superalgebra

and

supergeometry

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

For $v\in V$ any vector, write

$v\wedge :{\wedge }^{•}V\to {\wedge }^{•}V$v\wedge : \wedge^\bullet V \to \wedge^\bullet V

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

Let

$⟨-,-⟩:{\wedge }^{•}V\otimes {\wedge }^{•}V\to ℝ$\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

${\iota }_{v}:{\wedge }^{•}V\to {\wedge }^{•}V$\iota_v : \wedge^\bullet V \to \wedge^\bullet V

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

$\left[v\wedge ,w\wedge \right]=0$[v\wedge, w\wedge ] = 0
$\left[{\iota }_{v},{\iota }_{w}\right]=0$[\iota_v, \iota_w] = 0
$\left[{\iota }_{v},w\wedge \right]=⟨v,w⟩$[\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

$\rho :\mathrm{Cl}\left(V\right)\otimes {\wedge }^{•}V\to {\wedge }^{•}V$\rho : Cl(V)\otimes \wedge^\bullet V \to \wedge^\bullet V

given by

$\left({\gamma }_{v},\varphi \right)↦\left(v\wedge +{\iota }_{v}\right)\varphi \phantom{\rule{thinmathspace}{0ex}}.$(\gamma_v, \phi) \mapsto (v \wedge + \iota_v) \phi \,.

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

$\sigma :=\rho \left(-,1\right):\mathrm{Cl}\left(V\right)\to {\wedge }^{•}V\phantom{\rule{thinmathspace}{0ex}}.$\sigma := \rho(-,1) : Cl(V) \to \wedge^\bullet V \,.

This is an isomorphism of ${ℤ}_{2}$-graded vector space.

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

${\sigma }^{-1}:v\wedge w↦\frac{1}{2}\left({\gamma }_{v}\cdot {\gamma }_{w}-{\gamma }_{w}\cdot {\gamma }_{v}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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)