Alternative body

The term alternative body is a generalization of the algebraic concept of the body in mathematics. When defining the alternative field, the commutative law and the associative law for multiplication are dispensed with . Instead, it is required that multiplication has the property of being an alternative .

Every oblique body is an alternative body, every alternative body is at the same time a left and a right quasi-body . Finite alternative bodies are always bodies. (→ See also: Moufang level ).

The alternative bodies find an important application in synthetic geometry . Ruth Moufang proved in 1933 that every Moufang plane is isomorphic to a projective coordinate plane over an alternative body . ${\displaystyle \mathbb {P} ^{2}(A)}$ ${\displaystyle A}$

Definitions

A set with two links and is an alternative field if: ${\displaystyle M}$ ${\displaystyle +}$ ${\displaystyle \cdot }$

• ${\displaystyle (M,+)}$is an Abelian group , the neutral element of which is denoted as 0;
• ${\displaystyle (M\setminus \lbrace 0\rbrace ,\cdot )}$is a quasi-group with a neutral element denoted as 1;
• The alternative applies to the link : ${\displaystyle \cdot }$

${\displaystyle (A_{1})\quad a\cdot (a\cdot b)=(a\cdot a)\cdot b}$ and

${\displaystyle (A_{2})\quad a\cdot (b\cdot b)=(a\cdot b)\cdot b}$,

• both distributive laws apply : and . ${\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c}$ ${\displaystyle (a+b)\cdot c=a\cdot c+b\cdot c}$

Core of an alternative body

→ Main article : Affine translation plane

Each alternative body is a left and a right quasi-body . Analogous to quasi-bodies, one can define its core for each alternative body : ${\displaystyle A}$

${\displaystyle S=\operatorname {Kern} (A)=\lbrace x\in A:\forall a,b\in A\quad x(ab)=(xa)b\rbrace }$.

This core is clearly determined by the definition and, with the connections from the alternative field, fulfills the axioms of an oblique field. The alternative body is a sloping body exactly when it coincides with its core. Note that the core generally does not have to be a maximum inclined body (in the sense of inclusion) in the alternative body. ${\displaystyle S}$ ${\displaystyle A}$

properties

The flexibility law still follows from the alternative

${\displaystyle (F)\!\,\quad a\cdot (b\cdot a)=(a\cdot b)\cdot a}$.

The two alternatives and and the flexibility law are “cyclical” laws: If two of these laws apply, then the third follows. ${\displaystyle (A_{1})}$ ${\displaystyle (A_{2})}$ ${\displaystyle (F)}$

In an alternative body, the Moufang identities also apply for multiplication:

${\displaystyle [a\cdot (b\cdot a)]\cdot c=a\cdot [b\cdot (a\cdot c)]}$

and

${\displaystyle (a\cdot b)\cdot (c\cdot a)=a\cdot [(b\cdot c)\cdot a]}$.

Ruth Moufang showed in 1934 that any three different elements a, b, c from an alternative field that satisfy the relation create a skew . This is a tightening of a sentence by Artin . Artin's theorem says that any two different elements create a sloping body. The oblique bodies generated in this way are subsets of the core of if and only if the generating elements lie in this core. ${\displaystyle A}$ ${\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}$ ${\displaystyle A}$

Each alternative body is both a left and a right module over each inclined body contained in its core, i.e. in particular over the core itself.

Examples

• The best-known example of a "real" alternative body, which is not a sloping body, are the (real) octonions . The core of this alternative field is the field of real numbers . In addition it contains infinitely many fields isomorphic to the complex numbers . ${\displaystyle \mathbb {O} }$ ${\displaystyle \mathbb {O} }$
• Every body, and more generally every oblique body, is an example of an alternative body.

literature

• Ruth Moufang: The introduction of the ideal elements into plane geometry with the help of the theorem of the complete four-sided . In: Mathematical Annals . Volume 105, No. 1 . Hamburg 1931, p. 759-778 , doi : 10.1007 / BF01455845 .
• Ruth Moufang: Alternative body and the sentence of the complete four-page . In: Dep. Math. Sem . tape 8 . Hamburg 1933, p. 416-430 .
• Ruth Moufang: On the structure of alternative bodies . In: Mathematical Annals . Volume 110, Number 1, 1935, pp. 416-430 .
• Max August Zorn : Theory of Alternative Rings . In: Dep. Math. Sem . Volume 8, Number 1. Hamburg 1930, p. 123-147 , doi : 10.1007 / BF02940993 .

Individual evidence

1. ↑ Wrath (1930)
2. ↑ Moufang (1933)
3. ↑ Alternative fields by Hauke ​​Klein HTML (Eng.)