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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Hurwitz's theorem is a
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
of
Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, mathematical analysis, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a ...
(1859–1919), published posthumously in 1923, solving the Hurwitz problem for
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
unital real
non-associative algebra A non-associative algebra (or distributive algebra) is an algebra over a field where the binary operation, binary multiplication operation is not assumed to be associative operation, associative. That is, an algebraic structure ''A'' is a non-ass ...
s endowed with a nondegenerate
positive-definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite ...
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
. The theorem states that if the quadratic form defines a
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
into the positive real numbers on the non-zero part of the algebra, then the algebra must be
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s, the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s, or the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...
s, and that there are no other possibilities. Such algebras, sometimes called Hurwitz algebras, are examples of
composition algebra In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution ...
s. The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. 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 . Subsequent proofs of the restrictions on the dimension have been given by using the
representation theory of finite groups The representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are ...
and by and using
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...
s. Hurwitz's theorem has been applied in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
to problems on vector fields on spheres and the
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s of the
classical group In mathematics, the classical groups are defined as the special linear groups over the reals \mathbb, the complex numbers \mathbb and the quaternions \mathbb together with special automorphism groups of Bilinear form#Symmetric, skew-symmetric an ...
s and in
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
to the classification of simple Jordan algebras.


Euclidean Hurwitz algebras


Definition

A Hurwitz algebra or composition algebra is a finite-dimensional not necessarily associative algebra with identity endowed with a nondegenerate quadratic form such that . If the underlying coefficient field is the reals and is positive-definite, so that is an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
, then is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra. If is a Euclidean Hurwitz algebra and is in , define the
involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...
and right and left multiplication operators by :a^* = -a + 2(a,1)1,\quad L(a)b = ab,\quad R(a)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 In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an invertible antihomomorphism, i.e. an antiisomorphism, from a set to itself. Fro ...
, i.e. * * , , so that the involution on the algebra corresponds to taking adjoints * if * * , , so that is an
alternative algebra In abstract algebra, an alternative algebra is an algebra over a field, algebra in which multiplication need not be associative, only alternativity, alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the a ...
. These properties are proved starting from the polarized version of the identity : :\displaystyle Setting or yields and . Hence . Similarly . Hence , so that . By the polarized identity so . Applied to 1 this gives . Replacing by gives the other identity. Substituting the formula for in gives . The formula is proved analogously.


Classification

It is routine to check that the real numbers , the complex numbers and the quaternions are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusions . Analysing such an inclusion leads to the
Cayley–Dickson construction In mathematics, the Cayley–Dickson construction, sometimes also known as the Cayley–Dickson process or the Cayley–Dickson procedure produces a sequence of algebra over a field, algebras over the field (mathematics), field of real numbers, eac ...
, formalized by A.A. Albert. Let be a Euclidean Hurwitz algebra and a proper unital subalgebra, so a Euclidean Hurwitz algebra in its own right. Pick a
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
in orthogonal to . Since , it follows that and hence . Let be subalgebra generated by and . It is unital and is again a Euclidean Hurwitz algebra. It satisfies the following Cayley–Dickson multiplication laws: :\displaystyle and are orthogonal, since is orthogonal to . If is in , then , since by orthogonal . The formula for the involution follows. To show that is closed under multiplication . Since is orthogonal to 1, . * since so that, for in , . * taking adjoints above. * since = 0, so that, for in , . Imposing the multiplicativity of the norm on for and gives: :\displaystyle which leads to :\displaystyle Hence , so that ''must be associative''. This analysis applies to the inclusion of in and in . Taking with the product and inner product above gives a noncommutative nonassociative algebra generated by . This recovers the usual definition of the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...
s or Cayley numbers. If is a Euclidean algebra, it must contain . If it is strictly larger than , the argument above shows that it contains . If it is larger than , it contains . If it is larger still, it must contain . But there the process must stop, because is not associative. In fact is not commutative and in . The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.


Other proofs

The proofs of and use
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure of a distinguished subspace. As -algebras, they generalize the real number ...
s to show that the dimension of must be 1, 2, 4 or 8. In fact the operators with satisfy and so form a real Clifford algebra. If is a unit vector, then is skew-adjoint with square . So must be either even or 1 (in which case contains no unit vectors orthogonal to 1). The real Clifford algebra and its complexification act on the complexification of , an -dimensional complex space. If is even, is odd, so the Clifford algebra has exactly two complex
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
s of dimension . So this power of 2 must divide . It is easy to see that this implies can only be 1, 2, 4 or 8. The proof of uses the
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
of
finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
s, 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 In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
of the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W^\perp of all vectors in V that are orthogonal to every vector in W. I ...
of 1 gives rise to operators satisfying :U_i^2 = -I,\quad U_i U_j = -U_j U_i \,\, (i \ne j). This is a
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where G ...
of a direct product of groups of
order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...
2. ( is assumed to be greater than 1.) The operators 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 . Assume that there is a composition law for two forms :\displaystyle where is bilinear in and . Thus :\displaystyle where the
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
is linear in . The relations above are equivalent to :\displaystyle Writing :\displaystyle the relations become :\displaystyle Now set . Thus and the are skew-adjoint, orthogonal satisfying exactly the same relations as the 's: :\displaystyle Since is an
orthogonal matrix In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identi ...
with square on a real
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
, is even. Let be the finite group generated by elements such that :\displaystyle where is central of order 2. The
commutator subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...
is just formed of 1 and . If is odd this coincides with the center while if is even the center has order 4 with extra elements and . If in is not in the center its
conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
is exactly and . Thus there are conjugacy classes for odd and for even. has 1-dimensional complex representations. The total number of irreducible complex representations is the number of conjugacy classes. So since is even, there are two further irreducible complex representations. Since the sum of the squares of the dimensions equals and the dimensions divide , the two irreducibles must have dimension . When 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 . The space on which the 's act can be complexified. It will have complex dimension . It breaks up into some of complex irreducible representations of , all having dimension . In particular this dimension is , so is less than or equal to 8. If , the dimension is 4, which does not divide 6. So ''N'' can only be 1, 2, 4 or 8.


Applications to Jordan algebras

Let be a Euclidean Hurwitz algebra and let be the algebra of -by- matrices over . It is a unital nonassociative algebra with an involution given by :\displaystyle The trace is defined as the sum of the diagonal elements of and the real-valued trace by . The real-valued trace satisfies: :\operatorname_ XY = \operatorname_ YX, \qquad \operatorname_ (XY)Z = \operatorname_ X(YZ). These are immediate consequences of the known identities for . In define the ''
associator In abstract algebra, the term associator is used in different ways as a measure of the associativity, non-associativity of an algebraic structure. Associators are commonly studied as triple systems. Ring theory For a non-associative ring or non ...
'' by :\displaystyle It is trilinear and vanishes identically if is associative. Since is an
alternative algebra In abstract algebra, an alternative algebra is an algebra over a field, algebra in which multiplication need not be associative, only alternativity, alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the a ...
and . Polarizing it follows that the associator is antisymmetric in its three entries. Furthermore, if , or lie in then . These facts imply that has certain commutation properties. In fact if is a matrix in with real entries on the diagonal then :\displaystyle with in . In fact if , then :\displaystyle Since the diagonal entries of are real, the off-diagonal entries of vanish. Each diagonal entry of is a sum of two associators involving only off diagonal terms of . Since the associators are invariant under
cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has ''k'' elements, it may be called a ''k ...
s, the diagonal entries of are all equal. Let be the space of self-adjoint elements in with product and inner product . is a Euclidean Jordan algebra if is associative (the real numbers, complex numbers or quaternions) and or if is nonassociative (the octonions) and . The
exceptional Exception(s), The Exception(s), or exceptional may refer to: Arts and entertainment * '' The Exception'', a 2016 British film * ''The Exception'' (2006 novel), a Danish novel (orig. ''Undtagelsen'', 2004) by Christian Jungersen * ''The Excep ...
Jordan algebra is called the Albert algebra after A.A. Albert. To check that satisfies the axioms for a Euclidean Jordan algebra, the real trace defines a
symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a biline ...
with . So it is an inner product. It satisfies the associativity property because of the properties of the real trace. The main axiom to check is the Jordan condition for the operators defined by : :\displaystyle This is easy to check when is associative, since is an associative algebra so a Jordan algebra with . When and a special argument is required, one of the shortest being due to . See: * * In fact if is in with , then :\displaystyle defines a skew-adjoint derivation of . Indeed, :\operatorname(T(X(X^2)) -T(X^2(X))) = \operatorname T(aI) = \operatorname(T)a=0, so that :(D(X),X^2) = 0. Polarizing yields: :(D(X),Y\circ Z) + (D(Y),Z\circ X) + (D(Z),X\circ Y) = 0. Setting shows that is skew-adjoint. The derivation property follows by this and the associativity property of the inner product in the identity above. With and as in the statement of the theorem, let be the group of
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
s of leaving invariant the inner product. It is a closed subgroup of so a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
. Its
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
consists of skew-adjoint derivations. showed that given in there is an automorphism in such that is a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagon ...
. (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 for any non-associative algebra . To prove the diagonalization theorem, take in . By compactness can be chosen in minimizing the sums of the squares of the norms of the off-diagonal terms of . Since 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 . Replacing by , it can be assumed that the maximum is attained at . Since the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
, acting by permuting the coordinates, lies in , if is not diagonal, it can be supposed that and its adjoint are non-zero. Let be the skew-adjoint matrix with entry , entry and 0 elsewhere and let be the derivation ad of . Let in . Then only the first two diagonal entries in differ from those of . The diagonal entries are real. The derivative of at is the coordinate of , i.e. . This derivative is non-zero if . On the other hand, the group preserves the real-valued trace. Since it can only change and , it preserves their sum. However, on the line constant, has no local maximum (only a global minimum), a contradiction. Hence must be diagonal.


See also

* Multiplicative quadratic form * Radon–Hurwitz number * Frobenius Theorem


Notes


References

* * * * * * * * (reprint of 1951 article) * * * * * * * * * * * * *


Further reading

* * * * Max Koecher &
Reinhold Remmert Reinhold Remmert (22 June 1930 – 9 March 2016) was a German mathematician. Born in Osnabrück, Lower Saxony, he studied mathematics, mathematical logic and physics in Münster. He established and developed the theory of complex-analytic space ...
(1990) "Composition Algebras. Hurwitz's Theorem — Vector-Product Algebras", chapter 10 of ''Numbers'' by
Heinz-Dieter Ebbinghaus Heinz-Dieter Ebbinghaus (born 22 February 1939 in Hemer, Province of Westphalia) is a German mathematician and logician. He received his PhD in 1967 at the University of Münster under Hans Hermes and Dieter Rödding. Ebbinghaus has written v ...
et al., Springer, *{{citation , first=T. A. , last=Springer , author-link=T. A. Springer , author2=F. D. Veldkamp , year=2000 , title=Octonions, Jordan Algebras and Exceptional Groups , publisher=
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, isbn= 978-3-540-66337-9 Composition algebras Non-associative algebras Quadratic forms Representation theory Theorems about algebras 1923 introductions Hypercomplex numbers Octonions