superalgebra

and

supergeometry

# Contents

(…)

## 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 $Y\left(G\right):{\mathrm{SDiff}}^{\mathrm{op}}\to \mathrm{Set}$ 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

$\begin{array}{ccc}& & \mathrm{Grp}\\ & {}^{\left(G,\cdot \right)}↗& ↓\\ {\mathrm{SDiff}}^{\mathrm{op}}& \stackrel{Y\left(G\right)}{\to }& \mathrm{Set}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && Grp \\ & {}^{(G,\cdot)}\nearrow & \downarrow \\ SDiff^{op} &\stackrel{Y(G)}{\to}& Set } \,.

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

${\cdot }_{S}:G\left(S\right)×G\left(S\right)\to G\left(S\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccc}G\left(S\right)×G\left(S\right)& \stackrel{{\cdot }_{S}}{\to }& G\left(S\right)\\ {↑}^{G\left(f\right)×G\left(f\right)}& & {↑}^{G\left(f\right)}\\ G\left(T\right)×G\left(T\right)& \stackrel{{\cdot }_{T}}{\to }& G\left(T\right)\end{array}$\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

$Y\left(G×G\right)\to Y\left(G\right)$Y(G \times G) \to Y(G)

of representable presheaves. By the Yoneda lemma, this uniquely comes from a morphism $\cdot :G×G\to G$, 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 $G\left(A\right)$ or Lie algebra and to a morphism of Grassmann algebras $A\to B$ covariantly a morphism of groups $G\left(A\right)\to G\left(B\right)$. 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

${ℝ}^{1\mid 1}×{ℝ}^{1\mid 1}\to {ℝ}^{1\mid 1}$\mathbb{R}^{1|1} \times \mathbb{R}^{1|1} \to \mathbb{R}^{1|1}
$\left({t}_{1},{\theta }_{1}\right),\left({t}_{2},{\theta }_{2}\right)↦\left({t}_{1}+{t}_{2}+{\theta }_{1}{\theta }_{2},{\theta }_{1}+{\theta }_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$(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 $↪$ $\mathrm{Fun}\left({\mathrm{SDiff}}^{\mathrm{op}},\mathrm{Set}\right)$

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 $\mathrm{SDiff}\left(S,{ℝ}^{1\mid 1}\right)\simeq {C}^{\infty }\left(S{\right)}^{\mathrm{ev}}×{C}^{\infty }\left(X{\right)}^{\mathrm{odd}}$ given by precisely the above formula for this given $S$

${ℝ}^{1\mid 1}\left(S\right)×{ℝ}^{1\mid 1}\left(S\right)\to {ℝ}^{1\mid 1}\left(S\right)$\mathbb{R}^{1|1}(S) \times \mathbb{R}^{1|1}(S) \to \mathbb{R}^{1|1}(S)
$\left({t}_{1},{\theta }_{1}\right),\left({t}_{2},{\theta }_{2}\right)↦\left({t}_{1}+{t}_{2}+{\theta }_{1}{\theta }_{2},{\theta }_{1}+{\theta }_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$(t_1, \theta_1), (t_2, \theta_2) \mapsto (t_1 + t_2 + \theta_1 \theta_2, \theta_1 + \theta_2) \,.

where $\left({t}_{i},{\theta }_{i}\right)\in {C}^{\infty }\left(S{\right)}^{\mathrm{ev}}×{C}^{\infty }\left(S{\right)}^{\mathrm{odd}}$ etc and where the addition and product on the right takes place in the function super algebra ${C}^{\infty }\left(S\right)$.

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 $\left(1\mid 1\right)$-dimensional case of the super Euclidean group.

## References

Revised on August 30, 2013 02:49:30 by Urs Schreiber (89.204.137.78)