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 ordinary 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.
also called the super-Heisenberg group
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.
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-translation group is the -dimensional case of the super Euclidean group.
There is a finite analog for super-groups that does not quite fit in the framework presented here:
A finite super-group is a tuple , where is a finite group and is central and squares to .
The representations of a finite super-group are -graded: An irreducible representation has odd degree if acts by negation, and even degree if it acts as the identity.
This definition is found e.g. in:
Deligne's theorem on tensor categories (see there for details) says that every suitably well-behave linear tensor category is the category of representations of an algebraic supergroup. In particular the Hopf algebra of functions on an affine algebraic supergroup is a triangular Hopf algebra.
|monoid/associative algebra||category of modules|
|sesquialgebra||2-ring = monoidal presentable category with colimit-preserving tensor product|
|bialgebra||strict 2-ring: monoidal category with fiber functor|
|Hopf algebra||rigid monoidal category with fiber functor|
|hopfish algebra (correct version)||rigid monoidal category (without fiber functor)|
|weak Hopf algebra||fusion category with generalized fiber functor|
|quasitriangular bialgebra||braided monoidal category with fiber functor|
|triangular bialgebra||symmetric monoidal category with fiber functor|
|quasitriangular Hopf algebra (quantum group)||rigid braided monoidal category with fiber functor|
|triangular Hopf algebra||rigid symmetric monoidal category with fiber functor|
|supercommutative Hopf algebra (supergroup)||rigid symmetric monoidal category with fiber functor and Schur smallness|
|form Drinfeld double||form Drinfeld center|
|trialgebra||Hopf monoidal category|
|monoidal category||2-category of module categories|
|Hopf monoidal category||monoidal 2-category (with some duality and strictness structure)|
|monoidal 2-category||3-category of module 2-categories|
Groeger, Super Lie groups and super Lie algebras, lecture notes 2011 (pdf)
Discussion of group extensions of supergroups includes
Discussion as Hopf-superalgebras includes