Algebraic super groups

An affine algebraic super group is the formal dual of a super Hopf algebra which is supercommutative.

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 GG in the category SDiffSDiff of supermanifold is by first sending GG with the Yoneda embedding to a presheaf on SDiffSDiff and then imposing a lift of Y(G):SDiff opSetY(G) : SDiff^{op} \to Set through the forgetful functor Grp \to Set that sends a (ordinary) group to its underlying set.

So a group object structure on GG is a diagram

Grp (G,) SDiff op Y(G) Set. \array{ && Grp \\ & {}^{(G,\cdot)}\nearrow & \downarrow \\ SDiff^{op} &\stackrel{Y(G)}{\to}& Set } \,.

This gives for each supermanifold SS an ordinary group (G(S),)(G(S), \cdot), so in particular a product operation

S:G(S)×G(S)G(S). \cdot_S : G(S) \times G(S) \to G(S) \,.

Moreover, since morphisms in GrpGrp are group homomorphisms, it follows that for every morphism f:STf : S \to T of supermanifolds we get a commuting diagram

G(S)×G(S) S G(S) G(f)×G(f) G(f) G(T)×G(T) T G(T) \array{ G(S) \times G(S) &\stackrel{\cdot_S}{\to}& G(S) \\ \uparrow^{G(f)\times G(f)} && \uparrow^{G(f)} \\ G(T) \times G(T) &\stackrel{\cdot_T}{\to}& G(T) }

Taken together this means that there is a morphism

Y(G×G)Y(G) Y(G \times G) \to Y(G)

of representable presheaves. By the Yoneda lemma, this uniquely comes from a morphism :G×GG\cdot : G \times G \to G, which is the product of the group structure on the object GG 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 AA over an ordinmary vector space an ordinary group G(A)G(A) or Lie algebra and to a morphism of Grassmann algebras ABA \to B covariantly a morphism of groups G(A)G(B)G(A) \to G(B). But the Grassmann algebra on an nn-dimensional vector space is naturally isomorphic to the function ring on the supermanifold 0|n\mathbb{R}^{0|n }. 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 1|1\mathbb{R}^{1|1} 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

1|1× 1|1 1|1 \mathbb{R}^{1|1} \times \mathbb{R}^{1|1} \to \mathbb{R}^{1|1}
(t 1,θ 1),(t 2,θ 2)(t 1+t 2+θ 1θ 2,θ 1+θ 2). (t_1, \theta_1), (t_2, \theta_2) \mapsto (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2) \,.

Recall how the notation works here: by the Yoneda embedding we have a full and faithful functor

SDiff \hookrightarrow Fun(SDiff op,Set)Fun(SDiff^{op}, Set)

and we also have the theorem, discussed at supermanifolds, that maps from some SSDiffS \in SDiff into p|q\mathbb{R}^{p|q} is given by a tuple of pp even section t it_i and qq odd sections θ j\theta_j. The above notation specifies the map of supermanifolds by displaying what map of sets of maps from some test object SS it corresponds to under the Yoneda embedding.

Now, or each SS \in SDiff there is a group structure on the hom-set SDiff(S, 1|1)C (S) ev×C (X) oddSDiff(S, \mathbb{R}^{1|1}) \simeq C^\infty(S)^{ev} \times C^\infty(X)^{odd} given by precisely the above formula for this given SS

1|1(S)× 1|1(S) 1|1(S) \mathbb{R}^{1|1}(S) \times \mathbb{R}^{1|1}(S) \to \mathbb{R}^{1|1}(S)
(t 1,θ 1),(t 2,θ 2)(t 1+t 2+θ 1θ 2,θ 1+θ 2). (t_1, \theta_1), (t_2, \theta_2) \mapsto (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2) \,.

where (t i,θ i)C (S) ev×C (S) odd(t_i, \theta_i) \in C^\infty(S)^{ev} \times C^\infty(S)^{odd} etc and where the addition and product on the right takes place in the function super algebra C (S)C^\infty(S).

Since the formula looks the same for all SS, one often just writes it without mentioning SS as above.

The super Euclidean group

The super-translaton group is the (1|1)(1|1)-dimensional case of the super Euclidean group.

General linear supergroup

general linear supergroup

Orthosymplectic supergroup

orthosymplectic supergroup


Representations, Tannaka duality and Deligne’s theorem

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.

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module


Discussion of group extensions of supergroups includes

Revised on March 31, 2015 17:02:27 by Urs Schreiber (