Dietrich Burde |
30.01.2019 16:15
Crystallographic groups are groups acting by isometries on some n-dimensional Euclidean
space with compact quotient. The origin of the name *crystallographic* comes from the
symmetry groups of 3-dimensional crystals in real life. We discuss both the geometric and
the algebraic aspects of the theory of crystallographic groups and its generalizations. The
theory of affine crystallographic groups leads us to etale affine representations of Lie
algebras and algebraic groups. The latter is a special case of prehomogeneous
representations. We discuss a conjecture of V. Popov in the context of linearizable
subgroups of the Cremona group on affine space, which can be reformulated in terms of
etale affine representations. We present a counterexample to this conjecture.
