nLab bivector

Idea

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

A bivector is a multivector of degree 2.

Definition

For $V$ a $k$-vector space, a bivector in $V$ is a decomposable element $b = b_1\wedge b_2 \in \wedge^2 V$ of the second exterior power of $V$. If the dimension of $V$ is $3$ or less every element in $\Lambda^2 V$ is decomposable, hence bivectors form the vector space, namely $\Lambda^2 V$ itself. The plane associated to a nonzero bivector $b_1\wedge b_2$ is $Span\{b_1,b_2\}$.

Alternatively, a bivector is a class of equivalence of pairs $(b_1,b_2)\in V\times V$ by the smallest equivalence relation for which $(b_1,b_2)\sim (b_1+\lambda b_2,b_2)$ for any $\lambda\in k$, $(b_1,b_2)\sim (b_1,\lambda b_1+b_2)$ for any $\lambda\in k$, and $(b_1,b_2)\sim(\frac{b_1}\mu,\mu b_2)$ for any $\mu\in k\backslash \{0\}$. This axiomatics is parallel (and related) to the axiomatic definition of a $2\times 2$ determinant.

Some authors (including English wikipedia) by a bivector mean any element $\Sigma_i b_{1 i}\wedge b_{2 i}\in\Lambda^2 V$, regardless the decomposability property (and in particular, regardless the dimension of $V$ which guarantees decomposability it $dim V\leq 3$), while they call the decomposable elements “simple bivectors”. This terminology may be inconvenient in analytic geometry as only the nonzero simple bivectors define a 2-plane and most standard constructions which are used in the Euclidean geometry with bivectors need the restriction, for example the notion of a vector being parallel to a bivector (vector $v$ is parallel to a (simple) bivector $b$ iff $b = v\wedge w$ for some vector $w$, or equivalently when $v$ is in the span of some pair $b_1,b_2\in V$ such that we can write $b = b_1\wedge b_2$).

A bivector is canonically identified with an element of degree 2 in the Grassmann algebra $\wedge^\bullet V$.

Properties

If $V$ is equipped with a non-degenerate inner product then the space $\Lambda^2 V$ is also canonically identified with a subspace of the Clifford algebra $Cl(V)$.

If we write $\gamma_v \in Cl(V)$ for the Clifford algebra element corresponding to a vector $v \in V$, then this identification is given by the map

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

(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 $\mathfrak{so}(V)$ 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