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.

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

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

with the counit

${\displaystyle ϵ((-1{)}^F)=1}$

and the coproduct

${\displaystyle \mathrm{\Delta }(-1{)}^F=(-1{)}^F\otimes (-1{)}^F}$

and the antipode

${\displaystyle S(-1{)}^F=(-1{)}^F}$

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

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

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 ${\displaystyle {\mathbb{\mathbb{Z}}}_2}$ 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,

${\displaystyle \mathrm{\Delta }x=x\otimes 1+1\otimes x}$

${\displaystyle \mathrm{\Delta }y=y\otimes 1+(-1{)}^F\otimes y}$

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

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

• Introduction to quantum mechanics
• Group theory
• Symmetry group

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