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SUMMARY:What is an ind-group? Examples and applications
DTSTART:20151110T151500
DTEND:20151110T170000
DTSTAMP:20260413T205114Z
UID:8d71d818ed33a48655ede41955ee1dc5df65847dee71a100e61cab6d
CATEGORIES:Conferences - Seminars
DESCRIPTION:Hanspeter Kraft (Basel)\n\n	In 1966 Shafarevich introduced the
  notion of “infinite dimensional algebraic group”\, shortly “ind-gro
 up”. His main application was the automorphism group of affine $n$-space
  $A^n$ for which he claimed some interesting properties. Recently\, joint
 ly with J.-Ph. Furter we showed that the automorphism group of any finitel
 y generated (general) algebra has a natural structure of an ind-group\, a
 nd we further developed the theory.\n	It turned out that some properties w
 ell-known for algebraic groups carry over to ind-groups\, but others do no
 t. E.g. every ind-group has a Lie algebra\, but the relation between the g
 roup 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. \n	An interestin
 g “test case” is $Aut(A^2)$\, the automorphism group of affine 2-space
 \, because this group is the amalgamated product of two closed subgroups w
 hich 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 hi
 gher dimensions would have very strong and deep consequences\, e.g. for t
 he so-called linearisation problem.\n\n	 \n\n	One of the highlights is th
 e 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 isomor
 phic to $A^n$ as a variety.
LOCATION:CHB331
STATUS:CONFIRMED
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