Abstract
We classify maximal antipodal subgroups of the group \(\mathrm {Aut}(\mathfrak {g})\) of automorphisms of a compact classical Lie algebra \(\mathfrak {g}\). A maximal antipodal subgroup of \(\mathrm {Aut}(\mathfrak {g})\) gives us as many mutually commutative involutions of \(\mathfrak {g}\) as possible. For the classification we use our former results of the classification of maximal antipodal subgroups of quotient groups of compact classical Lie groups. We also use canonical forms of elements in a compact Lie group which is not connected.
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Notes
- 1.
We have \(\{\tau g \mid g \in U(n),\, (\tau g)^2=1_n\}\) \(=\) \(\bigcup _{g \in U(n)} g\tau 1_ng^{-1}\). It is remarkable in contrast to \(\{g \in U(n) \mid g^2 = 1_n\}\) \(=\) \(\bigcup _{g \in U(n)}\) \(g\varDelta _ng^{-1}\).
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Acknowledgements
The authors would like to thank Osamu Ikawa for his providing information about Proposition 3. The first author was partly supported by the Grant-in-Aid for Science Research (C) (No. 15K04855), Japan Society for the Promotion of Science. The second author was partly supported by the Grant-in-Aid for Science Research (C) (No. 15K04835), Japan Society for the Promotion of Science.
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Tanaka, M.S., Tasaki, H. (2017). Maximal Antipodal Subgroups of the Automorphism Groups of Compact Lie Algebras. In: Suh, Y., Ohnita, Y., Zhou, J., Kim, B., Lee, H. (eds) Hermitian–Grassmannian Submanifolds. Springer Proceedings in Mathematics & Statistics, vol 203. Springer, Singapore. https://doi.org/10.1007/978-981-10-5556-0_4
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DOI: https://doi.org/10.1007/978-981-10-5556-0_4
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