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Bivectors are an elegant choice for dealing with rotations be cause they are isomorphic to skew-symmetric matrices, which are the Lie algebra of the special ort
by dllu 6y ago
Bivectors are an elegant choice for dealing with rotations be cause they are isomorphic to skew-symmetric matrices, which are the Lie algebra of the special orthogonal group SO(n). The Lie group SO(n) is also known as n-dimensional rotation matrices.
In general, using Lie groups for this sort of thing is great. Things like time-derivatives become very natural in any dimension.
- centimeter 6y agoCorrect me if I'm wrong - I'm just trying to make sure I understand the generality of your statement. 3-bivectors are commonly referred to as the "axis angle" representation, and have an obvious embedding as the lie algebra to the lie group of rotation quaternions. [x,y,z] -> 0+xi+yj+zk -exp-> rotator quaternion. Does such a thing exist at higher dimensions? I vaguely recall something about having complex numbers for 2D rotation, quaternions for 3D rotation, and octonions for (I'm guessing) 4D rotation, but I'm curious if the loss of associativity with octonions screws with this relationship somehow.
- h-cobordism 6y ago> Does such a thing exist at higher dimensions? Yes, everything in your first paragraph extends to any number of dimensions (replacing "quaternion" with "rotor"). > I vaguely recall something about having complex numbers for 2D rotation, quaternions for 3D rotation, and octonions for (I'm guessing) 4D rotation Bivectors and rotors faithfully represent rotations in any number of dimensions. The octonion product can't, because as you said, it's not associative, but rotations obviously have to compose associatively.