The Quaternions

7/29/2019 The Quaternions

http://slidepdf.com/reader/full/the-quaternions 1/4

Section 5.1:

The Quaternions is the set . Addition and multiplication are defined as follows:

o ⃗ ⃗ ⃗ ⃗

o ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗

We will make more definitions that are analogous to the complex numbers:

o ( ) ( )

o

Now some analogous facts follow:

Fact 1: Let , then ⃗

Fact 2:

is a division ring

(To prove associativity, note that (⃗ ) ⃗ ( ⃗ )

Fact 3: The function is a multiplicative norm on

(Hint: Verify that ̅)

Fact 4: There is a homeomorphism of metric spaces:

Some interesting facts:

It is possible to prove that the product is a sum of 2 squares by using thefact that . Similarly, one can show that the product

is a sum of 4 squares using the fact that the quaternion

norm is multiplicative.

Questions:

Is there an analogue for the equation in the division ring of quaternions?

More generally, is there “quaternion analysis” ?

Section 5.2:

Reminder 1: Let K be a division ring. An absolute value is a function such that:

o , - o , - o , -



Reminder 2: Let V be a right vector space over a division ring K with an absolute value . Anorm is a function ‖ ‖

o ,‖‖ - o , ‖‖- o ,‖ ‖ ‖‖ ‖‖-

In all of the next facts, K is a division ring and belongs to {R,C,H}

Fact 1: If V is a finite dimensional normed right vector space over K, then all

norms on V are equivalent. (My understanding is that the metrics induced by the norms are

induce the same topology on V)

Proof idea:

Show that any norm (‖ ‖ is equivalent to the Euclidean norm(‖ ‖. Show that ,‖‖ ‖‖- by using the basis of . Show that ,‖‖ ‖‖- by

verifying that ‖ ‖ ‖ ‖ is continuous and by using the Heine Borel property of

Euclidean spaces.

Fact 2: Let V,W be normed right vector spaces such that V is finite dimensional over K. Let

be a right K-module homomorphism. Then, ‖ ‖ ‖ ‖ is

continuous.

Proof Idea: Let ‖ ‖ denote the Euclidean metric. Using Fact 1, deduce that ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ are homeomorphisms ( denote the identities in

the category Set). Use a basis of V to show that ‖ ‖ ‖ ‖ is continuous. Thus,

is continuous.

Fact 3: Let V be a normed n-dimensional vector space over K. There are homeomorphisms:

o ()

o ()

o ()

Proof Idea: Use Fact 1.

Fact 4:

Fact 5: There is a homeomorphism

Proof Idea:

Definition 1:



Questions:

Does fact 1 hold for for infinite dimensional vector spaces over R? If the answer is yes, this

would imply that any two norms on an infinite dimensional vector space over R induce the samemetric topology.

Let V be a normed vector space over K (not necessarily finite dimensional) and W be a subspace

of V. What is a natural way of defining a norm on V/W (algebraic not topological notation) using

the norm of V. If possible, does the topology induced by the natural norm on V/W coincide with

the final topology with respect to the map that sends x to x+W ? (Due to fact 2, we

know that the final topology is finer than the topology induced by a norm defined on V/W).

Section 5.3:

Defintion 1: P(V)

Defintion 2: The fundamental map is (add a note about equivalence classes)

Fact 1: The fundamental map is open

Proof Idea: By applying the definition of the final topology, it suffices to show that” is

open” implies that “[,-] is open”. Use the properties of norm to show that:

[,-] ,

[,-]- Fact 2: Let U be a K-subspace of V. Then, P(U) is closed in P(V)

Proof Idea: Apply the definition of the final topology along with the fact that U-{0} is p-

saturated.

Fact 3: Let U,V,W be vector spaces over K such that V=U+W and . Then there is

a homeomorphism .

Proof Idea: Let ⃗ be a nonzero element of W. Verify that the function

with the action ⃗ ⃗ ⃗ is a homeomorphism. Fact 2 may be helpful.

Corollary: * +

Fact 4:



Documents

The Quaternions