  Actions of algebraic groups on DG categories provide a convenient, unifying
framework in some parts of geometric representation theory, especially the
representation theory of reductive Lie algebras. We extend this theory to loop
groups and affine Lie algebras, extending previous work of Beraldo, Gaitsgory
and the author.
  Along the way, we introduce some subjects of independent interest:
ind-coherent sheaves of infinite type indschemes, topological DG algebras, and
weak actions of group indschemes on categories. We also present a new
construction of semi-infinite cohomology for affine Lie algebras, based on a
modular character for loop groups.
  As an application of our methods, we establish an important technical result
for Kac-Moody representations at the critical level, showing that the
appropriate ("renormalized") derived category of representations admits a large
class of symmetries coming from the adjoint action of the loop group and from
the center of the enveloping algebra.
