Schur Multipliers of Nilpotent Lie Algebras.

摘要

The multiplier of a group was first discovered by I. Schur in the early twentieth century. It can be defined using the second cohomology group, a free presentation, or a central extension. We examine the Schur multiplier for Lie algebras which are nilpotent. We compute multipliers for a particular type of nilpotent Lie algebra, categorizing them in regard to a certain invariant. Our computations indicate the existence of a bound on the dimension of the multiplier in terms of the dimension of the algebra. We prove this conjecture is a theorem. The analogue of this result is also shown to hold for a certain type of p--group. We use results from Berkovich [5], Ellis [8], and Zhou [21] to prove the result. Additionally, we develop another bound for the dimension of the multiplier in terms of its class and number of generators. We compare this bound to a known result in [12]. There are many results concerning the Schur multiplier of a group being trivial. See [13], [18], [6], and [19]. We examine sufficient conditions for making the Schur multiplier of a Lie algebra nontrivial proving an elegant theorem involving the dimension of the Lie algebra.