Defining Addition Operator for Algebra of Rotations in d-Dimensions

[Gerenuk]

Suppose I have objects which are rotations of d-dimensional real vectors with an additional optional scaling.
Concatenating means multiplication of these objects.

I want to define an addition operator, so that the "sum" of two rotations gives another unique rotation with scalings only.

Which other assumptions do I need to show that only certain dimensions for the vectors are possible (under some conditions for division algebras d=2,4?; i.e. complex numbers and quaternions)? I suppose rotations already have some of the required properties?!

Will the addition operation be neccessarily the one of complex numbers and quaternions?

 

[Cantab Morgan]

Rotations are transformations that preserve orientation and distance. Their matrices all have determinant one. If you add two such matrices together, the resulting matrix will not have determinant one. So the sum of two rotations is not a rotation.

Put another way, rotations form a group under multiplication, but not under addition. You can use addition to "break out" of the set.

Sorry if I didn't quite understand your question, I'm just offering one perspective.

 

[Gerenuk]

Thanks for your suggestion. I actually want to *define* addition so that I don't break out of this set. Is that possible?

I notice that I actually also want to include scaling - I will edit my quesiton. Then for example complex numbers and quaternions would be solution. The question is if other dimensions are also possible to solve this problem.

 

[Cantab Morgan]

Ahhhh. Okay, cool. If these transformations are linear, then you can talk about the set of linear transformations, which can be matrices with real coefficients. These include rotations, shears, and so forth. You can add them without leaving the set of matrices.

I think I'm starting to see what you mean. Rotations in the plane are like complex numbers. Certain transformations in R^4 are like quaternions. But are there number-like objects for arbitrary dimension?

By the way, I asked my professor once if there were any such things as "hexadeconians" or some such, and he said "no."

However the set of complex numbers is actually isomorphic to the set of conformal matrices in the plane (if I recall correctly). And similarly, the set of quaternions is isomorphic to a certain group of 4-D matrices.

But actually, more general matrices are a lot like numbers in that you can add and multiply them. They do not in general commute, but neither do quaternions. So perhaps the objects you seek are really the set of matrices? (But without the additional structure that are afforded to complex numbers or quaternions.)

 

[Gerenuk]

Wikipedia says there are division algebras with some additional contraints for dimensions 1,2,4,8 only.

The important properties of the object I'm searching for is that they behave like rotations combined with scaling only. So no shearing and so.

 

[dx]

I think what you're looking for are Clifford algebras.

 

[Gerenuk]

Are clifford algebras isomorphic to the d-dimensional rotation group plus a 1d real variable (i.e. for scaling)?

 

[vkroom]

in two dimension the clifford algebra is same as su(2) algebra of pauli matrices. quotiented with Z_2]/tex] it is isomorphic to so(3) algebra. therefore the clifford algebra fails to meet ur proposition