symbol map

**superalgebra** and (synthetic ) **supergeometry**

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 $V$.

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

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

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

For instance section 2.5 of

- Eckhard Meinrenken,
*Clifford algebras and Lie groups*(pdf)

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