∞-Lie theory

superalgebra

and

supergeometry

# Contents

## 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 $\mathrm{Cl}\left(V\right)$.

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

$v\wedge w↦\frac{1}{2}\left({\gamma }_{v}\cdot {\gamma }_{w}-{\gamma }_{w}\cdot {\gamma }_{v}\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $\mathrm{𝔰𝔬}\left(V\right)$ of $V$.

## References

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