Complements
Complements are unary operations in geometric algebra that perform a specific type of dualization.
Every basis element $$\mathbf u$$ has a right complement, which we denote by $$\overline{\mathbf u}$$, that satisfies the equation
- $$\mathbf u \wedge \overline{\mathbf u} = {\large\unicode{x1D7D9}}$$ .
There is also a left complement, which we denote by $$\underline{\mathbf u}$$, that satisfies the equation
- $$\underline{\mathbf u} \wedge \mathbf u = {\large\unicode{x1D7D9}}$$ .
Complements exchange full and empty dimensions, and the left and right complements can differ only by sign according to the relationship
- $$\underline{\mathbf u} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}\,\overline{\mathbf u}$$ .
This shows that the left and right complements of an element $$\mathbf u$$ are always the same if either its grade $$\operatorname{gr}(\mathbf u)$$ or its antigrade $$\operatorname{ag}(\mathbf u)$$ is even. If the number of dimensions is odd, then it is always true that one of these is even, so left and right complements are the same for all elements in an odd-dimensional algebra. As shown in the table below, applying the right or left complement twice can negate the operand in even numbers of dimensions. However, the right and left complements are inverse operations, so we always have $$\overline{\underline{\mathbf u}} = \mathbf u$$.
Taking the right or left complement twice causes the sign to change according to the formula
- $$\underline{\underline{\mathbf u}} = \overline{\overline{\mathbf u}} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}\,\mathbf u$$ .
The right and left complements under the wedge product are also the right and left complements under the antiwedge product, so we can write
- $$\mathbf u \vee \overline{\mathbf u} = \mathbf 1$$
- $$\underline{\mathbf u} \vee\mathbf u = \mathbf 1$$ .
To extend the complements to all elements of an algebra, we simply require that they are linear operations. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the right complement,
- $$\overline{(a\mathbf x + b\mathbf y)} = a\overline{\mathbf x} + b\overline{\mathbf y}$$ ,
and similarly for the left complement.
The following table lists the left and right complements for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.
Explicit Formula
In an $$n$$-dimensional algebra, suppose the volume element is given by
- $${\large\unicode{x1D7D9}} = m (\mathbf e_1 \wedge \mathbf e_2 \wedge \cdots \wedge \mathbf e_n)$$ ,
where $$m = \pm 1$$. Let $$\mathbf u$$ be a basis element with grade $$k$$. Then $$\mathbf u$$ can be expressed as
- $$\mathbf u = \mathbf e_{a_1} \wedge \mathbf e_{a_2} \wedge \cdots \wedge \mathbf e_{a_k}$$ ,
where each $$a_i$$ is a unique index satisfying $$1 \le a_i \le n$$. The right complement of $$\mathbf u$$ is given by
- $$\overline{\mathbf u} = m \operatorname{sgn}(a_1, a_2, \dots, a_k) \left(\prod_{i=1}^k (-1)^{a_i-k}\right) \left(\bigwedge_{j=1}^n \varphi(j)\right)$$ ,
where $$\operatorname{sgn}$$ is the signature of the permutation $$(a_1, a_2, \dots, a_k)$$, and $$\varphi(j)$$ is defined as
- $$\varphi(j) = \begin{cases}\mathbf e_j, & \text{if } j \notin \{a_1, a_2, \dots, a_k\}; \\ \mathbf 1, & \text{otherwise.}\end{cases}$$
In the Book
- Complements are introduced in Section 2.2.