Linear mapping of quaternion algebra

Linear automorphism of quaternion algebra is a liner over real field mapping $$f:H\rightarrow H$$ such that $$f(ab)=f(a)f(b)$$ Evidently, linear mapping $$E(x)=x$$ is linear automorphism. In quaternion algebra there are nontrivial linear automorphisms. For instance
E₁(x)=x⁰ +x²i +x³j +x¹k E₃(x)=x⁰ +x²i +x¹j -x³k where
x=x⁰ +x¹i +x²j +x³k

Similarly, the mapping

I(x)=x⁰ -x¹i -x²j -x³k is antilinear automorphism because $$I(ab)=I(b)I(a)$$

In the paper eprint arXiv:1107.1139, Linear Mappings of Quaternion Algebra, I proved following statement. For any linear over real field function $f$ there is unique expansion $$ f(x)=a_0E(x)+a_1E_1(x)+a_2E_2(x)+a_3I(x) $$

Representation of Universal Algebra

I published my new book: Representation Theory: Representation of Universal Algebra. In this book I consider morphism of representation, consept of generating set and basis of representation. This allows me to consider basis manifold of representation, active and passive transformations, concept of geometrical object in representation of universal algebra. Similar way I consider tower of representations.

My first published book

This year I published my first book Linear Mappings of Free Algebra: First Steps in Noncommutative Linear Algebra Book was published by Lambert Academic Publishing.

About Cauchy-Riemann Equation

Recently playing with complex numbers I discovered that transformation caused by matrix $$ \left( \begin{array}{cc} a_0 & -a_1 \\ a_1 & a_0 \end{array} \right) $$ is equivalent to multiplication over complex number $z=a_0+a_1i$. The Caushy-Riemann equation follows from this matrix.

Initially I assumed similar transformation for quaternion. However appropriate matrix for quaternion looks too restrictive. I assumed less restrictive condition for derivative of quaternion function $$ \frac {\partial y^0} {\partial x^0} = \frac{\partial y^1}{\partial x^1} = \frac{\partial y^2}{\partial x^2} = \frac{\partial y^3}{\partial x^3} $$ \[ \frac{\partial y^i}{\partial x^j} =- \frac{\partial y^j}{\partial x^i} \quad i\ne j \] I see that such functions like $y=ax$, $y=xa$, $y=x^2$ satisfy to this equations. The same time conjugation does not satisfy it. A lot of questions should be answered to understand if this is set of function that generalize set of complex functions in case of quaternions. You can find details in my paper eprint arXiv:0909.0855 Quaternion Rhapsody, 2010

Matrix of Mappings

I recently published new paper
eprint arXiv:1001.4852 The Matrix of Linear Mappings, 2010

In this paper, I consider change in notation of mapping. Instead of notation
y=f(x)
I use notation
y = f ° x
Let us consider ring of mappings of R-algebra A with product f ° g
This point of view allows me to consider ring R as representation in algebra A. If f is additive mapping of algebra A then I can write value of map in form f ° h. In particular I can write derivative of mapping of algebra in form
∂f(x)(h)=∂f(x) ° h
This allow me to separate derivative and increment of argument. In this case I need to use tensor notation for derivative. For instance, if f ° x = x² then
$$\partial f(x)\circ dx=(x\otimes 1+1\otimes x)\circ dx=x\ dx+dx\ x$$

Regular function of quaternion variable

Because I explore calculus over division ring I put attention on different papers related to quaternion calculus. These papers mainly discuss definition of regular function. According to definition regular function is function which satisfy to differential equation


^∂f/_∂x0+i^∂f/_∂x1+j^∂f/_∂x2+k^∂f/_∂x3=0

Recently Daniel Alayon-Solarz sent me interesting example of regular function.

f(x₀+x₁i+x₂j+x₃k)=x₀-x₁i+x₂j+x₃k

Higgs boson

The time when we see the answer on question does Higgs boson exist is close. I think that independently on answer physics will face new questions to answer.

If we will not find the Higgs boson then we need to reconsider model which predicted them.

More question will arise if we do find Higgs boson. The problem here is that Higgs boson is responsible for mass. Or to be more particular, Higgs boson is responsible for inertial mass. However we know two masses. We know inertial mass and we know gravitational mass. And these masses are equal. We yet not know the mechanism of this equality. Discovery of Higgs boson will refresh this question. If different interactions create inertial and gravitational masses, then third interaction should exist. And the third interaction is responsible for equality of masses. However if some interaction determines equality of inertial and gravitational masses, there is posibility when this equality will be broken.

How it may happens nobody knows. I can assume one of possibilities. Few years ago I explored tidal forces. I came to conclusion that these forces as fundamental as gravitational. I cannot tell which forces are primary. Do they interact one with another. Or one force is the reason of another. Feature research will show the answer on this question.