# 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.