Bare super groups
Super Lie groups
A super Lie group is a group object in the category SDiff of supermanifolds, that is a super Lie group.
In terms of generalized group elements
One useful way to characterize group objects in the category of supermanifold is by first sending with the Yoneda embedding to a presheaf on and then imposing a lift of through the forgetful functor Grp Set that sends a (ordinary) group to its underlying set.
So a group object structure on is a diagram
This gives for each supermanifold an ordinary group , so in particular a product operation
Moreover, since morphisms in are group homomorphisms, it follows that for every morphism of supermanifolds we get a commuting diagram
Taken together this means that there is a morphism
of representable presheaves. By the Yoneda lemma, this uniquely comes from a morphism , which is the product of the group structure on the object that we are after.
This way of thinking about supergroups is often explicit in some parts of the literature on supergeometry: some authors define a supergroup or super Lie algebra as a rule that assigns to every Grassmann algebra over an ordinmary vector space an ordinary group or Lie algebra and to a morphism of Grassmann algebras covariantly a morphism of groups . But the Grassmann algebra on an -dimensional vector space is naturally isomorphic to the function ring on the supermanifold . So the definition of supergroups in terms of Grassmann algebras is secretly the same as the above definition in terms of the Yoneda embedding.
The super-translation group
also called the super-Heisenberg group
The additive group structure on is given on generalized elements in (i.e. in the logic internal to) the topos of sheaves on the category SCartSp? of cartesian superspaces by
Recall how the notation works here: by the Yoneda embedding we have a full and faithful functor
and we also have the theorem, discussed at supermanifolds, that maps from some into is given by a tuple of even section and odd sections . The above notation specifies the map of supermanifolds by displaying what map of sets of maps from some test object it corresponds to under the Yoneda embedding.
Now, or each SDiff there is a group structure on the hom-set given by precisely the above formula for this given
where etc and where the addition and product on the right takes place in the function super algebra .
Since the formula looks the same for all , one often just writes it without mentioning as above.
The super Euclidean group
The super-translaton group is the -dimensional case of the super Euclidean group.