The Lie algebra of derivations of a block of a finite group

Lie groups and Lie algebras arise in Physics as symmetry groups of
physical systems and their tangent spaces, which may be regarded as
infinitesimal symmetry motions.
These notions have long been of interest in many areas of Mathematics.
Lie algebras arise, for instance, as operators on algebras respecting
Leibniz' product rule, called derivations.
The derivations on an algebra can be interpreted as representatives of
the first Hochschild cohomology of an algebra. The Lie algebra structure
on this space extends to a graded Lie algebra structure on Hochschild
cohomology - this goes back to pioneering work of Gerstenhaber, in the
context of the deformation theory of algebras.

The use of this technology in Physics tends to be over fields of
characteristic zero, but the underlying concepts have analogues over
fields of prime characteristic, and makes this technology available
for investigations in the modular representation theory of finite group
algebras over local rings and fields. In fact, there are `many more'
finite-dimensional Lie algebras over fields of prime characteristic than
over the complex numbers.

A particular feature of modular representation theory of finite groups
is that it is driven by a great number of conjectures, some of which
predict remarkable structural connection between various direct factors
of finite group algebras, and other simply predicting mysterious numerical
coincidences.

Hochschild cohomology in general has turned out to be useful for
reformulations and variations of those conjectures. Expectations are
high that investigating the (graded and restricted) Lie algebra structure
of Hochschild cohomology in the context of finite group algebras and their
direct factors should contribute to an understanding of some parts of those
conjectures.


The present proposal takes precisely these expectations as a starting
point, putting the focus on higher structural aspects of
Hochschild cohomology and their impact on invariants of finite group
algebras and their blocks. We set out describing this programme in a
sequence of nine conjectures, ranging from basic questions - such as
the non-vanishing of the first Hochschild cohomology of blocks - via
explicit calculations in certain classes of finite groups to
currently elusive conjectures on numerical and structural aspects of
finite groups and their blocks.