A note on semidirect sum of Lie algebras

Tadeusz Ostrowski

Discussiones Mathematicae - General Algebra and Applications (2013)

  • Volume: 33, Issue: 2, page 233-247
  • ISSN: 1509-9415

Abstract

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In the paper there are investigated some properties of Lie algebras, the construction which has a wide range of applications like computer sciences (especially to computer visions), geometry or physics, for example. We concentrate on the semidirect sum of algebras and there are extended some theoretic designs as conditions to be a center, a homomorphism or a derivative. The Killing form of the semidirect sum where the second component is an ideal of the first one is considered as well.

How to cite

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Tadeusz Ostrowski. "A note on semidirect sum of Lie algebras." Discussiones Mathematicae - General Algebra and Applications 33.2 (2013): 233-247. <http://eudml.org/doc/270203>.

@article{TadeuszOstrowski2013,
abstract = {In the paper there are investigated some properties of Lie algebras, the construction which has a wide range of applications like computer sciences (especially to computer visions), geometry or physics, for example. We concentrate on the semidirect sum of algebras and there are extended some theoretic designs as conditions to be a center, a homomorphism or a derivative. The Killing form of the semidirect sum where the second component is an ideal of the first one is considered as well.},
author = {Tadeusz Ostrowski},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {Lie algebra; subalgebra; ideal; center; semidirect sum; homomorphism; derivation; Killing form; killing form},
language = {eng},
number = {2},
pages = {233-247},
title = {A note on semidirect sum of Lie algebras},
url = {http://eudml.org/doc/270203},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Tadeusz Ostrowski
TI - A note on semidirect sum of Lie algebras
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2013
VL - 33
IS - 2
SP - 233
EP - 247
AB - In the paper there are investigated some properties of Lie algebras, the construction which has a wide range of applications like computer sciences (especially to computer visions), geometry or physics, for example. We concentrate on the semidirect sum of algebras and there are extended some theoretic designs as conditions to be a center, a homomorphism or a derivative. The Killing form of the semidirect sum where the second component is an ideal of the first one is considered as well.
LA - eng
KW - Lie algebra; subalgebra; ideal; center; semidirect sum; homomorphism; derivation; Killing form; killing form
UR - http://eudml.org/doc/270203
ER -

References

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  1. [1] V.I. Arnold, Theorem on altitudes of a triangle in the Lobachevsky geometry as the Jacobi identity in the Lie algebras of quadratic forms in the simplectic plane, Math. Prosveschenie. Third Series 9 (2005), 93-99. 
  2. [2] M.R. Bremer, I.R. Hentzel and L.A. Peresi, Dimension formulas for the free nonassociative algebra, Communications in Algebra 33 (2005), 4063-4081. doi: 10.1080/00927870500261389 Zbl1145.17300
  3. [3] H. Bass, J. Oesterle and A. Weinstein, Poisson Structures and Their Normal Forms (Birkhauser Basel, 2005). doi: 10.1007/b137493 
  4. [4] J.M. Ancochea Bermudez, R. Campoamor-Stursberg and L. Garcia Vergnolle, Indecomposable Lie algebras with nontrivial Levi decomposition cannot have filiform radical, Internat. Math. Forum 7 (2006), 309-316. Zbl1142.17300
  5. [5] R.W. Carter, Lie Algebras of Finite and Affine Type (Cambridge University Press, 2005). doi: 10.1017/CBO9780511614910 
  6. [6] K. Erdmann and M.J. Windon, Introduction to Lie Algebras (Springer-Verlag London, 2006). 
  7. [7] A. Figula and K. Strambach, Loops which are semidirect products of groups, Acta Mathematica Hungarica 114 (2007), 247-266. doi: 10.1007/s10474-006-0529-3 Zbl1123.20061
  8. [8] E.A. de Kerf, G.A. Bauerle and A.P. Kroode, Lie Algebras, E. van Groesen, E.M. de Jager (Editors), Elsevier Science (1997). 
  9. [9] B.C. Hall, Lie Groups, Lie Algebras, and Representations (Springer-Verlag, NY, 2004). 
  10. [10] D.D. Holm, J.E. Marsden and T.S. Ratiu, The Euler-Poincare equations and semidirect products with applications to continuum theories, Advances in Mathematics 137 (1998), 1-81. doi: 10.1006/aima.1998.1721 Zbl0951.37020
  11. [11] F. Iachello, Lie Algebras and Applications (Springer-Verlag, Berlin, Heidelberg, 2006). Zbl1156.17006
  12. [12] A. Kirillov, Jr., An Introduction to Lie Groups and Lie Algebras (Cambridge University Press, 2008). doi: 10.1017/CBO9780511755156 
  13. [13] J.E. Marsden, G. Misiolek, M. Perlmutter and T.S. Ratiu, Sympletic reduction for semidirect products and central extensions, Differential Geometry and its Applications 9 (1998), 173-212. doi: 10.1016/S0926-2245(98)00021-7 Zbl0973.53069
  14. [14] E. Mundt, Constant Young-Mills potentials, Journal of Lie Theory 3 (1993), 107-115. Zbl0776.53055
  15. [15] A.L. Onishchik and E.B. Vinberg (Editors), Lie Groups and Lie Algebras (Springer, New York, 1994). doi: 10.1007/978-3-662-03066-0 Zbl0813.53015
  16. [16] D.I. Panyushev, Semi-direct products of Lie algebras and their invariants, Publications of the Research Institute for Mathematical Sciences, Kyoto University 43 (2007), 1199-1257. doi: 10.2977/prims/1201012386 Zbl1151.14036
  17. [17] V.M. Petrogradsky, Y.P. Razmyslov and E.O. Shishkin, Wreath products and Kaluzhnin-Krasner embedding for Lie algebras, Proc. Amer. Math. Soc. 135 (2007), 625-636. doi: 10.1090/S0002-9939-06-08502-9 Zbl1173.17008
  18. [18] H. Samelson, Notes on Lie Algebras (Springer-Verlag Berlin and Heidelberg, 1990). doi: 10.1007/978-1-4613-9014-5 
  19. [19] V.I. Suhchansky and N.V. Netreba, Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of infinite symmetric groups, Algebra and Discrete Math. 1 (2005), 122-132. Zbl1122.17006
  20. [20] W.H. Steeb, Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra (World Scientific Publishing Co., 2007). PMid:17361179 
  21. [21] D.H. Sattinger and O.L. Weaver, Lie groups and Algebras with Applications to Physics, Geometry and Mechanics (Springer-Verlag, 1993). Zbl0595.22017
  22. [22] V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations (Springer-Verlag, New York Inc., 1984). doi: 10.1007/978-1-4612-1126-6 

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