The four fundamental frameworks of physical theories are classical machanics, quantum mechanics, classical field theory, and quantum field theory. In my talk I will discuss the related algebraic structures and connections between them. This general scheme will be exemplified in the two subsequent lectures on Saturday and Sunday.
--------------------------
In joint work with Ivan Penkov, we have begun a systematic study of (generalized) Harish-Chandra modules for certain symmetric subalgebras of classical infinite dimensional Lie algebras, such as gl , sp , and so. In particular, we have adapted the Vogan-Zuckerman theory of cohomological induction to the construction of series of irreducible Harish-Chandra modules. The subtle point is the proof that under very general conditions we have constructed modules which are integrable only with respect to the symmetric subalgebra. Until now, the literature has mainly concentrated on highest weight modules, which are integrable with respect to a maximal parabolic subalgebra.
Penkov and I have extended our methods to the following context: g is an infinite dimensional locally finite and reductive Lie algebra and k is a subalgebra of the form k_0 + C_g(k_0) , where k_0 is a finite dimensional reductive in g subalgebra and C_g(k_0) is the centralizer of k_0 in g . For example, g = gl and k_0 = gl(n) with the standard embedding. Another example is the case of a diagonal locally simple Lie algebra, which we will define explicitly in the lecture.