Rotor (mathematics)
A rotor is an object in the geometric algebra of a vector space that represents a rotation about the origin. The term originated with William Kingdon Clifford, in showing that the quaternion algebra is just a special case of Hermann Grassmann's "theory of extension". Hestenes defined a rotor to be any element of a geometric algebra that can be written as the product of an even number of unit vectors and satisfies, where is the "reverse" of —that is, the product of the same vectors, but in reverse order.
Definition
In mathematics, a rotor in the geometric algebra of a vector space V is the same thing as an element of the spin group Spin. We define this group below.Let V be a vector space equipped with a positive definite quadratic form q, and let Cl be the geometric algebra associated to V. The algebra Cl is the quotient of the tensor algebra of V by the relations for all. The Z-grading on the tensor algebra of V descends to a Z/2Z-grading on Cl, which we denote by Here, Cleven is generated by even-degree blades and Clodd is generated by odd-degree blades.
There is a unique antiautomorphism of Cl which restricts to the identity on V: this is called the transpose, and the transpose of any multivector a is denoted by. On a blade, it simply reverses the order of the factors. The spin group Spin is defined to be the subgroup of Cleven consisting of multivectors R such that That is, it consists of multivectors that can be written as a product of an even number of unit vectors.
Action as rotation on the vector space
Reflections along a vector in geometric algebra may be represented as sandwiching a multivector M between a non-null vector v perpendicular to the hyperplane of reflection and that vector's inverse v−1:and are of even grade. Under a rotation generated by the rotor R, a general multivector M will transform double-sidedly as
This action gives a surjective homomorphism presenting Spin as a double cover of SO.
Restricted alternative formulation
For a Euclidean space, it may be convenient to consider an alternative formulation, and some authors define the operation of reflection as the sandwiching of a unit multivector:forming rotors that are automatically normalised:
The derived rotor action is then expressed as a sandwich product with the reverse:
For a reflection for which the associated vector squares to a negative scalar, as may be the case with a pseudo-Euclidean space, such a vector can only be normalized up to the sign of its square, and additional bookkeeping of the sign of the application the rotor becomes necessary. The formulation in terms of the sandwich product with the inverse as above suffers no such shortcoming.