Donald S. Passman plans to work on three main problems during the next few years along with his usual slew of miscellaneous ones. The main problems concern (1) the semiprimitivity of groups rings of finitely generated groups, (2) the structure of the lattice of two-sided ideals of a group algebra, and (3) the simplicity of Lie algebras of generalized Cartan type. The proposer returned to the study of group rings in 1990 and during the past few years, in a long series of papers, he was able to describe the Jacobson radicals of group rings of locally finite groups. In some sense, this solves half of the semiprimitivity problem, and he now proposes to work on the other half, namely the case of finitely generated groups. The second problem is motivated by the visit of Alex Zalesskii to Madison for the Spring semester of 1999. He and the proposer plan to study variations of the question of when group algebras have only trivial ideals. This will hopefully be the start of a long term collaboration. Finally, the proposer has recently taken a ring theoretic approach to study the simplicity of Lie algebras of generalized Cartan type. He has already settled the problem for Witt type algebras, is presently working with Jeff Bergen on the special algebras, and hopes to deal with the Hamiltonian and contact types at some later time.
Historically, groups first arose as the set of all symmetries of a geometric object. Since the product (composition) of two symmetries is again a symmetry, we see that a group is a collection of elements with a ``nice'' multiplication. Conversely, if we start with an abstract group, then we can better understand it by allowing it to ``act'' as symmetries on certain algebraic objects. The appropriate objects to consider here are built out of vectors and are called vector spaces. The action is achieved by forming the group algebra, which is an algebraic object determined by the group and having both an addition and a multiplication. The vector spaces of interest are then what are known as the irreducible modules for the group algebra, irreducible because we would like them to be as small as possible. As it turns out, there is a fundamental obstacle to this procedure, namely the Jacobson radical of the group algebra. Indeed, the procedure works if and only if this radical is trivial, in which case we say that the group algebra is semiprimitive. Thus, it is a natural and important problem to decide when group algebras are semiprimitive, and this is the goal of proposal (1), described above. Proposal (2) is concerned with taking an even closer look at the group algebra, but under more special circumstances.
