Cayley algebra (octonions or octaves)

Lecture

The Käli algebra is a system of hypercomplex numbers, an 8-dimensional algebra over the field of real numbers. Usually denoted by Cayley algebra (octonions or octaves) , because its elements ( Cayley numbers ) are sometimes called octonions or octaves .

Cayley number is a linear combination of elements. Cayley algebra (octonions or octaves) . Each octave x can be written in the form

with real coefficients Cayley algebra (octonions or octaves) . Octonions are used in physics: for example, in STR and string theory ^[1] . Octave multiplication table:

one	i ( e1 )	j ( e2 )	k ( e3 )	l ( e4 )	il ( e5 )	jl ( e6 )	kl ( e7 )
i ( e1 )	−1	k	- j	il	- l	- kl	jl
j ( e2 )	- k	−1	i	jl	kl	- l	- il
k ( e3 )	j	- i	−1	kl	- jl	il	- l
l ( e4 )	- il	- jl	- kl	−1	i	j	k
il ( e5 )	l	- kl	jl	- i	−1	- k	j
jl ( e6 )	kl	l	- il	- j	k	−1	- i
kl ( e7 )	- jl	il	l	- k	- j	i	−1

Fano plane for memorizing the multiplication table

Table (Cayley) octonion multiplication ^[2]

e ₀	e ₁	e ₂	e ₃	e ₄	e ₅	e ₆	e ₇
e ₁	-one	e ₃	−e ₂	e ₅	−e ₄	−e ₇	e ₆
e ₂	−e ₃	-one	e ₁	e ₆	e ₇	−e ₄	−e ₅
e ₃	e ₂	−e ₁	-one	e ₇	−e ₆	e ₅	−e ₄
e ₄	−e ₅	−e ₆	−e ₇	-one	e ₁	e ₂	e ₃
e ₅	e ₄	−e ₇	e ₆	−e ₁	-one	−e ₃	e ₂
e ₆	e ₇	e ₄	−e ₅	−e ₂	e ₃	-one	−e ₁
e ₇	−e ₆	e ₅	e ₄	−e ₃	−e ₂	e ₁	-one

Often numbers can be replaced by letter designation:

Number	one	2	3	four	five	6	7
Letters	i	j	k	l	il	jl	kl
Replacement	i	j	k	l	m	n	o

Content

1 Properties
2 Conjugation and rate
3 History
4 References

Properties

By the Frobenius theorem, Cayley algebra is the only 8-dimensional real alternative algebra without zero divisors.
Cayley algebra is a unique algebra and with an alternative, but non-associative and non-commutative unit.