| This dissertation consists of three separate pieces, each concerning nilpotent Lie algebras. In the first piece we investigate topics related to the 5-sequence of cohomology. We discuss how, under various conditions, the sequence can be extended exactly by one term. These results are used to obtain inequalities bounding the dimension of the Schur Multiplier of a Lie algebra. We then show it is possible to obtain equality in each instance. The second piece investigates nilpotent extensions. We consider two extensions of M by L, C and C ', and their corresponding lifts, f and f' . We show if f and f' are lifts of the same homomorphism F : L → Out(M), then C is nilpotent if and only if C' is nilpotent However, it turns out that the nilpotency classes of the extensions need not be the same and we discuss instances where they are and are not. We also use the aforementioned result to obtain an alternate proof of the Lie algebra analogue to a theorem of Phillip Hall from group theory. The final piece concerns an exact sequence involving the Schur Multiplier of a Lie algebra. We prove exactness of the sequence and use the result to obtain an alternate construction for the Schur Multiplier of a Lie algebra, L, when the nilpotency class of L is 2. Using this construction we get another way to compute the dimension of the Schur Multiplier for the 2n + 1 dimensional Heisenberg Lie algebra when n > 1. |
