Supersymmetry as a quantum group

The concept in theoretical physics of supersymmetry can be reinterpreted in the language of noncommutative geometry and quantum groups. In particular, it involves a mild form of noncommutativity, namely supercommutativity.

## Unitary (–1)F operator

Following is the essence of supersymmetry, which is encapsulated within the following minimal quantum group. We have the two dimensional Hopf algebra generated by (-1)F subject to

[math]\displaystyle{ {(-1)^F}^2=1 }[/math]

with the counit

[math]\displaystyle{ \epsilon((-1)^F)=1 }[/math]

and the coproduct

[math]\displaystyle{ \Delta (-1)^F=(-1)^F \otimes (-1)^F }[/math]

and the antipode

[math]\displaystyle{ S(-1)^F=(-1)^F }[/math]

Thus far, there is nothing supersymmetric about this Hopf algebra at all; it is isomorphic to the Hopf algebra of the two element group [math]\displaystyle{ \mathbb{Z}_2 }[/math]. Supersymmetry comes in when introducing the nontrivial quasitriangular structure

[math]\displaystyle{ \mathcal{R}=\frac{1}{2}\left[ 1 \otimes 1 + (-1)^F \otimes 1 + 1 \otimes (-1)^F - (-1)^F \otimes (-1)^F\right] }[/math]

where +1 eigenstates of (-1)F are called bosons and -1 eigenstates are called fermions.

This describes a fermionic braiding; don't pick up a phase factor when interchanging two bosons or a boson and a fermion, but multiply by -1 when interchanging two fermions. This provides the essence of the boson/fermion distinction.

## Fermionic operators

The previous analysis only introduced the concept of fermions, and is not actual supersymmetry. The Hopf algebra is [math]\displaystyle{ \mathbb{Z}_2 }[/math] graded and contains even and odd elements. Even elements commute with (-1)F; odd ones anticommute. The subalgebra not containing (-1)^F is supercommutative.

Let's say we are dealing with a super Lie algebra with even generators x and odd generators y.

Then,

[math]\displaystyle{ \Delta x = x \otimes 1 + 1 \otimes x }[/math]
[math]\displaystyle{ \Delta y = y \otimes 1 + (-1)^F \otimes y }[/math]

This is compatible with [math]\displaystyle{ \mathcal{R} }[/math].

Supersymmetry is the symmetry over systems where interchanging two fermions attain a minus sign.

## See also

* Introduction to quantum mechanics
* Group theory
* Symmetry group

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