Hurwitz's theorem (composition algebras)

Hurwitz's theorem (composition algebras) (Redirected from Hurwitz's theorem (normed division algebras)) Aller à la navigation Aller à la recherche En mathématiques, 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, les nombres complexes, 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 et 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also in Radon (1922). Subsequent proofs of the restrictions on the dimension have been given by Eckmann (1943) using the representation theory of finite groups and by Lee (1948) and Chevalley (1954) 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] Contenu 1 Euclidean Hurwitz algebras 1.1 Définition 1.2 Classification 2 Other proofs 3 Applications to Jordan algebras 4 Voir également 5 Remarques 6 Références 7 Further reading Euclidean Hurwitz algebras Definition A Hurwitz algebra or composition algebra is a finite-dimensional not necessarily associative algebra A with identity endowed with a nondegenerate quadratic form q such that q(a b) = q(un)q(b). If the underlying coefficient field is the reals and q is positive-definite, pour que (un, b) = 1 / 2 [q(un + b) − q(un) − q(b)] is an inner product, then A is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra.[4] If A is a Euclidean Hurwitz algebra and a is in A, define the involution and right and left multiplication operators by {style d'affichage a^{*}=-a+2(un,1)1,,,,L(un)b=ab,,,,R(un)b=ba.} Evidently the involution has period two and preserves the inner product and norm. These operators have the following properties: the involution is an antiautomorphism, c'est à dire. (a b)* = b* a* a a* = ‖ a ‖2 1 = a* a L(a*) = L(un)*, R(a*) = R(un)*, so that the involution on the algebra corresponds to taking adjoints Re (a b) = Re (b a) if Re x = (X + X*)/2 = (X, 1)1 Concernant (a b) c = Re a(b c) L(a2) = L(un)2, R(a2) = R(un)2, so that A is an alternative algebra.

These properties are proved starting from the polarized version of the identity (a b, a b) = (un, a)(b, b): {style d'affichage style d'affichage {2(un,b)(c,ré)=(ac,bd)+(un d,avant JC).}} Setting b = 1 or d = 1 yields L(a*) = L(un)* et R(c*) = R(c)*.

Hence Re(a b) = (a b, 1)1 = (un, b*)1 = (b a, 1)1 = Re(b a).

Similarly Re (a b)c = ((a b)c,1)1 = (a b, c*)1 = (b, a* c*)1 = (avant JC,a*)1 = (un(avant JC),1)1 = Re a(b c).

Ainsi ((un B)*,c) = (un B,c*) = (b,a*c*) = (1,b*(a*c*)) = (1,(b*a*)c*) = (b*a*,c), pour que (un B)* = b*a*.

By the polarized identity ‖ a ‖2 (c, d) = (a c, a d) = (a* (a c), d) so L(a*) L(un) = L(‖ a ‖2). Applied to 1 this gives a* a = ‖ a ‖2 1. Replacing a by a* gives the other identity.

Substituting the formula for a* in L(a*) L(un) = L(a* a) gives L(un)2 = L(a2). The formula R(a2) = R(un)2 is proved analogically.

Classification It is routine to check that the real numbers R, the complex numbers C and the quaternions H are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusions R ⊂ C ⊂ H.

Analysing such an inclusion leads to the Cayley–Dickson construction, formalized by A.A. Albert. Let A be a Euclidean Hurwitz algebra and B a proper unital subalgebra, so a Euclidean Hurwitz algebra in its own right. Pick a unit vector j in A orthogonal to B. Depuis (j, 1) = 0, it follows that j* = −j and hence j2 = −1. Let C be subalgebra generated by B and j. It is unital and is again a Euclidean Hurwitz algebra. It satisfies the following Cayley–Dickson multiplication laws: {style d'affichage style d'affichage {C=Boplus Bj,,,,(a+bj)^{*}=un^{*}-bj,,,,(a+bj)(c+dj)=(ac-d^{*}b)+(bc^{*}+da)j.}} B and B j are orthogonal, since j is orthogonal to B. If a is in B, then j a = a* j, since by orthogonal 0 = 2 (j, a*) = j a − a* j. The formula for the involution follows. To show that B ⊕ B j is closed under multiplication Bj = j B. Since B j is orthogonal to 1, (b j)* = −b j.

b(c j) = (c b)j since (b, j) = 0 pour que, for x in A, (b(c j), x) = (b(j x), j(c j)) = −(b(j x), c*) = −(c b, (j x)*) = −((c b)j, x*) = ((c b)j, x). (j c)b = j(b c) taking adjoints above. (b j)(c j) = −c* b since (b, c j) = 0, pour que, for x in A, ((b j)(c j), x) = −((c j)x*, b j) = (b x*, (c j)j) = −(c* b, x).

Imposing the multiplicativity of the norm on C for a + b j and c + d j gives: {style d'affichage style d'affichage {(|un|^{2}+|b|^{2})(|c|^{2}+|ré|^{2})=|ac-d^{*}b|^{2}+|bc^{*}+da|^{2},}} which leads to {style d'affichage style d'affichage {(ac,d^{*}b)=(bc^{*},da).}} Hence d(a c) = (d a)c, so that B must be associative.

This analysis applies to the inclusion of R in C and C in H. Taking O = H ⊕ H with the product and inner product above gives a noncommutative nonassociative algebra generated by J = (0, 1). This recovers the usual definition of the octonions or Cayley numbers. If A is a Euclidean algebra, it must contain R. If it is strictly larger than R, the argument above shows that it contains C. If it is larger than C, it contains H. If it is larger still, it must contain O. But there the process must stop, because O is not associative. In fact H is not commutative and a(b j) = (b a) j ≠ (a b)j in O.[5] Théorème. The only Euclidean Hurwitz algebras are the real numbers, les nombres complexes, the quaternions and the octonions.

Other proofs The proofs of Lee (1948) and Chevalley (1954) use Clifford algebras to show that the dimension N of A must be 1, 2, 4 ou 8. In fact the operators L(un) avec (un, 1) = 0 satisfy L(un)2 = −‖ a ‖2 and so form a real Clifford algebra. If a is a unit vector, then L(un) is skew-adjoint with square −I. So N must be either even or 1 (in which case A contains no unit vectors orthogonal to 1). The real Clifford algebra and its complexification act on the complexification of A, an N-dimensional complex space. If N is even, N − 1 est impair, so the Clifford algebra has exactly two complex irreducible representations of dimension 2N/2 − 1. So this power of 2 must divide N. It is easy to see that this implies N can only be 1, 2, 4 ou 8.

The proof of Eckmann (1954) 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. En effet, taking an orthonormal basis ei of the orthogonal complement of 1 gives rise to operators Ui = L(ei) satisfaisant {style d'affichage style d'affichage {U_{je}^{2}=-I,,,,U_{je}U_{j}=-U_{j}U_{je},,(ineq j).}} This is a projective representation of a direct product of N − 1 groups of order 2. (N is assumed to be greater than 1.) The operators Ui 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 Hurwitz (1923).[6] Assume that there is a composition law for two forms {style d'affichage style d'affichage {(X_{1}^{2}+cdots +x_{N}^{2})(y_{1}^{2}+cdots +y_{N}^{2})=z_{1}^{2}+cdots +z_{N}^{2},}} where zi is bilinear in x and y. Ainsi {style d'affichage style d'affichage {z_{je}=somme _{j=1}^{N}un_{ij}(X)y_{j}}} where the matrix T(X) = (aij) is linear in x. The relations above are equivalent to {style d'affichage style d'affichage {J(X)J(X)^{t}=x_{1}^{2}+cdots +x_{N}^{2}.}} L'écriture {style d'affichage style d'affichage {J(X)=T_{1}X_{1}+cdots +T_{N}X_{N},}} the relations become {style d'affichage style d'affichage {T_{je}T_{j}^{t}+T_{j}T_{je}^{t}=2delta _{ij}JE.}} Now set Vi = (TN)t Ti. Thus VN = I and the V1, ... , VN − 1 are skew-adjoint, orthogonal satisfying exactly the same relations as the Ui's: {style d'affichage style d'affichage {V_{je}^{2}=-I,,,,V_{je}V_{j}=-V_{j}V_{je},,(ineq j).}} Since Vi is an orthogonal matrix with square −I on a real vector space, N is even.

Let G be the finite group generated by elements vi such that {style d'affichage style d'affichage {v_{je}^{2}=varepsilon ,,,,v_{je}v_{j}=varepsilon v_{j}v_{je},,(ineq j),}} where ε is central of order 2. The commutator subgroup [g, G] is just formed of 1 and ε. If N is odd this coincides with the center while if N is even the center has order 4 with extra elements γ = v1 ... vN − 1 and ε γ. If g in G is not in the center its conjugacy class is exactly g and ε g. Thus there are 2N − 1 + 1 conjugacy classes for N odd and 2N − 1 + 2 for N even. G has | G / [g, G] | = 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 = 6, the dimension is 4, which does not divide 6. So N can only be 1, 2, 4 ou 8.

Applications to Jordan algebras Let A be a Euclidean Hurwitz algebra and let Mn(UN) be the algebra of n-by-n matrices over A. It is a unital nonassociative algebra with an involution given by {style d'affichage style d'affichage {(X_{ij})^{*}=(X_{ji}^{*}).}} The trace Tr(X) is defined as the sum of the diagonal elements of X and the real-valued trace by TrR(X) = Re Tr(X). The real-valued trace satisfies: {nom de l'opérateur de style d'affichage {Tr} _{mathbf {R} }XY=operatorname {Tr} _{mathbf {R} }YX,qquad operatorname {Tr} _{mathbf {R} }(XY)Z=operatorname {Tr} _{mathbf {R} }X(YZ).} These are immediate consequences of the known identities for n = 1.

In A define the associator by {style d'affichage style d'affichage {[un,b,c]=un(avant JC)-(un B)c.}} It is trilinear and vanishes identically if A is associative. Since A is an alternative algebra [un, a, b] = 0 et [b, a, a] = 0. Polarizing it follows that the associator is antisymmetric in its three entries. Par ailleurs, if a, b or c lie in R then [un, b, c] = 0. These facts imply that M3(UN) has certain commutation properties. In fact if X is a matrix in M3(UN) with real entries on the diagonal then {style d'affichage style d'affichage {[X,X^{2}]=aI,}} with a in A. In fact if Y = [X, X2], alors {style d'affichage style d'affichage {y_{ij}=somme _{k,aune }[X_{ik},X_{kell },X_{ell j}].}} Since the diagonal entries of X are real, the off diagonal entries of Y vanish. Each diagonal entry of Y is a sum of two associators involving only off diagonal terms of X. Since the associators are invariant under cyclic permutations, the diagonal entries of Y are all equal.

Let Hn(UN) be the space of self-adjoint elements in Mn(UN) with product X∘Y = 1 / 2 (X Y + Y X) and inner product (X, Y) = TrR(X Y).

Théorème. Hn(UN) is a Euclidean Jordan algebra if A is associative (the real numbers, complex numbers or quaternions) and n ≥ 3 or if A is nonassociative (the octonions) et n = 3.

The exceptional Jordan algebra H3(O) is called the Albert algebra after A.A. Albert.

To check that Hn(UN) satisfies the axioms for a Euclidean Jordan algebra, the real trace defines a symmetric bilinear form with (X, X) = Σ ‖ xij ‖2. So it is an inner product. It satisfies the associativity property (Z∘X, Y) = (X, Z∘Y) because of the properties of the real trace. The main axiom to check is the Jordan condition for the operators L(X) defined by L(X)Y = X∘Y: {style d'affichage style d'affichage {[L(X),L(X^{2})]=0.}} This is easy to check when A is associative, since Mn(UN) is an associative algebra so a Jordan algebra with X∘Y = 1 / 2 (X Y + Y X). When A = O and n = 3 a special argument is required, one of the shortest being due to Freudenthal (1951).[7] In fact if T is in H3(O) with Tr T = 0, alors {style d'affichage style d'affichage {ré(X)=TX-XT}} defines a skew-adjoint derivation of H3(O). En effet, {nom de l'opérateur de style d'affichage {Tr} (J(X(X^{2}))-J(X^{2}(X)))=nomopérateur {Tr} J(aI)=nomopérateur {Tr} (J)a=0,} pour que {style d'affichage (ré(X),X^{2})=0.} Polarizing yields: {style d'affichage (ré(X),Ycirc Z)+(ré(Oui),Zcirc X)+(ré(Z),Xcirc Y)=0.} Setting Z = 1, shows that D is skew-adjoint. The derivation property D(X∘Y) =D(X)∘Y + X∘D(Oui) follows by this and the associativity property of the inner product in the identity above.

With A and n as in the statement of the theorem, let K be the group of automorphisms of E = Hn(UN) leaving invariant the inner product. It is a closed subgroup of O(E) so a compact Lie group. Its Lie algebra consists of skew-adjoint derivations. Freudenthal (1951) showed that given X in E there is an automorphism k in K such that k(X) 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 Mn(UN) for any non-associative algebra A.

To prove the diagonalization theorem, take X in E. By compactness k can be chosen in K minimizing the sums of the squares of the norms of the off-diagonal terms of k(X). Since K 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 k(X). Replacing X by k X, it can be assumed that the maximum is attained at X. Since the symmetric group Sn, acting by permuting the coordinates, lies in K, if X is not diagonal, it can be supposed that x12 and its adjoint x21 are non-zero. Let T be the skew-adjoint matrix with (2, 1) entry a, (1, 2) entry −a* and 0 elsewhere and let D be the derivation ad T of E. Let kt = exp tD in K. Then only the first two diagonal entries in X(t) = ktX differ from those of X. The diagonal entries are real. The derivative of x11(t) at t = 0 est le (1, 1) coordinate of [J, X], c'est à dire. a* x21 + x12 a = 2(x21, a). This derivative is non-zero if a = x21. D'autre part, the group kt preserves the real-valued trace. Since it can only change x11 and x22, it preserves their sum. Cependant, on the line x + y =constant, x2 + y2 has no local maximum (only a global minimum), une contradiction. Hence X must be diagonal.

See also Multiplicative quadratic form Radon–Hurwitz number Frobenius Theorem Notes ^ See: Lam 2005 Rajwade 1993 Shapiro 2000 ^ See: Eckmann 1989 Eckmann 1999 ^ Jordan, von Neumann & Wigner 1934 ^ Faraut & Koranyi 1994, p. 82 ^ Faraut & Koranyi 1994, pp. 81–86 ^ See: Hurwitz 1923, p. 11 Herstein 1968, pp. 141–144 ^ See: Faraut & Koranyi 1994, pp. 88–91 Postnikov 1986 References Albert, UN. UN. (1934), "On a certain algebra of quantum mechanics", Anne. of Math., 35 (1): 65–73, est ce que je:10.2307/1968118, JSTOR 1968118 Chevalley, C. (1954), The algebraic theory of spinors and Clifford algebras, Columbia University Press Eckmann, Beno (1943), "Gruppentheoretischer Beweis des Satzes von Hurwitz–Radon über die Komposition quadratischer Formen", Commentaire. Math. Helv., 15: 358–366, est ce que je:10.1007/bf02565652, S2CID 123322808 Eckmann, Beno (1989), "Hurwitz–Radon matrices and periodicity modulo 8", Enseign. Math., 35: 77–91, archivé à partir de l'original sur 2013-06-16 Eckmann, Beno (1999), "Topologie, algebra, analysis—relations and missing links", Notices Amer. Math. Soc., 46: 520–527 Faraut, J; Koranyi, UN. (1994), Analyse sur cônes symétriques, Monographies mathématiques d'Oxford, Presse universitaire d'Oxford, ISBN 978-0198534778 Freudenthal, Hans (1951), Oktaven, Ausnahmegruppen und Oktavengeometrie, Mathematisch Instituut der Rijksuniversiteit te Utrecht Freudenthal, Hans (1985), "Oktaven, Ausnahmegruppen und Oktavengeometrie", Géom. Dedicata, 19: 7–63, est ce que je:10.1007/bf00233101, S2CID 121496094 (réimpression de 1951 article) Herstein, je. N. (1968), Noncommutative rings, Carus Mathematical Monographs, volume. 15, Association mathématique d'Amérique, ISBN 978-0883850152 Hurwitz, UN. (1898), "Über die Composition der quadratischen Formen von beliebig vielen Variabeln", Goett. Nachr.: 309–316 Hurwitz, UN. (1923), "Über die Komposition der quadratischen Formen", Math. Ann., 88 (1–2): 1–25, est ce que je:10.1007/bf01448439, S2CID 122147399 Jacobson, N. (1968), Structure et représentations des algèbres de Jordan, Publications du colloque de l'American Mathematical Society, volume. 39, American Mathematical Society Jordan, P; par Neumann, J; Wigner, E. (1934), "Sur une généralisation algébrique du formalisme de la mécanique quantique", Anne. of Math., 35 (1): 29–64, est ce que je:10.2307/1968117, JSTOR 1968117 Lam, Tsit Yuen (2005), Introduction to Quadratic Forms over Fields, Études supérieures en mathématiques, volume. 67, Société mathématique américaine, ISBN 978-0-8218-1095-8, M 2104929, Zbl 1068.11023 Lee, H. C. (1948), "Sur le théorème de Hurwitz-Radon pour la composition des formes quadratiques", Commentaire. Math. Helv., 21: 261–269, est ce que je:10.1007/bf02568038, S2CID 121079375, archivé à partir de l'original sur 2014-05-03 Porteous, I.R. (1969), Topological Geometry, Van Nostrand Reinhold, ISBN 978-0-442-06606-2, Zbl 0186.06304 Postnikov, M. (1986), Lie groups and Lie algebras. Lectures in geometry. Semester V, Mir Radon, J. (1922), "Lineare scharen orthogonaler matrizen", Documents du séminaire de mathématiques de l'Université de Hambourg, 1: 1–14, est ce que je:10.1007/bf02940576, S2CID 120583389 Rajwade, UN. R. (1993), Squares, Série de notes de cours de la London Mathematical Society, volume. 171, la presse de l'Universite de Cambridge, ISBN 978-0-521-42668-8, Zbl 0785.11022 Schafer, Richard D.. (1995) [1966], An introduction to non-associative algebras, Publications de Douvres, ISBN 978-0-486-68813-8, Zbl 0145.25601 Shapiro, Daniel B. (2000), Compositions of quadratic forms, De Gruyter Expositions in Mathematics, volume. 33, Walter de Gruyter, ISBN 978-3-11-012629-7, Zbl 0954.11011 Further reading Baez, Jean C. (2002), "The octonions", Taureau. Amer. Math. Soc., 39 (2): 145–205, arXiv:math/0105155, est ce que je:10.1090/S0273-0979-01-00934-X, S2CID 586512 Conway, John H.; Forgeron, Derek A. (2003), On quaternions and octonions: their geometry, arithmetic, and symmetry, AK Peters, ISBN 978-1568811345 Kantor, I.L.; Solodovnikov, A.S. (1989), "Normed algebras with an identity. Hurwitz's theorem.", Hypercomplex numbers. An elementary introduction to algebras, Trans. UN. Shenitzer (2sd éd.), Springer Verlag, p. 121, ISBN 978-0-387-96980-0, Zbl 0669.17001 Max Koecher & Reinhold Remmert (1990) "Composition Algebras. Hurwitz's Theorem — Vector-Product Algebras", chapitre 10 of Numbers by Heinz-Dieter Ebbinghaus et al., Springer, ISBN 0-387-97202-1 Springer, J. UN.; F. ré. Camp sur le terrain (2000), Octonions, Jordan Algebras and Exceptional Groups, Springer Verlag, ISBN 978-3-540-66337-9 Catégories: Composition algebrasNon-associative algebrasQuadratic formsRepresentation theoryTheorems about algebras1923 introductions