Семинар НУЛ АГП "Ind-groups and their Lie algebras"

Ind-groups are a natural way to study polynomial automorphism groups of affine varieties. In this talk, we will recall their topology and Lie theory, and see why the Lie algebra of Aut(Aⁿ) consists precisely of polynomial vector fields with constant divergence.

We will highlight the differences between algebraic groups and ind-groups. Most striking example, due to Furter and Kraft, is a connected subgroup of Aut(A³), preserving the third coordinate, that contains a proper closed ind-subgroup with the same Lie algebra. The construction relies on topological properties of the subgroup of tame automorphisms.