Revista de la
Unión Matemática Argentina
The full group of isometries of some compact Lie groups endowed with a bi-invariant metric
Alberto Dolcetti and Donato Pertici

Volume 65, no. 2 (2023), pp. 245–262    

Published online: October 23, 2023

https://doi.org/10.33044/revuma.2737

Download PDF

Abstract

We describe the full group of isometries of absolutely simple, compact, connected real Lie groups, of $S\mathcal{O}(4)$, and of $U(n)$, endowed with suitable bi-invariant Riemannian metrics.

References

  1. Killing form, in Encyclopedia of Mathematics. Available at http://encyclopediaofmath.org/index.php?title=Killing_form&oldid=12372.
  2. Lie algebra, semi-simple, in Encyclopedia of Mathematics. Available at http://encyclopediaofmath.org/index.php?title=Lie_algebra,_semi-simple&oldid=44225.
  3. T. Abe, S. Akiyama, and O. Hatori, Isometries of the special orthogonal group, Linear Algebra Appl. 439 no. 1 (2013), 174–188.  DOI  MR  Zbl
  4. M. M. Alexandrino and R. G. Bettiol, Lie Groups and Geometric Aspects of Isometric Actions, Springer, Cham, 2015.  DOI  MR  Zbl
  5. A. Dolcetti and D. Pertici, Some differential properties of $GL_n(\mathbb{R})$ with the trace metric, Riv. Mat. Univ. Parma (N.S.) 6 no. 2 (2015), 267–286.  MR  Zbl
  6. A. Dolcetti and D. Pertici, Skew symmetric logarithms and geodesics on ${\mathrm{O}}_n(\mathbb{R})$, Adv. Geom. 18 no. 4 (2018), 495–507.  DOI  MR  Zbl
  7. A. Dolcetti and D. Pertici, Differential properties of spaces of symmetric real matrices, Rend. Semin. Mat. Univ. Politec. Torino 77 no. 1 (2019), 25–43.  MR  Zbl
  8. A. Dolcetti and D. Pertici, Real square roots of matrices: differential properties in semi-simple, symmetric and orthogonal cases, Riv. Mat. Univ. Parma (N.S.) 11 no. 2 (2020), 315–333.  MR  Zbl
  9. A. Dolcetti and D. Pertici, Elliptic isometries of the manifold of positive definite real matrices with the trace metric, Rend. Circ. Mat. Palermo (2) 70 no. 1 (2021), 575–592.  DOI  MR  Zbl
  10. B. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, second ed., Graduate Texts in Mathematics 222, Springer, Cham, 2015.  DOI  MR  Zbl
  11. O. Hatori and L. Molnár, Isometries of the unitary group, Proc. Amer. Math. Soc. 140 no. 6 (2012), 2127–2140.  DOI  MR  Zbl
  12. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics 80, Academic Press, New York-London, 1978.  MR  Zbl
  13. M. Hunt, G. Mullineux, R. J. Cripps, and B. Cross, Characterizing isoclinic matrices and the Cayley factorization, Proc. Institution Mechanical Engineers, Part C: J. Mech. Eng. Sci. 230 no. 18 (2016), 3267–3273.  DOI
  14. A. W. Knapp, Lie Groups Beyond an Introduction, second ed., Progress in Mathematics 140, Birkhäuser Boston, Boston, MA, 2002.  MR  Zbl
  15. J. E. Mebius, A matrix-based proof of the quaternion representation theorem for four-dimensional rotations, 2005. arXiv:math/0501249 [math.GM].
  16. B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics 103, Academic Press, New York, 1983.  MR  Zbl
  17. A. L. Onishchik, The group of isometries of a compact Riemannian homogeneous space, in Differential Geometry and Its Applications (Eger, 1989), Colloq. Math. Soc. János Bolyai 56, North-Holland, Amsterdam, 1992, pp. 597–616.  MR  Zbl
  18. A. Perez-Gracia and F. Thomas, On Cayley's factorization of 4D rotations and applications, Adv. Appl. Clifford Algebr. 27 no. 1 (2017), 523–538.  DOI  MR  Zbl
  19. G. de Rham, Sur la reductibilité d'un espace de Riemann, Comment. Math. Helv. 26 (1952), 328–344.  DOI  MR  Zbl
  20. M. R. Sepanski, Compact Lie Groups, Graduate Texts in Mathematics 235, Springer, New York, 2007.  DOI  MR  Zbl
  21. K. Shankar, Isometry groups of homogeneous spaces with positive sectional curvature, Differential Geom. Appl. 14 no. 1 (2001), 57–78.  DOI  MR  Zbl
  22. F. Thomas, Approaching dual quaternions from matrix algebra, IEEE Trans. Robotics 30 no. 5 (2014), 1037–1048.  DOI
  23. F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics 94, Springer-Verlag, New York-Berlin, 1983.  MR  Zbl