Division algebra

Division algebra

touches the specialties

mathematics Abstract algebra Linear Algebra

is a special case of

algebra Quasi-body (for division algebra with one)

includes as special cases

Oblique body Octaves

Division algebra is a term from the mathematical branch of abstract algebra . Roughly speaking, a division algebra is a vector space in which elements can be multiplied and divided.

Definition and example

A division algebra is not necessarily an associative algebra in which the equations and always have unique solutions for every two elements . "·" Denotes the vector multiplication in algebra. This means that algebra is free of zero divisors . ${\displaystyle D\neq \{0\}}$ ${\displaystyle a,b\in D,a\neq 0,}$ ${\displaystyle a\cdot x=b}$ ${\displaystyle y\cdot a=b}$ ${\displaystyle x,y\in D}$

Contains the division algebra an element 1, such that for all rule, so it is called a division algebra with unity. ${\displaystyle a\in D}$ ${\displaystyle a\cdot 1=1\cdot a=a}$

Example of a division algebra without a unit element with the two units and , which can be multiplied by any real numbers: ${\displaystyle e_{1}}$ ${\displaystyle e_{2}}$

${\displaystyle {\begin{matrix}e_{1}\cdot e_{1}&=&e_{1}\\e_{1}\cdot e_{2}&=&-e_{2}\\e_{2}\cdot e_{1}&=&-e_{2}\\e_{2}\cdot e_{2}&=&-e_{1}\end{matrix}}}$

Theorems about real division algebras

A finite-dimensional division algebra over the real numbers always has the dimension 1, 2, 4 or 8. This was proven in 1958 using topological methods by John Milnor and Michel Kervaire .

The four real, normalized , division algebras with one are (except for isomorphism ):

• the real numbers themselves
• the complex numbers
• the quaternions
• the octaves also have octonions or Cayley numbers.

This result is known as the Hurwitz Theorem (1898). All but the octaves satisfy the associative law of multiplication.

Every real, finite-dimensional and associative division algebra is isomorphic to real numbers, complex numbers or to quaternions; this is the theorem of Frobenius (1877).

Every real, finite-dimensional commutative division algebra has a maximum of dimension 2 as a vector space over the real numbers (theorem of Hopf, Heinz Hopf 1940). Associativity is not required.

Topological proofs of the existence of division algebras over the real numbers

Heinz Hopf showed in 1940 that the dimension of a division algebra must be a power of 2. In 1958, Michel Kervaire and John Milnor independently showed , using Raoul Bott's periodicity theorem on homotopy groups of the unitary and orthogonal groups, that the dimensions must be 1, 2, 4 or 8 (corresponding to the real numbers, the complex numbers, the quaternions and Octonions). The latter statement could not yet be proven purely algebraically. The proof was formulated by Michael Atiyah and Friedrich Hirzebruch with the help of the K-theory .

For this purpose, according to Hopf, one considers the multiplication of a division algebra of dimension n over the real numbers as a continuous mapping or restricted to elements of length 1 (divide by the norm of the elements, this is not equal to zero for elements not equal to zero because a division algebra is zero divisor-free) as Illustration . Hopf proved that such an odd mapping (that is ) only exists if n is a power of 2. To do this, he used the homology groups of projective space. There are other equivalent formulations for the existence of division algebras of dimension n: ${\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle S^{n-1}\times S^{n-1}\to S^{n-1}}$ ${\displaystyle f(-x,y)=-f(x,y)=f(x,-y)}$

• The sphere (or the projective space ) can be parallelized (i.e. for every point x there are vectors that are linearly independent of (n-1) and that depend continuously on x and are perpendicular to x). ${\displaystyle S^{n-1}}$ ${\displaystyle \mathbb {P} ^{n-1}}$ ${\displaystyle S^{n-1}}$
• there are vector space bundles E over with a Stiefel-Whitney cohomology class not equal to zero ${\displaystyle S^{n-1}}$ ${\displaystyle w_{n}(E)}$
• there is a map with an odd Hopf invariant (see Hopf link ). Frank Adams showed that such mappings only exist for n = 2,4,8. ${\displaystyle f:S^{2n-1}\to S^{n}}$

application

• Division algebras with one element are quasi-bodies (not necessarily the other way around). Hence, every example of a division algebra in synthetic geometry provides an example of an affine translation plane . ${\displaystyle D}$ ${\displaystyle D^{2}}$

See also

• Hypercomplex number

literature

• Ebbinghaus et al .: Numbers . Berlin: Springer, 1992, ISBN 3-540-55654-0
• Stefaan Caenepeel, A. Verschoren Rings, Hopf Algebras, and Brauer Groups , CRC Press, 1998, ISBN 0-82470-153-4

Individual evidence

1. ↑ z. B. Shafarevich, Grundzüge der Algebraischen Geometrie, Vieweg 1972, p. 201. The linear mapping (analogous for right multiplication) maps D to itself and is injective, the kernel then consists only of the zero. ${\displaystyle \phi (x)=ax}$
2. ^ Hopf, A Topological Contribution to Real Algebra, Comm. Math. Helvetici, Volume 13, 1940/41, pp. 223-226
3. ↑ Milnor, Some consequences of a theorem of Bott, Annals of Mathematics, Volume 68, 1958, pp. 444-449
4. ↑ Atiyah, Hirzebruch, Bott periodicity and the parallelisability of the spheres, Proc. Cambridge Phil. Soc., Vol. 57, 1961, pp. 223-226
5. ↑ The presentation of the topological proofs follows Friedrich Hirzebruch, Divisionsalgebren und Topologie (Chapter 10), in Ebbinghaus u. a. Numbers, Springer, 1983
6. ^ Adams, On the non-existence of elements of Hopf invariant one, Annals of Mathematics, Volume 72, 1960, pp. 20-104
7. ↑ A proof with K-Theory is in Atiyah, K-Theory, Benjamin 1967