Hurwitz's theorem (normed division algebras) | Wikipedia audio article

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00:01:45 1 Euclidean Hurwitz algebras
00:01:56 1.1 Definition
00:02:19 1.2 Classification
00:04:03 2 Other proofs
00:04:16 3 Applications to Jordan algebras
00:04:38 4 See also
00:05:01 5 Notes
00:06:03 6 References
00:06:16 7 Further reading
00:06:33 (a b, 1)1
00:06:47 (b a, 1)1
00:07:03 ((a b)c,1)1
00:07:17 (b, a* c*)1
00:07:31 (a(bc),1)1
00:07:48 (ab,c*)
00:08:00 (1,b*(a*c*))
00:08:14 (b*a*,c), so that (ab)*
00:08:32 (a c, a d)
00:08:48 ‖ a ‖2. Applied to 1 this gives a* a
00:09:13 L(a* a) gives L(a)2
00:09:29 Classification
00:10:18 0, it follows that j*
00:12:14 a* j, since by orthogonal 0
00:12:29 j a − a* j. The formula for the involution follows. To show that B ⊕ B j is closed under multiplication note that Bj
00:13:01 (c b)j since (b, j)
00:13:18 (b(j x), j(c j)
00:13:34 −(c b, (j x)*)
00:14:02 −c* b since (b, c j)
00:14:20 −((c j)x*, b j)
00:16:21 H ⊕ H with the product and inner product above gives a noncommutative nonassociative algebra generated by J
00:17:08 (b a) j ≠ (a b)j in O.Theorem. The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.
00:17:23 Other proofs
00:17:33 The proofs of Lee (1948) and Chevalley (1954) use Clifford algebras to show that the dimension N of A must be 1, 2, 4 or 8. In fact the operators L(a) with (a, 1)
00:24:21 (TN)t Ti. Thus VN
00:27:13 2N − 1 1-dimensional complex representations. The total number of irreducible complex representations is the number of conjugacy classes. So since N is even, there are two further irreducible complex representations. Since the sum of the squares of the dimensions equals | G | and the dimensions divide | G |, the two irreducibles must have dimension 2(N − 2)/2. When N is even, there are two and their dimension must divide the order of the group, so is a power of two, so they must both have dimension 2(N − 2)/2. The space on which the Vi's act can be complexified. It will have complex dimension N. It breaks up into some of complex irreducible representations of G, all having dimension 2(N − 2)/2. In particular this dimension is ≤ N, so N is less than or equal to 8. If N
00:28:41 Applications to Jordan algebras
00:31:24 0 and [b, a, a]
00:33:25 1/2(X Y + Y X) and inner product (X, Y)
00:34:17 ∑ ‖ xij ‖2. So it is an inner product. It satisfies the associativity property (Z∘X, Y)
00:35:18 1/2(X Y + Y X). When A
00:35:24 3 a special argument is required, one of the shortest being due to Freudenthal (1951).In fact if T is in H3(O) with Tr T
00:38:00 1, shows that D is skew-adjoint. The derivation property D(X∘Y)
00:40:07 exp tD in K. Then only the first two diagonal entries in X(t)
00:40:31 0 is the (1, 1) coordinate of [T, X], i.e. a* x21 + x12 a
00:40:54 x21. On the other hand, the group kt preserves the real-valued trace. Since it can only change x11 and x22, it preserves their sum. However, on the line x + y
00:41:31 See also



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SUMMARY
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In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions or the octonions. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras.
The theory of composition algebras has subsequently been generalized ...