nLab supergroup

Idea

A super-group is the analog in supergeometry of Lie groups in differential geometry.

Algebraic super groups

An affine algebraic super group is the formal dual of a super-commutative Hopf algebra.

Super Lie groups

Definition

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

So a group object structure on $G$ is a diagram

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

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

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

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

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

Taken together this means that there is a morphism

$$Y(G \times G) \to Y(G)$$

of representable presheaves. By the Yoneda lemma, this uniquely comes from a morphism $\cdot : G \times G \to G$, which is the product of the group structure on the object $G$ that we are after.

etc.

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 $A$ over an ordinary vector space an ordinary group $G(A)$ or Lie algebra and to a morphism of Grassmann algebras $A \to B$ covariantly a morphism of groups $G(A) \to G(B)$. But the Grassmann algebra on an $n$-dimensional vector space is naturally isomorphic to the function ring on the supermanifold $\mathbb{R}^{0|n }$. So the definition of supergroups in terms of Grassmann algebras is secretly the same as the above definition in terms of the Yoneda embedding.

Examples

The super-translation group

also called the super-Heisenberg group

The additive group structure on $\mathbb{R}^{1|1}$ 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

$$\mathbb{R}^{1|1} \times \mathbb{R}^{1|1} \to \mathbb{R}^{1|1}$$

$$(t_1, \theta_1), (t_2, \theta_2) \mapsto (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2) \,.$$

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

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

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

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

$$\mathbb{R}^{1|1}(S) \times \mathbb{R}^{1|1}(S) \to \mathbb{R}^{1|1}(S)$$

$$(t_1, \theta_1), (t_2, \theta_2) \mapsto (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2) \,.$$

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

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

The super Euclidean group

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

General linear supergroup

general linear supergroup

…

Orthosymplectic supergroup

orthosymplectic supergroup

…

Finite super-groups

There is a finite analog for super-groups that does not quite fit in the framework presented here:

Definition

A finite super-group is a tuple $(G, z \in G)$, where $G$ is a finite group and $z$ is central and squares to $1$.

The representations of a finite super-group are $\mathbb{Z}_2$-graded: An irreducible representation has odd degree if $z$ acts by negation, and even degree if it acts as the identity.

This definition is found e.g. in:

• Paul Bruillard, Cesar Galindo, Tobias Hagge, Siu-Hung Ng, Julia Yael Plavnik, Eric C. Rowell, Zhenghan Wang, Fermionic Modular Categories and the 16-fold Way (pdf)

Properties

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 algebra	category of modules
$A$	$Mod_A$
$R$-algebra	$Mod_R$-2-module
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

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	$Mod_A$
$R$-2-algebra	$Mod_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

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

monoidal 2-category	3-category of module 2-categories
$A$	$Mod_A$
$R$-3-algebra	$Mod_R$-4-module

References

Discussion via functorial geometry:

• Claudio Carmeli, Lauren Caston, Rita Fioresi, Mathematical Foundations of Supersymmetry, EMS Series of Lectures in Mathematics Volume: 15; 2011; 263 pp; ( ISBN:978-3-03719-097-5, arXiv:0710.5742)

See also

• Katsumi Yagi, Super Lie Groups, Adv. Stud. Pure Math. Progress in Differential Geometry (1993) 407 - 412 (euclid:1534359537)

• Dennis Westra, Superrings and supergroups, 2009 (pdf)

• Groeger, Super Lie groups and super Lie algebras, lecture notes 2011 (pdf)

• Veeravalli Varadarajan, section 6.1 of: Supersymmetry for mathematicians: An introduction

Discussion of group extensions of supergroups includes

• Christopher Drupieski, Strict polynomial superfunctors and universal extension classes for algebraic supergroups (arXiv:1408.5764)

Discussion as Hopf-superalgebras includes

• Nicolás Andruskiewitsch, Iván Angiono, Hiroyuki Yamane, On pointed Hopf superalgebras, Contemp. Math. vol. 544, pp. 123–140, 2011 (arXiv:1009.5148)