Теоремы конечности для алгебраических групп и групп Ли

Успехи математических наук. 2025. Т. 80. № 3(483). С. 189–190.

Popov V.

It is proved that for any positive integers d and c, the set of isomorphism classes of all d-dimensional reductive algebraic groups with exactly c connected components is finite. As a corollary, the set of isomorphism classes of all d-dimensional compact real Lie groups with exactly c connected components is proved to be finite. To obtain these results, a theorem on the finiteness of the cohomology of commutative algebraic groups whose connected component of the identity is a semi-Abelian variety is proved, as well as a theorem on the finiteness of the number of orbits of the action of the group of algebraic automorphisms of a connected reductive algebraic group R on the group of homomorphisms of a given finite group into the group of outer algebraic automorphisms of R.

Keywords: orbit, орбита, cohomology, когомологии, reductive group, редуктивная группа