- #1

BruceG

- 40

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For some time I've been trying to get a geometric appreciation of why normed division algebras only exist in dimensions 1,2,4,8 (namely R,C,H,O).

As always Baez provides the most elegant answer:

http://math.ucr.edu/home/baez/octonions/node6.html"

Allow me to descibe the key point of the proof in case anyone has any better insight to add.

The "division" property of an algebra (ab=0 iff a=0 or b=0) gaurantees that multiplication by unit numbers generates all possible rotations of the unit sphere (e.g. any 2 points on a circle can be reached by rotation by a unit complex number).

From this Baez goes on to show that an n-dimensional normed division algebra must be an irreducible representation of the clifford algebra of dimension n-1.

The result can then be read from the following table:

n -> irreduclible rep of Cliff(n)

0 ->

3 ->

5 -> 8

6 -> 8

7 ->

9 -> 32

10 -> 64

11 -> 64

12 -> 128

13 -> 128

14 -> 128

15 -> 128

We see that the clifford algebras matrices rapidly become too large to allow the formation of a division algebra.

As always Baez provides the most elegant answer:

http://math.ucr.edu/home/baez/octonions/node6.html"

Allow me to descibe the key point of the proof in case anyone has any better insight to add.

The "division" property of an algebra (ab=0 iff a=0 or b=0) gaurantees that multiplication by unit numbers generates all possible rotations of the unit sphere (e.g. any 2 points on a circle can be reached by rotation by a unit complex number).

From this Baez goes on to show that an n-dimensional normed division algebra must be an irreducible representation of the clifford algebra of dimension n-1.

The result can then be read from the following table:

n -> irreduclible rep of Cliff(n)

0 ->

**1****1**->**2****2**-> 43 ->

**4****4**-> 85 -> 8

6 -> 8

7 ->

**8****8**-> 169 -> 32

10 -> 64

11 -> 64

12 -> 128

13 -> 128

14 -> 128

15 -> 128

We see that the clifford algebras matrices rapidly become too large to allow the formation of a division algebra.

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