On the uniqueness of the (2,2)-dimensional supertorus associated to a nontrivial representation of its underlying 2-torus, and having nontrivial odd brackets

Abstract

It is proved that up to isomorphism there is only one (2,2)-dimensional supertorus associated to a nontrivial representation of its underlying 2-torus, and that it has nontrivial odd brackets. This supertorus is obtained by finding out first a canonical form for its Lie superalgebra, and then using Lie's technique to represent it faithfully as supervector fields on a supermanifold. Those supervector fields can be integrated, and through their various integral flows the composition law for the supergroup is straightforwardly deduced. It turns out that this supertorus is precisely the supergroup described by Guhr (1993) following a formal analogy with the classical unitary group U(2) but with no further intrinsic characterization.