algebraic structure or
There are two ways to extend to
- Linear Algebra
Example Real 2-dimensional space.
and the distributive law. The construction of property-linear-algebra uses the property-real-algebra of
- Algebra
Eaxmple [complex-number] Complex number.
Addition is the same as . Multiplication uses or and the distributive law
one of motivation of complex-number or is the characteristic polynomial equation of harmonic-oscillator
Eaxmple [split-complex-number] Split-complex number.
also cf. complex-numbler-geometric-meaning
Addition is the same as . Multiplication uses or and the distributive law
[linear] linear structure
struct homomorphism := a mapping that preserves struct
Example linear struct hom
- map to
- map to
so map to or abbreviated as
linear struct hom is also called linear mapping
This homomorphism can also be considered similar to the distributive law of scalar multiplication. Vector addition followed by linear mapping is equivalent to linear mapping followed by vector addition.
bijection to itself + hom = isomorphism
linear isomorphism of :=
algebraic structure
Example
- complex numbers, quaternions, octonions
- Matrix algebra. But conceptually and meaningfully, it doesn't seem like a good generalization of algebra. So we need other restrictions, e.g. normed algebra, composition algebra
[normed-algebra]
The multiplication of has the property
The spatial quadratic form has the property
For algebra, we expect the property
Example
def complex conjugate
. This is spatial
by
. This is spacetime
by
null elements have no multiplicative inverse
-
give
-
give
New imaginary unit construction method
Eaxmple [quaternion]
Use a new imaginary unit in the complex number with
-
Define other imaginary units
-
Different imaginary units anticommute
-
Invert the imaginary unit conjugate or
Anti-commutation + conjugate inversion makes , and also gives
Imaginary unit associativity
Satisfies norm multiplication
Example If split complex is used, then , so both give split quaternion
-
give
-
give
Eaxmple [octonion] Using a new imaginary unit in quaternion
Define other imaginary units
Anti-commutation of different imaginary units
Anti-associativity of different imaginary units if
Imaginary unit conjugate inversion
Anti-commutation + conjugate inversion makes , and also gives
Question Anti-associativity is needed for norm multiplication
gives octonion . split octonion similarly, with signature
-
give (Question)
-
give
What is obtained from and the associative law of imaginary units is another algebra , which does not satisfy
Question Anti-combination cannot be further extended to sixteen dimensions and beyond?
[imaginary-automorphism] The new imaginary unit construction method is not coordinate-free, so we need to consider the automorphism of imaginary units with . Since it preserves multiplication, it automatically preserves distance
Example for it's symmetric
Question (ref-21, p.35) (ref-22, p.85)
- for
- for .
as automorphism of illustrates that, without additional structure, such as multiplication and , only a simple linear space structure, it is impossible to give special groups like . (Although it is said that all compact groups can have matrix representations.)
[affine] affine structure
Change the origin, translate
hom additionally keeps abbreviated as