Derivative of function of division ring

When we learn derivative in calculus, we know that derivative at point is number. When we move to vector function we understand that derivative is linear operator. This allows use Jacobian as expression for derivative. However in this case we again have option to separate derivative and increment of argument.

When we start to work with function of division ring (quaternion valued function for instance) we expect something similar. However it does not work or makes set of differentiable functions very limitted. Derivative of function of division ring is not number, but additive map and it is easy to see that this is the Gateaux derivative. When we learn derivative of function of division ring we cannot separate derivative itself and increment of argument.

For instance derivative of function f(x)=axb has form

∂f(x)(h)=ahb

Derivative of function f(x)=x² has form

∂f(x)(h)=xh+hx

Derivative of function f(x)=x^-1 has form

∂f(x)(h)=-x^-1hx^-1

Additive map of division ring

I created this blog to share and discus ideas related to my research in geometry.

Recently one question arised. What happens when in geometry that I know to put division ring instead of field. When I started research in 2005 it was not clear how far I can go. However I discovered that linear algebra has a lot of common, at least when we work with linear space or try to solve system of linear equations.

I met first severe problems when decided to work with tensor product. I saw that non-commutativity may cause strong problems. Moving to calculus I realised that there exists new type of map between vector spaces. This map holds sum of vectors, however it does not hold product of vector over scalar. This is why I called such map additive.

When I returned back to linear algebra additive map helped me to define tensor product of vector spaces. Because we can consider division ring as vector space we can define tensor product of division rings. Tensor product of division rings is not division ring and tensor product of vector spaces is not vector space. However this module has basis over tensor product of rings.

Quaternion algebra is an example of division ring.

You can read my papers in arXiv

eprint arXiv:math.GM/0701238 Lectures on Linear Algebra over Division Ring, 2007

eprint arXiv:0812.4763 Introduction into Calculus over Division Ring, 2008