Английская Википедия:Frobenius theorem (real division algebras)
Шаблон:Short description Шаблон:For In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following:
- Шаблон:Math (the real numbers)
- Шаблон:Math (the complex numbers)
- Шаблон:Math (the quaternions).
These algebras have real dimension Шаблон:Math, and Шаблон:Math, respectively. Of these three algebras, Шаблон:Math and Шаблон:Math are commutative, but Шаблон:Math is not.
Proof
The main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem of algebra.
Introducing some notation
- Let Шаблон:Math be the division algebra in question.
- Let Шаблон:Math be the dimension of Шаблон:Math.
- We identify the real multiples of Шаблон:Math with Шаблон:Math.
- When we write Шаблон:Math for an element Шаблон:Mvar of Шаблон:Mvar, we imply that Шаблон:Mvar is contained in Шаблон:Math.
- We can consider Шаблон:Mvar as a finite-dimensional Шаблон:Math-vector space. Any element Шаблон:Mvar of Шаблон:Mvar defines an endomorphism of Шаблон:Mvar by left-multiplication, we identify Шаблон:Mvar with that endomorphism. Therefore, we can speak about the trace of Шаблон:Mvar, and its characteristic- and minimal polynomials.
- For any Шаблон:Mvar in Шаблон:Math define the following real quadratic polynomial:
- <math>Q(z; x) = x^2 - 2\operatorname{Re}(z)x + |z|^2 = (x-z)(x-\overline{z}) \in \mathbf{R}[x].</math>
- Note that if Шаблон:Math then Шаблон:Math is irreducible over Шаблон:Math.
The claim
The key to the argument is the following
- Claim. The set Шаблон:Mvar of all elements Шаблон:Mvar of Шаблон:Mvar such that Шаблон:Math is a vector subspace of Шаблон:Mvar of dimension Шаблон:Math. Moreover Шаблон:Math as Шаблон:Math-vector spaces, which implies that Шаблон:Mvar generates Шаблон:Mvar as an algebra.
Proof of Claim: Pick Шаблон:Mvar in Шаблон:Mvar with characteristic polynomial Шаблон:Math. By the fundamental theorem of algebra, we can write
- <math>p(x) = (x-t_1)\cdots(x-t_r) (x-z_1)(x - \overline{z_1}) \cdots (x-z_s)(x - \overline{z_s}), \qquad t_i \in \mathbf{R}, \quad z_j \in \mathbf{C} \setminus \mathbf{R}.</math>
We can rewrite Шаблон:Math in terms of the polynomials Шаблон:Math:
- <math>p(x) = (x-t_1)\cdots(x-t_r) Q(z_1; x) \cdots Q(z_s; x).</math>
Since Шаблон:Math, the polynomials Шаблон:Math are all irreducible over Шаблон:Math. By the Cayley–Hamilton theorem, Шаблон:Math and because Шаблон:Mvar is a division algebra, it follows that either Шаблон:Math for some Шаблон:Mvar or that Шаблон:Math for some Шаблон:Mvar. The first case implies that Шаблон:Mvar is real. In the second case, it follows that Шаблон:Math is the minimal polynomial of Шаблон:Mvar. Because Шаблон:Math has the same complex roots as the minimal polynomial and because it is real it follows that
- <math>p(x) = Q(z_j; x)^k = \left(x^2 - 2\operatorname{Re}(z_j) x + |z_j|^2 \right)^k</math>
Since Шаблон:Math is the characteristic polynomial of Шаблон:Mvar the coefficient of Шаблон:Math in Шаблон:Math is Шаблон:Math up to a sign. Therefore, we read from the above equation we have: Шаблон:Math if and only if Шаблон:Math, in other words Шаблон:Math if and only if Шаблон:Math.
So Шаблон:Mvar is the subset of all Шаблон:Mvar with Шаблон:Math. In particular, it is a vector subspace. The rank–nullity theorem then implies that Шаблон:Mvar has dimension Шаблон:Math since it is the kernel of <math>\operatorname{tr} : D \to \mathbf{R}</math>. Since Шаблон:Math and Шаблон:Mvar are disjoint (i.e. they satisfy <math>\mathbf R \cap V = \{0\}</math>), and their dimensions sum to Шаблон:Mvar, we have that Шаблон:Math.
The finish
For Шаблон:Math in Шаблон:Mvar define Шаблон:Math. Because of the identity Шаблон:Math, it follows that Шаблон:Math is real. Furthermore, since Шаблон:Math, we have: Шаблон:Math for Шаблон:Math. Thus Шаблон:Mvar is a positive-definite symmetric bilinear form, in other words, an inner product on Шаблон:Mvar.
Let Шаблон:Mvar be a subspace of Шаблон:Mvar that generates Шаблон:Mvar as an algebra and which is minimal with respect to this property. Let Шаблон:Math be an orthonormal basis of Шаблон:Mvar with respect to Шаблон:Math. Then orthonormality implies that:
- <math>e_i^2 =-1, \quad e_i e_j = - e_j e_i.</math>
If Шаблон:Math, then Шаблон:Mvar is isomorphic to Шаблон:Math.
If Шаблон:Math, then Шаблон:Mvar is generated by Шаблон:Math and Шаблон:Math subject to the relation Шаблон:Math. Hence it is isomorphic to Шаблон:Math.
If Шаблон:Math, it has been shown above that Шаблон:Mvar is generated by Шаблон:Math subject to the relations
- <math>e_1^2 = e_2^2 =-1, \quad e_1 e_2 = - e_2 e_1, \quad (e_1 e_2)(e_1 e_2) =-1.</math>
These are precisely the relations for Шаблон:Math.
If Шаблон:Math, then Шаблон:Mvar cannot be a division algebra. Assume that Шаблон:Math. Let Шаблон:Math. It is easy to see that Шаблон:Math (this only works if Шаблон:Math). If Шаблон:Mvar were a division algebra, Шаблон:Math implies Шаблон:Math, which in turn means: Шаблон:Math and so Шаблон:Math generate Шаблон:Mvar. This contradicts the minimality of Шаблон:Mvar.
- The fact that Шаблон:Mvar is generated by Шаблон:Math subject to the above relations means that Шаблон:Mvar is the Clifford algebra of Шаблон:Math. The last step shows that the only real Clifford algebras which are division algebras are Шаблон:Math and Шаблон:Math.
- As a consequence, the only commutative division algebras are Шаблон:Math and Шаблон:Math. Also note that Шаблон:Math is not a Шаблон:Math-algebra. If it were, then the center of Шаблон:Math has to contain Шаблон:Math, but the center of Шаблон:Math is Шаблон:Math. Therefore, the only finite-dimensional division algebra over Шаблон:Math is Шаблон:Math itself.
- This theorem is closely related to Hurwitz's theorem, which states that the only real normed division algebras are Шаблон:Math, and the (non-associative) algebra Шаблон:Math.
- Pontryagin variant. If Шаблон:Mvar is a connected, locally compact division ring, then Шаблон:Math, or Шаблон:Math.
References
- Ray E. Artz (2009) Scalar Algebras and Quaternions, Theorem 7.1 "Frobenius Classification", page 26.
- Ferdinand Georg Frobenius (1878) "Über lineare Substitutionen und bilineare Formen", Journal für die reine und angewandte Mathematik 84:1–63 (Crelle's Journal). Reprinted in Gesammelte Abhandlungen Band I, pp. 343–405.
- Yuri Bahturin (1993) Basic Structures of Modern Algebra, Kluwer Acad. Pub. pp. 30–2 Шаблон:ISBN .
- Leonard Dickson (1914) Linear Algebras, Cambridge University Press. See §11 "Algebra of real quaternions; its unique place among algebras", pages 10 to 12.
- R.S. Palais (1968) "The Classification of Real Division Algebras" American Mathematical Monthly 75:366–8.
- Lev Semenovich Pontryagin, Topological Groups, page 159, 1966.