Algebra Seminar

Steinberg groups
Monday, 20 March 2017 - 1:10 pm to 1:40 pm
Room number: 
FSS 8003
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Cost to attend: 
Free of charge
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SPEAKER: Erhard Neher (Ottawa) DATE: Monday, March 20, 2017 TIME: 1:10 pm ROOM: FSS 8003 ABSTRACT:

The Steinberg group St(C) of a Chevalley group C over a field F is defined by generators and relations involving commutators of root subgroups of C. A classical result of Steinberg says that St(C) is the universal central extension of C except when the rank of C and the size of F is small. In particular, St(C) is centrally closed (= its own universal central extension).

Several natural generalizations of this situation have been considered and will be discussed in the talk. One replaces C by a group G with commutator relations with respect to a family of subgroups and shows that the canonically defined Steinberg group associated with G is centrally closed, although it may in general no longer be a universal central extension of G. For example, this is so for G the elementary group of n x n matrices over a ring. . In characteristic 0, these generalizations are special cases of groups associated with the elementary automorphism group of a root-graded Lie algebra.