What is an ind-group? Examples and applications
Event details
| Date | 10.11.2015 |
| Hour | 15:15 › 17:00 |
| Speaker | Hanspeter Kraft (Basel) |
| Location |
CHB331
|
| Category | Conferences - Seminars |
In 1966 Shafarevich introduced the notion of “infinite dimensional algebraic group”, shortly “ind-group”. His main application was the automorphism group of affine $n$-space $A^n$ for which he claimed some interesting properties. Recently, jointly with J.-Ph. Furter we showed that the automorphism group of any finitely generated (general) algebra has a natural structure of an ind-group, and we further developed the theory.
It turned out that some properties well-known for algebraic groups carry over to ind-groups, but others do not. E.g. every ind-group has a Lie algebra, but the relation between the group and its Lie algebra still remains unclear. As another by-product of this theory we get new interpretations and a better understanding of some classical results, together with short and nice proofs.
An interesting “test case” is $Aut(A^2)$, the automorphism group of affine 2-space, because this group is the amalgamated product of two closed subgroups which implies a number of remarkable properties. E.g. a conjugacy class of an element $g \in Aut(A^2)$ is closed if and only if $g$ is semi-simple, a result well-known for algebraic groups. A generalisation of this to higher dimensions would have very strong and deep consequences, e.g. for the so-called linearisation problem. One of the highlights is the following result. If $X$ is a connected variety whose automorphism group $Aut(X)$ is isomorphic to $Aut(A^n)$ as an ind-group, then $X$ is isomorphic to $A^n$ as a variety.
It turned out that some properties well-known for algebraic groups carry over to ind-groups, but others do not. E.g. every ind-group has a Lie algebra, but the relation between the group and its Lie algebra still remains unclear. As another by-product of this theory we get new interpretations and a better understanding of some classical results, together with short and nice proofs.
An interesting “test case” is $Aut(A^2)$, the automorphism group of affine 2-space, because this group is the amalgamated product of two closed subgroups which implies a number of remarkable properties. E.g. a conjugacy class of an element $g \in Aut(A^2)$ is closed if and only if $g$ is semi-simple, a result well-known for algebraic groups. A generalisation of this to higher dimensions would have very strong and deep consequences, e.g. for the so-called linearisation problem. One of the highlights is the following result. If $X$ is a connected variety whose automorphism group $Aut(X)$ is isomorphic to $Aut(A^n)$ as an ind-group, then $X$ is isomorphic to $A^n$ as a variety.
Practical information
- Informed public
- Free
Organizer
- Chair of Geometry and Group-Testerman
Contact
- Donna Testerman