Abstract
By a quasi-connected reductive group (a term of Labesse) over an arbitrary field we mean an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group. We compute the first Galois cohomology set \(H^1({\mathbb {R}},G)\) of a quasi-connected reductive group G over the field \({\mathbb {R}}\) of real numbers in terms of a certain action of a subgroup of the Weyl group on the Galois cohomology of a fundamental quasi-torus of G.
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Acknowledgements
The authors are grateful to N. Q. Thǎńg for the reference to Labesse [11]. We thank B. È. Kunyavskiĭ and J. S. Milne for helpful comments. We also thank the anonymous referees for their helpful comments and suggestions, which permitted us to improve the exposition. In particular, it was suggested by a referee to consider quasi-connected reductive groups over a field of arbitrary characteristic rather than only over a field of characteristic 0.
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Borovoi and Gornitskii were supported by the Israel Science Foundation (Grant 870/16). Gornitskii was supported by the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement No. 075-15-2019-1621. Rosengarten was supported by a Zuckerman Postdoctoral Scholarship.
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Borovoi, M., Gornitskii, A.A. & Rosengarten, Z. Galois cohomology of real quasi-connected reductive groups. Arch. Math. 118, 27–38 (2022). https://doi.org/10.1007/s00013-021-01678-x
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DOI: https://doi.org/10.1007/s00013-021-01678-x