Electronic Thesis and Dissertation Repository


Doctor of Philosophy


Applied Mathematics


Gregory J. Reid


Superspaces are an extension of classical spaces that include certain (non-commutative) supervariables. Super differential equations are differential equations defined on superspaces, which arise in certain popular mathematical physics models. Supersymmetries of such models are superspace transformations which leave their sets of solutions invariant. They are important generalization of classical Lie symmetry groups of differential equations.

In this thesis, we consider finite-dimensional Lie supersymmetry groups of super differential equations. Such supergroups are locally uniquely determined by their associated Lie superalgebras, and in particular by the structure constants of those algebras. The main work of this thesis is providing an algorithmic method for finding the structure constants of such Lie superalgebras. The traditional method uses heuristic integrations to determine such structure constants. Two typical examples are used to demonstrate our algorithm for determining structure constants.

We also apply our method to a large class of super Lagrangians in 1+1 dimensional space time. The supersymmetry classification of such a large class is impossible for hand calculation since it requires analysis of thousands of cases. We will show how to find hidden supersymmetry for such a class of super differential equations by our algorithms and the Physics, DEtools, PDEtools packages of Maple 17.