nLab supergroup

Idea

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

Definition

Algebraic super groups

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

Super Lie groups

A super Lie group is a group object in the category SDiff of supermanifolds, that is a super Lie group.

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.

Super-Minkowski Lie group

We spell out the super-translation super Lie group-structure on the supermanifold $\mathbb{R}^{1,d\vert\mathbf{N}}$ underlying super Minkowski spacetime, hence equivalently of the quotient super Lie group of the super Poincaré group (the “supersymmetry” group) by its Lorentzian spin-subgroup:

(1)$$\mathbb{R}^{1,d\vert\mathbf{N}} \;\simeq\; Iso\big( \mathbb{R}^{1,d\vert\mathbf{N}} \big) \big/ Spin(1,d) \,.$$

Here

• $d \in \mathbb{N}$ is the spatial dimension (a natural number),

• $\mathbf{N} \in Rep_{\mathbb{R}}\big(Spin(1,d)\big)$ is a real spin representation equipped with a linear map

which is symmetric and $Spin(1,d)$-equivariant.

First, the super-Minkowski super Lie algebra structure on the super vector space

$$\mathbb{R}^{1,d\vert\mathbf{N}} \;\coloneqq\; \mathbb{R}^{1+d} \times \mathbb{R}^N_{odd}$$

is defined, dually, by the Chevalley-Eilenberg dgc-superalgebra with generators of $\mathbb{Z} \times \mathbb{Z}/2$ bidegree

generator	bidegree
$e^a$	$(1,evn)$
$\psi^\alpha$	$(1,odd)$

for $a \in \{0,1, \cdots, d\}$ indexing a linear basis of $\mathbb{R}^D$ and $\alpha \in \{1,\cdots, N\}$ indexing a linear basis of $\mathbf{N}$ by the differential equations

(2)$$\begin{array}{ccl} \mathrm{d}\, e^a &\coloneqq& \big( \overline{\psi} \,\Gamma^a\, \psi \big) \\ \mathrm{d}\, \psi &=& 0 \end{array}$$

The first differential is the linear dual of the archetypical super Lie bracket in the supersymmetry super Lie algebra which takes two odd elements to a spatial translation. The second differential is the linear dual of the fact that in the absence of rotational generators, no Lie bracket in the supersymmetry alegbra results in a non-vanishing odd element.

Next we regard $\mathbb{R}^{1,10\vert\mathbf{N}}$ not just as a super vector space but as a Cartesian supermanifold. As such it has canonical coordinate functions

generator	bidegree
$x^a$	$(0,evn)$
$\theta^\alpha$	$(0,odd)$

On this supermanifold, consider the super coframe field

$$(e,\psi) \;\colon\; T\mathbb{R}^{1,d\vert\mathbf{N}} \xrightarrow{\;} \mathbb{R}^{1,10\vert\mathbf{N}}$$

(where on the left we have the tangent bundle and on the right its typical fiber super vector space) given by

(3)$$\begin{array}{ccl} e^a &\coloneqq& \mathrm{d}x^a + \big(\overline{\theta} \,\Gamma^a \mathrm{d}\theta\big) \\ \psi &\coloneqq& \mathrm{d}\theta \end{array}$$

It is clear that this is a coframe field in that for all $x \in \mathbb{R}^{1,d\vert\mathbf{N}}$ it restricts to an isomorphism

$$T_{x}\mathbb{R}^{1,d\vert\mathbf{N}} \xrightarrow{\;\sim\;} \mathbb{R}^{1,d\vert\mathbf{N}}$$

and the peculiar second summand in the first line is chosen such that its de Rham differential has the same form as the differential in the Chevalley-Eilenberg algebra (2).

(Incidentally, a frame field linear dual to the coframe field (3) is

$$\begin{array}{ccl} D_a &\coloneqq& \partial_{x^a} \\ D_\alpha &\coloneqq& \partial_{\theta^\alpha} + \overline{\theta}\Gamma^a \partial_{v^a} \end{array} \;\;\;\;\;\;\; \text{such that} \;\;\;\; \begin{array}{ll} e^a(D_b) \,=\, \delta^a_b \,, & e^a(D_\alpha) \,=\, 0 \\ \psi^\alpha(D_a) \,=\, 0 \,, & \psi^\alpha(D_\beta) \,=\, \delta^\alpha_\beta \end{array}$$

which are the operators often stated right away in introductory texts on supersymmetry.)

This fact, that the Maurer-Cartan equations of a coframe field (3) coincide with the defining equations (2) of the Chevalley-Eilenberg algebra of a Lie algebra of course characterizes the left invariant 1-forms on a Lie group, and hence what remains to be done now is to construct a super Lie group-structure on the supermanifold $\mathbb{R}^{1,d\vert\mathbf{N}}$ with respect to which the coframe (3) is left invariant 1-form.

Recalling (from here) that a morphism of supermanifolds is dually given by a reverse algebra homomorphism between their function algebras, which in the present case are freely generated by the above coordinate functions, we denote the canonical coordinates on the Cartesian product $\mathbb{R}^{1,d\vert\mathbf{N}} \times \mathbb{R}^{1,d\vert\mathbf{N}}$ by $(x^a_{'}, \theta^\alpha_{'})$ for the first factor and $(x^a, \theta^\alpha)$ for the second, and declare a group product operation as follows:

(4)

(cf. CAIP99, (2.1) & (2.6))

Here the choice of notation for the coordinates on the left is adapted to thinking of this group operation equivalently as the left multiplication action of the group on itself, which makes the following computation nicely transparent.

Indeed, the induced left action of the super-group on its odd tangent bundle

is dually given by

and left-invariance of the coframe (2) means that it is fixed by this operation (so the differential $\mathrm{d}$ in the following computation is just that of the second factor, hence acting on unprimed coordinates only):

$$\begin{array}{ccl} \mathrm{act}^\ast e^a &=& \mathrm{act}^\ast \Big( \mathrm{d}x^a + \big(\overline{\theta} \,\Gamma^a\, \mathrm{d}\theta\big) \Big) \\ &=& \mathrm{d}\,\mathrm{act}^\ast x^a + \big(\overline{\mathrm{act}^\ast\theta} \,\Gamma^a\, \mathrm{d}\,\mathrm{act}^\ast\theta\big) \\ &=& \mathrm{d} \Big( x^a_{'} + x^a - \big( \overline{\theta}_{'} \,\Gamma^a\, \theta \big) \Big) + \big( (\overline{\theta}_{'} + \overline{\theta}) \,\Gamma^a\, \mathrm{d}( \theta_{'} + \theta ) \big) \\ &=& \mathrm{d}x^a - \big( \overline{\theta}_{'} \,\Gamma^a\, \mathrm{d}\theta \big) \,+\, \big( \overline{\theta}_{'} \,\Gamma^a\, \mathrm{d}\theta \big) \,+\, \big( \overline{\theta} \,\Gamma^a\, \mathrm{d}\theta \big) \\ &=& \mathrm{d}x^a \,+\, \big( \overline{\theta} \,\Gamma^a\, \mathrm{d}\theta \big) \\ &=& e^a \mathrlap{\,,} \end{array} \;\;\;\;\; \begin{array}{ccl} \mathrm{prd}^\ast \psi &=& \mathrm{prd}^\ast \mathrm{d}\theta \\ &=& \mathrm{d} \, \mathrm{prd}^\ast \theta \\ &=& \mathrm{d}\big( \theta_{'} + \theta \big) \\ &=& \mathrm{d}\theta \\ &=& \psi \mathrlap{\,.} \\ {} \end{array}$$

This shows that if (4) is the group product of a group object in SuperManifolds then the corresponding super Lie algebra is the super-Minkowski super translation Lie algebra and hence that this group object is the desired super-Minkowski super Lie group.

So, defining the remaining group object-operations as follows:

neutral element:

inverse elements:

we conclude by checking the group object-axioms:

For associativity we need to check that the following diagram commutes:

and indeed it does — the term $\big(\overline{\theta} \Gamma^a \theta\big)$ vanishes because the $\theta^\alpha$ anti-commute among themselves, while the pairing (1) is symmetric:

For unitality we need to check that the following diagram commutes:

and indeed it does:

And finally, for invertibility we need to check that the following diagram commutes:

and indeed it does:

$\Box$

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

References mentioning the definition of super Lie groups as internal groups in supermanifolds and mostly also the perspective of functorial geometry:

• Katsumi Yagi, Super Lie Groups, Adv. Stud. Pure Math. 22, Progress in Differential Geometry (1993) 407-412 [euclid:1534359537]

• Pierre Deligne, John Morgan, §2.10 in: Notes on Supersymmetry (following Joseph Bernstein), in: Quantum Fields and Strings, A course for mathematicians, 1, Amer. Math. Soc. Providence (1999) 41-97 [ISBN:978-0-8218-2014-8, web version, pdf]

• Veeravalli Varadarajan, section 7.1 of: Supersymmetry for mathematicians: An introduction, Courant Lecture Notes in Mathematics 11, American Mathematical Society (2004) [doi:10.1090/cln/011, pdf]

• Dennis Westra, Superrings and supergroups, PhD thesis (2009) [doi10.25365/thesis.6869, pdf, pdf]

• Claudio Carmeli, Lauren Caston, Rita Fioresi, Section 3.4 in: Mathematical Foundations of Supersymmetry, EMS Series of Lectures in Mathematics 15 (2011) [ISBN:978-3-03719-097-5, arXiv:0710.5742]

Discussion via dual superalgebra:

• Felix A. Berezin (edited by Alexandre A. Kirillov): Lie Supergroups, in: Introduction to Superanalysis, Mathematical Physics and Applied Mathematics 9, Springer (1987) [doi:10.1007/978-94-017-1963-6_8]

Discussion in a context of supergravity:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section II.2 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, chII.2: pdf, ch II.3: pdf]

Discussion of group extensions for supergroups:

• 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)