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

${(-1)^F}^2=1$

with the counit

$\epsilon((-1)^F)=1$

and the coproduct

$\Delta (-1)^F=(-1)^F \otimes (-1)^F$

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 $\mathbb{Z}_2$. Supersymmetry comes in when introducing the nontrivial quasitriangular structure

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

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 $\mathbb{Z}_2$ 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,

$\Delta x = x \otimes 1 + 1 \otimes x$

$\Delta y = y \otimes 1 + (-1)^F \otimes y$

This is compatible with $\mathcal{R}$.

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

See also

• Introduction to quantum mechanics
• Group theory
• Symmetry group