The Main Involutions

Let \(C\) be a Clifford algebra of a quadratic space \(V\) and suppose that \(e_1\), \(e_2\), …, \(e_n\) is a basis for \(V\). Let \(C_+\) be the subspace of linear combinations of products of an even number of basis elements and let \(C_-\) be the subspace of linear combinations of products of an odd number of basis elements. Then \(C_+\) is a subalgebra and \(C\) is the direct sum of \(C_+\) and \(C_-\). The main involution of \(C\) is the automorphism \(J\) such that \(J(u) = u\) if \(u\in C_+\) and \(J(u) = -u\) if \(u\in C_-\).

MainInvolution(C): AlgClff -> Map

The main involution of the Clifford algebra C.

The mapping \(C \to C\) which reverses the multiplication is an antiautomorphism whose square is the identity; it is called the main antiautomorphism of \(C\).

MainAntiautomorphism(C): AlgClff -> Map

The main antiautomorphism of \(C\). The first time this function is invoked it sets the attribute antiAutMat of \(C\) to the matrix defining this map.