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.

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

(
−
1
)
F
2
=
1
with the counit

ϵ
(
(
−
1
)
F
)
=
1
and the coproduct

Δ
(
−
1
)
F
=
(
−
1
)
F
⊗
(
−
1
)
F
and the antipode

S
(
−
1
)
F
=
(
−
1
)
F
Thus far, there is nothing supersymmetric about this Hopf algebra at all; it is isomorphic to the Hopf algebra of the two element group
Z
2
. Supersymmetry comes in when introducing the nontrivial quasitriangular structure

R
=
1
2
[
1
⊗
1
+
(
−
1
)
F
⊗
1
+
1
⊗
(
−
1
)
F
−
(
−
1
)
F
⊗
(
−
1
)
F
]
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.

The previous analysis only introduced the concept of fermions, and is not actual supersymmetry. The Hopf algebra is
Z
2
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,

Δ
x
=
x
⊗
1
+
1
⊗
x
Δ
y
=
y
⊗
1
+
(
−
1
)
F
⊗
y
This is compatible with
R
.

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