nLab symbol map

Idea

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

Definition

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

For $v \in V$ any vector, write

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

for the linear map given by exterior product with $v$.

Let

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

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

$$\iota_v : \wedge^\bullet V \to \wedge^\bullet V$$

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

$$[v\wedge, w\wedge ] = 0$$

$$[\iota_v, \iota_w] = 0$$

$$[\iota_v, w\wedge] = \langle v,w\rangle$$

(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

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

given by

$$(\gamma_v, \phi) \mapsto (v \wedge + \iota_v) \phi \,.$$

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

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

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

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

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

Related concepts

• The symbol map may be thought of as a special case of a symbol of a differential operator. See there for more.

References

For instance section 2.5 of

• Eckhard Meinrenken, Clifford algebras and Lie groups (pdf)