Lie Algebra Decompositions with Applications to Quantum Dynamics Table of Contents List of Figures Figure 2.1 a Schematic Representation of the Khaneja Glaser Decomposition . . . 14

Lie group decompositions are useful tools in the analysis and control of quantum systems. Several decompositions proposed in the literature are based on a recursive procedure that systematically uses the Cartan decomposition theorem. In this dissertation, we establish a link between Lie algebra gradings and recursive Lie algebra decompositions, and then we formulate a general scheme to generate Lie group decompositions. This scheme contains some procedures previously proposed as special cases and gives a virtually unbounded number of alternatives to factor elements of a Lie group.