nLab
bivector

Context

Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Super-Algebra and Super-Geometry

Contents

Idea

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

Definition

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

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

Properties

Clifford algebra and rotations

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

If we write γ vCl(V)\gamma_v \in Cl(V) for the CLifford algebra element corresponding to a vector vVv \in V, then this identification is given by the map

vw12(γ vγ wγ wγ 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 𝔰𝔬(V)\mathfrak{so}(V) of VV.

References

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

See also

Revised on August 30, 2011 01:42:31 by Urs Schreiber (131.211.238.184)