A note on semidirect sum of Lie algebras

• Autorzy: Tadeusz Ostrowski
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Lie algebra; subalgebra; ideal; center; semidirect sum; homomorphism; derivation; Killing form

Kategorie tematyczne

• 17B66: Lie algebras of vector fields and related (super) algebras
• 17B40: Automorphisms, derivations, other operators

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2013
• Tom: 33
• Numer: 2
• Strony: 233-247

Daty

wydano

2013

otrzymano

2013-07-02

poprawiono

2013-08-05

Twórcy

autor

Tadeusz Ostrowski

• The State Higher Vocational School in Gorzów Wlkp., Technical Institute, ul. Mysliborska 34, 66-400 Gorzów Wlkp., Poland

Bibliografia

• [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] 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
• [3] H. Bass, J. Oesterle and A. Weinstein, Poisson Structures and Their Normal Forms (Birkhauser Basel, 2005). doi: 10.1007/b137493
• [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.
• [5] R.W. Carter, Lie Algebras of Finite and Affine Type (Cambridge University Press, 2005). doi: 10.1017/CBO9780511614910
• [6] K. Erdmann and M.J. Windon, Introduction to Lie Algebras (Springer-Verlag London, 2006).
• [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
• [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] B.C. Hall, Lie Groups, Lie Algebras, and Representations (Springer-Verlag, NY, 2004).
• [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
• [11] F. Iachello, Lie Algebras and Applications (Springer-Verlag, Berlin, Heidelberg, 2006).
• [12] A. Kirillov, Jr., An Introduction to Lie Groups and Lie Algebras (Cambridge University Press, 2008). doi: 10.1017/CBO9780511755156
• [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
• [14] E. Mundt, Constant Young-Mills potentials, Journal of Lie Theory 3 (1993), 107-115.
• [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
• [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
• [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
• [18] H. Samelson, Notes on Lie Algebras (Springer-Verlag Berlin and Heidelberg, 1990). doi: 10.1007/978-1-4613-9014-5
• [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.
• [20] W.H. Steeb, Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra (World Scientific Publishing Co., 2007). PMid:17361179
• [21] D.H. Sattinger and O.L. Weaver, Lie groups and Algebras with Applications to Physics, Geometry and Mechanics (Springer-Verlag, 1993).
• [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

Identyfikatory

DOI

10.7151/dmgaa.1208