nLab symbol map



Super-Algebra and Super-Geometry



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.


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.


,: 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) \,.


For instance section 2.5 of

Last revised on September 2, 2022 at 17:13:12. See the history of this page for a list of all contributions to it.