Super-Algebra and Super-Geometry
For an inner product space, the symbol map constitutes an isomorphism of super vector spaces between the Clifford algebra of and the exterior algebra on .
Let be an inner product space. Write for its Clifford algebra and for its Grassmann algebra.
For any vector, write
for the linear map given by exterior product with .
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 . 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
The symbol map is the restriction of this action to the identity element :
This is an isomorphism of -graded vector space.
The inverse maps is on even-graded elements given by sending bivectors to their Clifford incarnation
For instance section 2.5 of
Revised on November 7, 2012 19:34:56
by Urs Schreiber