Rahul Sharma

Supersymmetry as a quantum group

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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.

Contents

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

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.

Fermionic operators

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.

References

Supersymmetry as a quantum group Wikipedia


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