Timeline for Chevalley Groups over an arbitrary ring.
Current License: CC BY-SA 4.0
9 events
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| Mar 29, 2022 at 1:22 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
fixed arxiv front-end link, gave title, and doi link to published version
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| Dec 17, 2013 at 1:37 | vote | accept | M.B | ||
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| Nov 14, 2013 at 17:35 | comment | added | Jim Humphreys | @M.B.: The definition of "Chevalley group" varies in different sources, so it's always important to pin it down. In his many papers, Vavilov works with the points of a Chevalley-Demazure group scheme but also with the usually smaller "elementary Chevalley group"; the difference is an important part of studying the groups over rings. See his long survey: Structure of Chevalley groups over commutative rings.* Nonassociative algebras and related topics (Hiroshima, 1990), 219–335, World Sci. Publ., River Edge, NJ, 1991. Your construction is much narrower, similar to Chevalley's. | |
| Nov 14, 2013 at 17:27 | comment | added | Jim Humphreys | @marguax: No, Steinberg works with an arbitrary lattice between the root lattice and the weight lattice. His "Chevalley groups" are generated by elementary (unipotent) matrices, but in his lectures and related papers he deals with related covering groups as well. The notes are online at www.math.ucla.edu/~rst/ Your further comments are helpful. | |
| Nov 14, 2013 at 16:02 | comment | added | M.B | Dear Jim: Many thanks for the answer. You have mentioned that the Chevalley group, constructed by the Chevalley basis, has been considered for commutative rings. So I would like to know if the construction is the same as I presented in my question. This is the reason I asked the question. | |
| Nov 14, 2013 at 15:11 | comment | added | Marguax | Steinberg's Yale notes build only simply connected forms, yes? (Section 4 of Borel's article which the OP looked at goes further.) For your #2, the "Chevalley groups" in Vavilov et al. are the ring-valued points of Chevalley-Deamzure group schemes, whereas what the OP and Borel call a "Chevalley group" is called there an "elementary subgroup". The link between them and "rare" cases of equality (and role of Bruhat decomposition in that) are addressed in the first 2 pages of the 1996 Acta Applicandae paper of Vavilov and Plotkin; also see the last paragraph of section 1. | |
| Nov 14, 2013 at 13:28 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
added 84 characters in body
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| Nov 14, 2013 at 12:50 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
added 263 characters in body
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| Nov 14, 2013 at 12:28 | history | answered | Jim Humphreys | CC BY-SA 3.0 |