Lie Groups I: Introduction and Examples

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


The concept of a group was described briefly in Chapter 1. This chapter serves as an introduction to a special class of groups, the Lie groups, which are named after Norwegian mathematician Sophus Lie.1 Furthermore, when referring to Lie groups, what will be meant in the context of this book is matrix Lie groups, where each element of the group is a square invertible matrix. Other books focusing specifically on matrix groups include [3, 9, 19].


Matrix Multiplication Structure Constant Binary Operation Heisenberg Group Semidirect Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Angeles, J., Rational Kinematics, Springer, New York, 1989.Google Scholar
  2. 2.
    Artin, M., Algebra, Prentice Hall, Upper Saddle River, NJ, 1991.Google Scholar
  3. 3.
    Baker, A., Matrix Groups: An Introduction to Lie Group Theory, Springer, New York, 2002.Google Scholar
  4. 4.
    Baker, H.F., “Alternants and continuous groups,” Proc. London Math. Soc. (Second Series),3, pp. 24–47, 1904.Google Scholar
  5. 5.
    Birkhoff, G., MacLane, S., A Survey of Modern Algebra, 4th ed., Macmillan Publishing Co., New York, 1977.Google Scholar
  6. 6.
    Bottema, O., Roth, B., Theoretical Kinematics, Dover, New York, 1990.Google Scholar
  7. 7.
    Campbell, J.E., “On a law of combination of operators,” Proc. London Math. Soc., 29, pp. 14–32, 1897.CrossRefGoogle Scholar
  8. 8.
    Chirikjian, G.S., Kyatkin, A.B., Engineering Applications of Noncommutative Harmonic Analysis, CRC Press, Boca Raton, FL, 2001.Google Scholar
  9. 9.
    Curtis, M.L., Matrix Groups, 2nd ed., Springer, New York, 1984.Google Scholar
  10. 10.
    Gilmore, R., Lie Groups, Lie Algebras, and Some of Their Applications, Dover, New York, 2006.Google Scholar
  11. 11.
    Hausdorff, F., “Die symbolische Exponentialformel in der Gruppentheorie,” Berich. der Sachsichen Akad. Wissensch., 58, pp. 19–48, 1906.Google Scholar
  12. 12.
    Inui, T., Tanabe, Y., Onodera, Y., Group Theory and Its Applications in Physics, 2nd ed., Springer-Verlag, New York, 1996.Google Scholar
  13. 13.
    Kol´aˇr, I.,Michor, P.W., Slov´ak, J., Natural Operations in Differential Geometry, Springer- Verlag, Berlin, 1993.Google Scholar
  14. 14.
    McCarthy, J.M., An Introduction to Theoretical Kinematics, MIT Press, Cambridge, MA, 1990.Google Scholar
  15. 15.
    Miller, W., Jr., Symmetry Groups and Their Applications, Academic Press, New York, 1972.Google Scholar
  16. 16.
    Murray, R.M., Li, Z., Sastry, S.S., A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994.Google Scholar
  17. 17.
    Park, F.C., The Optimal Kinematic Design of Mechanisms, Ph.D. thesis, Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 1991.Google Scholar
  18. 18.
    Selig, J.M., Geometrical Methods in Robotics, 2nd ed., Springer, New York, 2005.Google Scholar
  19. 19.
    Tapp, K., Matrix Groups for Undergraduates, American Mathematical Society, Providence, RI, 2005.Google Scholar
  20. 20.
    Varadarajan, V.S., Lie Groups, Lie Algebras, and their Representations, Springer-Verlag, New York, 1984.Google Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe Johns Hopkins UniversityBaltimoreUSA

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