We introduce the formalism of differential conformal superalgebras, which we show leads to the "correct" automorphism group functor and accompanying descent theory in the conformal setting. As an application, we classify forms of N = 2 and N = 4 conformal superalgebras by means of Galois cohomology.

We describe the finite-dimensional simple modules of all the (twisted and untwisted) multiloop algebras and classify them up to isomorphism.

For any finite-dimensional simple Lie algebra g and commutative associative algebra S of finite type, we give a complete classification of the simple weight modules of g ⊗ S with bounded weight multiplicities.

Thin coverings are a method of constructing graded-simple modules from simple (ungraded) modules. After a general discussion, we classify the thin coverings of (quasifinite) simple modules over associative algebras graded by finite abelian groups. The classification uses the representation theory of cyclotomic quantum tori. We close with an application to representations of multiloop Lie algebras.

We construct irreducible modules for twisted toroidal Lie algebras and extended affine Lie algebras. This is done by combining the representation theory of untwisted toroidal algebras with the technique of thin coverings of modules. We illustrate our method with examples of extended affine Lie algebras of Clifford type.Extended affine Lie algebras (EALAs) are natural generalizations of the affine Kac-Moody algebras. They come equipped with a non-degenerate symmetric invariant bilinear form, a finite-dimensional Cartan subalgebra, and a discrete root system. Originally introduced in the contexts of singularity theory and mathematical physics, their structure theory has been extensively studied for over 15 years. (See [2,3,20] and the references therein.)Their representations are much less well understood. Early attempts to replicate the highest weight theory of the affine setting were stymied by the lack of a triangular decomposition; later work considered only the untwisted toroidal Lie algebras and a few other isolated examples. As a result of major breakthroughs announced in [3] and [20], it is now clear that, except for (extensions of) matrix algebras over non-cyclotomic quantum tori, every extended affine Lie algebra can be constructed as