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bivector

Idea

Where a vector specifies a direction and a magnitude , a bivector specifies a plane and a magnitude.

Definition

For $V$ a vector space, a bivector in $V$ is an element $b\in {\wedge }^{2}V$ of the second exterior power of $V$.

This is canonically identified with an element of degree 2 in the Grassmann algebra ${\wedge }^{•}V$.

Properties

Clifford algebra and rotations

If $V$ is equipped with a non-degenerate inner product then the space of bivectors is also canonically identified with a subspace of the Clifford algebra $\mathrm{Cl}(V)$.

If we write ${\gamma }_v\in \mathrm{Cl}(V)$ for the CLifford algebra element corresponding to a vector $v\in V$, then this identification is given by the map

$$v\wedge w\mapsto \frac{1}{2}({\gamma }_v\cdot {\gamma }_w-{\gamma }_w\cdot {\gamma }_v).$$v \wedge w \mapsto \frac{1}{2}\left(\gamma_{v} \cdot \gamma_w - \gamma_w \cdot \gamma_v\right) \,.

(The inverse of this map is called the symbol map.)

Under the commutator in the Clifford algebra bivectors go to bivectors and hence form a Lie algebra. This Lie algebra is the special orthogonal Lie algebra $\mathrm{𝔰𝔬}(V)$ of $V$.

References

Discussion of Clifford algebra and exterior algebra that amplifies the role of bivectors is notably in the references at Geometric Algebra .

See also

• wikipedia, bivector