Английская Википедия:Hurwitz's theorem (composition algebras)

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Шаблон:Short description 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 to arbitrary quadratic forms and arbitrary fields.[1] Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also in Шаблон:Harvtxt. Subsequent proofs of the restrictions on the dimension have been given by Шаблон:Harvtxt using the representation theory of finite groups and by Шаблон:Harvtxt and Шаблон:Harvtxt using Clifford algebras. Hurwitz's theorem has been applied in algebraic topology to problems on vector fields on spheres and the homotopy groups of the classical groups[2] and in quantum mechanics to the classification of simple Jordan algebras.[3]

Euclidean Hurwitz algebras

Definition

A Hurwitz algebra or composition algebra is a finite-dimensional not necessarily associative algebra Шаблон:Mvar with identity endowed with a nondegenerate quadratic form Шаблон:Mvar such that Шаблон:Math. If the underlying coefficient field is the reals and Шаблон:Mvar is positive-definite, so that Шаблон:Math is an inner product, then Шаблон:Mvar is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra.[4]

If Шаблон:Mvar is a Euclidean Hurwitz algebra and Шаблон:Mvar is in Шаблон:Mvar, define the involution and right and left multiplication operators by

<math>a^* = -a + 2(a,1)1,\quad L(a)b = ab,\quad R(a)b = ba.</math>

Evidently the involution has period two and preserves the inner product and norm. These operators have the following properties:

These properties are proved starting from the polarized version of the identity Шаблон:Math:

<math>\displaystyle{2(a,b)(c,d) = (ac,bd) + (ad,bc).}</math>

Setting Шаблон:Math or Шаблон:Math yields Шаблон:Math and Шаблон:Math.

Hence Шаблон:Math.

Similarly Шаблон:Math.

Hence Шаблон:Math, so that Шаблон:Math.

By the polarized identity Шаблон:Math so Шаблон:Math. Applied to 1 this gives Шаблон:Math. Replacing Шаблон:Mvar by Шаблон:Math gives the other identity.

Substituting the formula for Шаблон:Math in Шаблон:Math gives Шаблон:Math. The formula Шаблон:Math is proved analogously.

Classification

It is routine to check that the real numbers Шаблон:Math, the complex numbers Шаблон:Math and the quaternions Шаблон:Math are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusions Шаблон:Math.

Analysing such an inclusion leads to the Cayley–Dickson construction, formalized by A.A. Albert. Let Шаблон:Mvar be a Euclidean Hurwitz algebra and Шаблон:Mvar a proper unital subalgebra, so a Euclidean Hurwitz algebra in its own right. Pick a unit vector Шаблон:Mvar in Шаблон:Mvar orthogonal to Шаблон:Mvar. Since Шаблон:Math, it follows that Шаблон:Math and hence Шаблон:Math. Let Шаблон:Mvar be subalgebra generated by Шаблон:Mvar and Шаблон:Mvar. It is unital and is again a Euclidean Hurwitz algebra. It satisfies the following Cayley–Dickson multiplication laws:

<math>\displaystyle{C=B\oplus Bj, \,\,\, (a+bj)^*=a^* - bj, \,\,\, (a+bj)(c+dj)=(ac -d^*b) +(bc^*+da)j.}</math>

Шаблон:Mvar and Шаблон:Math are orthogonal, since Шаблон:Mvar is orthogonal to Шаблон:Mvar. If Шаблон:Mvar is in Шаблон:Mvar, then Шаблон:Math, since by orthogonal Шаблон:Math. The formula for the involution follows. To show that Шаблон:Math is closed under multiplication Шаблон:Math. Since Шаблон:Math is orthogonal to 1, Шаблон:Math.

Imposing the multiplicativity of the norm on Шаблон:Mvar for Шаблон:Math and Шаблон:Math gives:

<math>\displaystyle{(\|a\|^2 + \|b\|^2)(\|c\|^2 + \|d\|^2) = \|ac - d^*b\|^2 + \|bc^* + da\|^2,}</math>

which leads to

<math>\displaystyle{(ac,d^*b) = (bc^*,da).}</math>

Hence Шаблон:Math, so that Шаблон:Mvar must be associative.

This analysis applies to the inclusion of Шаблон:Math in Шаблон:Math and Шаблон:Math in Шаблон:Math. Taking Шаблон:Math with the product and inner product above gives a noncommutative nonassociative algebra generated by Шаблон:Math. This recovers the usual definition of the octonions or Cayley numbers. If Шаблон:Mvar is a Euclidean algebra, it must contain Шаблон:Math. If it is strictly larger than Шаблон:Math, the argument above shows that it contains Шаблон:Math. If it is larger than Шаблон:Math, it contains Шаблон:Math. If it is larger still, it must contain Шаблон:Math. But there the process must stop, because Шаблон:Math is not associative. In fact Шаблон:Math is not commutative and Шаблон:Math in Шаблон:Math.[5]

Шаблон:Smallcaps The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.

Other proofs

The proofs of Шаблон:Harvtxt and Шаблон:Harvtxt use Clifford algebras to show that the dimension Шаблон:Mvar of Шаблон:Mvar must be 1, 2, 4 or 8. In fact the operators Шаблон:Math with Шаблон:Math satisfy Шаблон:Math and so form a real Clifford algebra. If Шаблон:Mvar is a unit vector, then Шаблон:Math is skew-adjoint with square Шаблон:Math. So Шаблон:Mvar must be either even or 1 (in which case Шаблон:Mvar contains no unit vectors orthogonal to 1). The real Clifford algebra and its complexification act on the complexification of Шаблон:Mvar, an Шаблон:Mvar-dimensional complex space. If Шаблон:Mvar is even, Шаблон:Math is odd, so the Clifford algebra has exactly two complex irreducible representations of dimension Шаблон:Math. So this power of 2 must divide Шаблон:Mvar. It is easy to see that this implies Шаблон:Mvar can only be 1, 2, 4 or 8.

The proof of Шаблон:Harvtxt uses the representation theory of finite groups, or the projective representation theory of elementary abelian 2-groups, known to be equivalent to the representation theory of real Clifford algebras. Indeed, taking an orthonormal basis Шаблон:Math of the orthogonal complement of 1 gives rise to operators Шаблон:Math satisfying

<math>U_i^2 = -I,\quad U_i U_j = -U_j U_i \,\, (i \ne j).</math>

This is a projective representation of a direct product of Шаблон:Math groups of order 2. (Шаблон:Mvar is assumed to be greater than 1.) The operators Шаблон:Math by construction are skew-symmetric and orthogonal. In fact Eckmann constructed operators of this type in a slightly different but equivalent way. It is in fact the method originally followed in Шаблон:Harvtxt.[6] Assume that there is a composition law for two forms

<math>\displaystyle{(x_1^2 + \cdots +x_N^2)(y_1^2 + \cdots + y_N^2) =z_1^2 + \cdots + z_N^2,}</math>

where Шаблон:Math is bilinear in Шаблон:Mvar and Шаблон:Mvar. Thus

<math>\displaystyle{z_i=\sum_{j=1}^N a_{ij}(x)y_j}</math>

where the matrix Шаблон:Math is linear in Шаблон:Mvar. The relations above are equivalent to

<math>\displaystyle{T(x)T(x)^t=x_1^2 +\cdots + x_N^2.}</math>

Writing

<math>\displaystyle{T(x)=T_1x_1 + \cdots + T_Nx_N,}</math>

the relations become

<math>\displaystyle{T_iT^t_j+T_jT_i^t =2\delta_{ij}I.}</math>

Now set Шаблон:Math. Thus Шаблон:Math and the Шаблон:Math are skew-adjoint, orthogonal satisfying exactly the same relations as the Шаблон:Math's:

<math>\displaystyle{V_i^2 = -I,\quad V_iV_j = -V_jV_i \,\, (i \ne j).}</math>

Since Шаблон:Math is an orthogonal matrix with square Шаблон:Math on a real vector space, Шаблон:Mvar is even.

Let Шаблон:Mvar be the finite group generated by elements Шаблон:Math such that

<math>\displaystyle{v_i^2 = \varepsilon,\quad v_iv_j = \varepsilon v_jv_i \,\, (i \ne j),}</math>

where Шаблон:Mvar is central of order 2. The commutator subgroup Шаблон:Math is just formed of 1 and Шаблон:Mvar. If Шаблон:Mvar is odd this coincides with the center while if Шаблон:Mvar is even the center has order 4 with extra elements Шаблон:Math and Шаблон:Math. If Шаблон:Mvar in Шаблон:Mvar is not in the center its conjugacy class is exactly Шаблон:Mvar and Шаблон:Math. Thus there are Шаблон:Math conjugacy classes for Шаблон:Mvar odd and Шаблон:Math for Шаблон:Mvar even. Шаблон:Mvar has Шаблон:Math 1-dimensional complex representations. The total number of irreducible complex representations is the number of conjugacy classes. So since Шаблон:Mvar is even, there are two further irreducible complex representations. Since the sum of the squares of the dimensions equals Шаблон:Math and the dimensions divide Шаблон:Math, the two irreducibles must have dimension Шаблон:Math. When Шаблон:Mvar 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 Шаблон:Math. The space on which the Шаблон:Math's act can be complexified. It will have complex dimension Шаблон:Mvar. It breaks up into some of complex irreducible representations of Шаблон:Mvar, all having dimension Шаблон:Math. In particular this dimension is Шаблон:Math, so Шаблон:Mvar is less than or equal to 8. If Шаблон:Math, the dimension is 4, which does not divide 6. So N can only be 1, 2, 4 or 8.

Applications to Jordan algebras

Let Шаблон:Mvar be a Euclidean Hurwitz algebra and let Шаблон:Math be the algebra of Шаблон:Mvar-by-Шаблон:Mvar matrices over Шаблон:Mvar. It is a unital nonassociative algebra with an involution given by

<math>\displaystyle{(x_{ij})^*=(x_{ji}^*).}</math>

The trace Шаблон:Math is defined as the sum of the diagonal elements of Шаблон:Mvar and the real-valued trace by Шаблон:Math. The real-valued trace satisfies:

<math>\operatorname{Tr}_{\mathbf{R}} XY = \operatorname{Tr}_{\mathbf{R}} YX, \qquad \operatorname{Tr}_{\mathbf{R}} (XY)Z = \operatorname{Tr}_{\mathbf{R}} X(YZ).</math>

These are immediate consequences of the known identities for Шаблон:Math.

In Шаблон:Mvar define the associator by

<math>\displaystyle{[a,b,c]=a(bc) - (ab)c.}</math>

It is trilinear and vanishes identically if Шаблон:Mvar is associative. Since Шаблон:Mvar is an alternative algebra Шаблон:Math and Шаблон:Math. Polarizing it follows that the associator is antisymmetric in its three entries. Furthermore, if Шаблон:Mvar, Шаблон:Mvar or Шаблон:Mvar lie in Шаблон:Math then Шаблон:Math. These facts imply that Шаблон:Math has certain commutation properties. In fact if Шаблон:Mvar is a matrix in Шаблон:Math with real entries on the diagonal then

<math>\displaystyle{[X,X^2] = aI,}</math>

with Шаблон:Mvar in Шаблон:Mvar. In fact if Шаблон:Math, then

<math>\displaystyle{y_{ij} = \sum_{k,\ell} [x_{ik},x_{k\ell},x_{\ell j}].}</math>

Since the diagonal entries of Шаблон:Mvar are real, the off-diagonal entries of Шаблон:Mvar vanish. Each diagonal entry of Шаблон:Mvar is a sum of two associators involving only off diagonal terms of Шаблон:Mvar. Since the associators are invariant under cyclic permutations, the diagonal entries of Шаблон:Mvar are all equal.

Let Шаблон:Math be the space of self-adjoint elements in Шаблон:Math with product Шаблон:Math and inner product Шаблон:Math.

Шаблон:Smallcaps Шаблон:Math is a Euclidean Jordan algebra if Шаблон:Mvar is associative (the real numbers, complex numbers or quaternions) and Шаблон:Math or if Шаблон:Mvar is nonassociative (the octonions) and Шаблон:Math.

The exceptional Jordan algebra Шаблон:Math is called the Albert algebra after A.A. Albert.

To check that Шаблон:Math satisfies the axioms for a Euclidean Jordan algebra, the real trace defines a symmetric bilinear form with Шаблон:Math. So it is an inner product. It satisfies the associativity property Шаблон:Math because of the properties of the real trace. The main axiom to check is the Jordan condition for the operators Шаблон:Math defined by Шаблон:Math:

<math>\displaystyle{[L(X),L(X^2)]=0.}</math>

This is easy to check when Шаблон:Mvar is associative, since Шаблон:Math is an associative algebra so a Jordan algebra with Шаблон:Math. When Шаблон:Math and Шаблон:Math a special argument is required, one of the shortest being due to Шаблон:Harvtxt.[7]

In fact if Шаблон:Mvar is in Шаблон:Math with Шаблон:Math, then

<math>\displaystyle{D(X) = TX -XT}</math>

defines a skew-adjoint derivation of Шаблон:Math. Indeed,

<math>\operatorname{Tr}(T(X(X^2)) -T(X^2(X))) = \operatorname{Tr} T(aI) = \operatorname{Tr}(T)a=0,</math>

so that

<math>(D(X),X^2) = 0.</math>

Polarizing yields:

<math>(D(X),Y\circ Z) + (D(Y),Z\circ X) + (D(Z),X\circ Y) = 0.</math>

Setting Шаблон:Math shows that Шаблон:Mvar is skew-adjoint. The derivation property Шаблон:Math follows by this and the associativity property of the inner product in the identity above.

With Шаблон:Mvar and Шаблон:Mvar as in the statement of the theorem, let Шаблон:Mvar be the group of automorphisms of Шаблон:Math leaving invariant the inner product. It is a closed subgroup of Шаблон:Math so a compact Lie group. Its Lie algebra consists of skew-adjoint derivations. Шаблон:Harvtxt showed that given Шаблон:Mvar in Шаблон:Mvar there is an automorphism Шаблон:Mvar in Шаблон:Mvar such that Шаблон:Math is a diagonal matrix. (By self-adjointness the diagonal entries will be real.) Freudenthal's diagonalization theorem immediately implies the Jordan condition, since Jordan products by real diagonal matrices commute on Шаблон:Math for any non-associative algebra Шаблон:Mvar.

To prove the diagonalization theorem, take Шаблон:Mvar in Шаблон:Mvar. By compactness Шаблон:Mvar can be chosen in Шаблон:Mvar minimizing the sums of the squares of the norms of the off-diagonal terms of Шаблон:Math. Since Шаблон:Mvar preserves the sums of all the squares, this is equivalent to maximizing the sums of the squares of the norms of the diagonal terms of Шаблон:Math. Replacing Шаблон:Mvar by Шаблон:Math, it can be assumed that the maximum is attained at Шаблон:Mvar. Since the symmetric group Шаблон:Math, acting by permuting the coordinates, lies in Шаблон:Mvar, if Шаблон:Mvar is not diagonal, it can be supposed that Шаблон:Math and its adjoint Шаблон:Math are non-zero. Let Шаблон:Mvar be the skew-adjoint matrix with Шаблон:Math entry Шаблон:Mvar, Шаблон:Math entry Шаблон:Math and 0 elsewhere and let Шаблон:Mvar be the derivation ad Шаблон:Mvar of Шаблон:Mvar. Let Шаблон:Math in Шаблон:Mvar. Then only the first two diagonal entries in Шаблон:Math differ from those of Шаблон:Mvar. The diagonal entries are real. The derivative of Шаблон:Math at Шаблон:Math is the Шаблон:Math coordinate of Шаблон:Math, i.e. Шаблон:Math. This derivative is non-zero if Шаблон:Math. On the other hand, the group Шаблон:Math preserves the real-valued trace. Since it can only change Шаблон:Math and Шаблон:Math, it preserves their sum. However, on the line Шаблон:Math constant, Шаблон:Math has no local maximum (only a global minimum), a contradiction. Hence Шаблон:Mvar must be diagonal.

See also

Notes

Шаблон:Reflist

References

Further reading

Партнерские ресурсы
Криптовалюты
Магазины
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Разное

  1. ↑ See:
  2. ↑ See:
  3. ↑ Шаблон:Harvnb
  4. ↑ Шаблон:Harvnb
  5. ↑ Шаблон:Harvnb
  6. ↑ See:
  7. ↑ See:
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