nLab
symbol map

Contents

Idea

For VV an inner product space, the symbol map constitutes an isomorphism of super vector spaces between the Clifford algebra of VV and the exterior algebra on VV.

Definition

Let VV be an inner product space. Write Cl(V)Cl(V) for its Clifford algebra and V\wedge^\bullet V for its Grassmann algebra.

For vVv \in V any vector, write

v: V V v\wedge : \wedge^\bullet V \to \wedge^\bullet V

for the linear map given by exterior product with vv.

Let

,: V V \langle -,-\rangle : \wedge^\bullet V \otimes \wedge^\bullet V \to \mathbb{R}

be the Hodge inner product on the exterior algebra induced from the inner product. With respect to this inner product the above multiplication operator has an adjoint operator?

ι v: V V \iota_v : \wedge^\bullet V \to \wedge^\bullet V

called contraction with VV. These operators satisfy the canonical anticommutation relations?

[v,w]=0 [v\wedge, w\wedge ] = 0
[ι v,ι w]=0 [\iota_v, \iota_w] = 0
[ι v,w]=v,w [\iota_v, w\wedge] = \langle v,w\rangle

(where all these are supercommutators?, hence in fact anticommutators? in the present case).

There is a canonical representation of the Clifford algebra on the exterior algebra induced by this construction

ρ:Cl(V) V V \rho : Cl(V)\otimes \wedge^\bullet V \to \wedge^\bullet V

given by

(γ v,ϕ)(v+ι v)ϕ. (\gamma_v, \phi) \mapsto (v \wedge + \iota_v) \phi \,.

The symbol map is the restriction of this action to the identity element 1 V1 \in \wedge^\bullet V:

σ:=ρ(,1):Cl(V) V. \sigma := \rho(-,1) : Cl(V) \to \wedge^\bullet V \,.

This is an isomorphism of 2\mathbb{Z}_2-graded vector space.

The inverse maps is on even-graded elements given by sending bivectors to their Clifford incarnation

σ 1:vw12(γ vγ wγ wγ v). \sigma^{-1} : v \wedge w \mapsto \frac{1}{2}\left(\gamma_v \cdot \gamma_w - \gamma_w \cdot \gamma_v\right) \,.

References

For instance section 2.5 of

Revised on November 7, 2012 19:34:56 by Urs Schreiber (82.169.65.155)