The ordinary Euclidean group of is the group generated from the rigid translation action of on itself and rotations about the origin.
The super Euclidean group is analogously the supergroup of translations and rotations of the supermanifold .
Its super Lie algebra should be the super Poincare Lie algebra (up to the signature of the metric).
incomplete for the moment, to be finished off tomorrow
The following description of the super Euclidean group (once it is finished, and polished) is due to Stephan Stolz and Peter Teichner.
The data needed to define the super Euclidean group is
a -dimensional inner product space
a spinor representation of
a -equivariant map
where is the Spin group (see Clifford algebra for the moment).
Here is the construction of for
remark is a multiple of
is a complex supermanifold of dimension
for this is
where the last factor is where is the spinor bundle
now define the multiplication
by sayin what it does on sets of probes by
here on the left we have the sets of sections
so we can map these as
if the data and is isomorphic we get compatible notions of structures
But if and then there is a unique such triple with non-degenerate pairing up to isomorphism.
The structure of a Euclidean supermanifold on a -dimensional supermanifold is a -structure. See there for details.
recall the Clifford algebra table:
the group structure on is that of the “translations” and “rotations”
it will be defined on generalized elements with domain by maps of sets
so here this is the super translation group.
the first map is multiplication by and then the isomorphism on the right sends
multiplication on -elements