, there are exactly four finite-dimensional normed division algebras over the real numbers:

Each of these is produced from the previous one by the Cayley–Dickson construction; when applied to 𝕆\mathbb{O}, this construction produces the algebra of sedenions, which do not form a division algebra.

The Cayley–Dickson construction applies to an algebra with involution; by the abstract nonsense of that construction, we can see that the four normed division algebras above have these properties:

  • \mathbb{R} is associative, commutative, and with trivial involution,
  • \mathbb{C} is associative and commutative but has nontrivial involution,
  • \mathbb{H} is associative but noncommutative and with nontrivial involution,
  • 𝕆\mathbb{O} is neither associative, commutative, nor with trivial involution.

However, these algebras do all have some useful algebraic properties; in particular, they are all alternative (a weak version of associativity). They are also all composition algebras.

A normed field is a commutative normed division algebra; it follows from the preceding that the only normed fields over \mathbb{R} are \mathbb{R} and \mathbb{C} (e.g. Tornheim 52).

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) somesenseSL(2,𝕆)Spin(9,1) \simeq_{some\,sense} SL(2,\mathbb{O})𝕆\mathbb{O} the octonionsheterotic/type II string

see division algebra and supersymmetry

References

Revised on July 11, 2014 03:12:21 by Urs Schreiber (88.128.80.67)