nLab
supergroup

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 $\mathrm{SDiff}$ of supermanifold is by first sending $G$ with the Yoneda embedding to a presheaf on $\mathrm{SDiff}$ and then imposing a lift of $Y(G):{\mathrm{SDiff}}^{\mathrm{op}}\to \mathrm{Set}$ 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

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

$${\cdot }_S:G(S)\times G(S)\to G(S).$$\cdot_S : G(S) \times G(S) \to G(S) \,.

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

$$\begin{array}{ccc}G(S)\times G(S)& \stackrel{{\cdot }_S}{\to }& G(S)\\ {\uparrow }^{G(f)\times G(f)}& & {\uparrow }^{G(f)}\\ G(T)\times G(T)& \stackrel{{\cdot }_T}{\to }& G(T)\end{array}$$\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

$$Y(G\times G)\to Y(G)$$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 ordinmary 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 ${ℝ}^{0\mid n}$. 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 ${ℝ}^{1\mid 1}$ 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

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

$$(t_1,{\theta }_1),(t_2,{\theta }_2)\mapsto (t_1+t_2+{\theta }_1{\theta }_2,{\theta }_1+{\theta }_2).$$(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 $\hookrightarrow$ $\mathrm{Fun}({\mathrm{SDiff}}^{\mathrm{op}},\mathrm{Set})$

and we also have the theorem, discussed at supermanifolds, that maps from some $S\in \mathrm{SDiff}$ into ${ℝ}^{p\mid q}$ 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, or each $S\in$ SDiff there is a group structure on the hom-set $\mathrm{SDiff}(S,{ℝ}^{1\mid 1})\simeq C^{\mathrm{\infty }}(S{)}^{\mathrm{ev}}\times C^{\mathrm{\infty }}(X{)}^{\mathrm{odd}}$ given by precisely the above formula for this given $S$

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

$$(t_1,{\theta }_1),(t_2,{\theta }_2)\mapsto (t_1+t_2+{\theta }_1{\theta }_2,{\theta }_1+{\theta }_2).$$(t_1, \theta_1), (t_2, \theta_2) \mapsto (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2) \,.

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

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-translaton group is the $(1\mid 1)$-dimensional case of the super Euclidean group.

$\mathrm{GL}(n\mid N)$

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$\mathrm{OSp}(2p\mid N)$

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