Contents

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 $\mathrm{Cl}(V)$ for its Clifford algebra and ${\wedge }^{•}V$ for its Grassmann algebra.

For $v\in V$ any vector, write

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

Let

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

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

(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

given by

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

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

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)