International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 71, Pages 3931-3939
On the uniqueness of the -dimensional supertorus associated to a nontrivial representation of its underlying -torus, and having nontrivial odd brackets
1Centro de Investigación en Matemáticas, Apartado Postal 402, Guanajuato 36000, Mexico
2Facultad de Matemáticas, Universidad Autónoma de Yucatán, Mérida, Yucatán, Mexico
3Department of Mathematics, Victoria University of Wellington, Wellington 6001, New Zealand
Received 12 May 2003
Copyright © 2004 R. Peniche et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
It is proved that up to isomorphism there is only one -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 but with no further intrinsic characterization.