superalgebra and (synthetic ) supergeometry
A super-group is the analog in supergeometry of Lie groups in differential geometry.
An affine algebraic super group is the formal dual of a super-commutative Hopf algebra.
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 in the category of supermanifold is by first sending with the Yoneda embedding to a presheaf on and then imposing a lift of through the forgetful functor Grp Set that sends a (ordinary) group to its underlying set.
So a group object structure on is a diagram
This gives for each supermanifold an ordinary group , so in particular a product operation
Moreover, since morphisms in are group homomorphisms, it follows that for every morphism of supermanifolds we get a commuting diagram
Taken together this means that there is a morphism
of representable presheaves. By the Yoneda lemma, this uniquely comes from a morphism , which is the product of the group structure on the object 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 over an ordinary vector space an ordinary group or Lie algebra and to a morphism of Grassmann algebras covariantly a morphism of groups . But the Grassmann algebra on an -dimensional vector space is naturally isomorphic to the function ring on the supermanifold . So the definition of supergroups in terms of Grassmann algebras is secretly the same as the above definition in terms of the Yoneda embedding.
also called the super-Heisenberg group
The additive group structure on 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
Recall how the notation works here: by the Yoneda embedding we have a full and faithful functor
and we also have the theorem, discussed at supermanifolds, that maps from some into is given by a tuple of even section and odd sections . The above notation specifies the map of supermanifolds by displaying what map of sets of maps from some test object it corresponds to under the Yoneda embedding.
Now, for each SDiff there is a group structure on the hom-set given by precisely the above formula for this given
where etc and where the addition and product on the right takes place in the function super algebra .
Since the formula looks the same for all , one often just writes it without mentioning as above.
The super-translation group is the -dimensional case of the super Euclidean group.
We spell out the super-translation super Lie group-structure on the supermanifold 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:
Here
is the spatial dimension (a natural number),
is a real spin representation equipped with a linear map
which is symmetric and -equivariant.
First, the super-Minkowski super Lie algebra structure on the super vector space
is defined, dually, by the Chevalley-Eilenberg dgc-superalgebra with generators of bidegree
generator | bidegree |
---|---|
for indexing a linear basis of and indexing a linear basis of by the differential equations
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 not just as a super vector space but as a Cartesian supermanifold. As such it has canonical coordinate functions
generator | bidegree |
---|---|
On this supermanifold, consider the super coframe field
(where on the left we have the tangent bundle and on the right its typical fiber super vector space) given by
It is clear that this is a coframe field in that for all it restricts to an isomorphism
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
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 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 by for the first factor and for the second, and declare a group product operation as follows:
(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 in the following computation is just that of the second factor, hence acting on unprimed coordinates only):
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:
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 vanishes because the 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:
…
…
There is a finite analog for super-groups that does not quite fit in the framework presented here:
A finite super-group is a tuple , where is a finite group and is central and squares to .
The representations of a finite super-group are -graded: An irreducible representation has odd degree if acts by negation, and even degree if it acts as the identity.
This definition is found e.g. in:
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 |
---|---|
-algebra | -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 |
---|---|
-2-algebra | -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 |
---|---|
-3-algebra | -4-module |
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:
Discussion in a context of supergravity:
Discussion of group extensions for supergroups:
Discussion as Hopf-superalgebras includes
Last revised on September 17, 2024 at 08:56:07. See the history of this page for a list of all contributions to it.