Geodesic
Inner automorphisms of Lie algebras related with generic 2×2 matrices
Algebra and discrete mathematics, Tome 14 (2012) no. 1, pp. 49-70.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let Fm=Fm(var(sl2(K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl2(K) over a field K of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion Fm^ of Fm with respect to the formal power series topology. Our results are more precise for m=2 when F2 is isomorphic to the Lie algebra L generated by two generic traceless 2×2 matrices. We give a complete description of the group of inner automorphisms of L^. As a consequence we obtain similar results for the automorphisms of the relatively free algebra Fm/Fmc+1=Fm(var(sl2(K))∩Nc) in the subvariety of var(sl2(K)) consisting of all nilpotent algebras of class at most c in var(sl2(K)).
Mots-clés : free Lie algebras, generic matrices, inner automorphisms, Baker–Campbell–Hausdorff formula. 
@article{ADM_2012_14_1_a5,
     author = {Vesselin Drensky and \c{S}ehmus F{\i}nd{\i}k},
     title = {Inner automorphisms of {Lie} algebras related with generic $2\times 2$ matrices},
     journal = {Algebra and discrete mathematics},
     pages = {49--70},
     publisher = {mathdoc},
     volume = {14},
     number = {1},
     year = {2012},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/ADM_2012_14_1_a5/}
}
TY  - JOUR
AU  - Vesselin Drensky
AU  - Şehmus Fındık
TI  - Inner automorphisms of Lie algebras related with generic $2\times 2$ matrices
JO  - Algebra and discrete mathematics
PY  - 2012
SP  - 49
EP  - 70
VL  - 14
IS  - 1
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/ADM_2012_14_1_a5/
LA  - en
ID  - ADM_2012_14_1_a5
ER  - 
%0 Journal Article
%A Vesselin Drensky
%A Şehmus Fındık
%T Inner automorphisms of Lie algebras related with generic $2\times 2$ matrices
%J Algebra and discrete mathematics
%D 2012
%P 49-70
%V 14
%N 1
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/ADM_2012_14_1_a5/
%G en
%F ADM_2012_14_1_a5
Vesselin Drensky; Şehmus Fındık. Inner automorphisms of Lie algebras related with generic $2\times 2$ matrices. Algebra and discrete mathematics, Tome 14 (2012) no. 1, pp. 49-70. https://geodesic-test.mathdoc.fr/item/ADM_2012_14_1_a5/

[1] VNU Science Press, Utrecht, 1987 | Zbl

[2] H. F. Baker, “On the exponential theorem for a simply transitive continuous group, and the calculation of the finite equations from the constants of structure”, Proc. London Math. Soc., 34 (1901), 91–129 | DOI | MR

[3] Math. USSR, Sb., 63 (1988), 363–374 | DOI | MR | Zbl

[4] N. Bourbaki, Éléments de mathématique. Fasc. XXXVII: Groupes et algèbres de Lie, v. II, Algèbres de Lie libres; v. III, Actualites scientifiques et industrielles, 1349, Groupes de Lie, Hermann, Paris, 1972 ; Translation: Elements of mathematics. Lie Groups and Lie Algebras, Chapters 1–3, Springer-Verlag, Berlin etc., 1989 | MR | Zbl | MR

[5] P. M. Cohn, “Subalgebras of free associative algebras”, Proc. London Math. Soc. (3), 14 (1964), 618–632 | DOI | MR | Zbl

[6] V. Drensky, Free Algebras and PI-Algebras, Springer-Verlag, Singapore, 2000 | MR | Zbl

[7] V. Drensky, “Defining relations for the algebra of invariants of $2\times 2$ matrices”, Algebr. Represent. Theory, 6 (2003), 193–214 | DOI | MR | Zbl

[8] V. Drensky, Ş. F{\i}nd{\i}k, “Inner and outer automorphisms of relatively free Lie algebras”, C. R. Acad. Bulg. Sci., 64 (2011), 315–324 | MR

[9] V. Drensky, Ş. F{\i}nd{\i}k, “Inner and outer automorphisms of free metabelian nilpotent Lie algebras”, Commun. Algebra (to appear) , arXiv: 1003.0350v1[math.RA]

[10] V. Drensky, C. K. Gupta, “Lie Automorphisms of the algebra of two generic $2\times 2$ matrices”, J. Algebra, 224 (2000), 336–355 | DOI | MR | Zbl

[11] V. S. Drensky, P. E. Koshlukov, “Weak polynomial identities for a vector space with a symmetric bilinear form”, Mathematics and Education in Mathematics, Proc. 16th Spring Conf. (Sunny Beach/Bulg., 1987), 213–219 | MR | Zbl

[12] V. Drensky, J. Szigeti, L. van Wyk, “Algebras generated by two quadratic elements”, Commun. Algebra, 39 (2011), 1344–1355 | DOI | MR | Zbl

[13] Ş. F{\i}nd{\i}k, “Outer automorphisms of Lie algebras related with generic $2\times 2$ matrices”, Serdica Math. J., 38 (2012), 273–296

[14] L. Gerritzen, “Taylor expansion of noncommutative power series with an application to the Hausdorff series”, J. Reine Angew. Math., 556 (2003), 113–125 | MR | Zbl

[15] C. E. Kofinas, A. I. Papistas, “Automorphisms of relatively free nilpotent Lie algebras”, Commun. Algebra, 38 (2010), 1575–1593 | DOI | MR | Zbl

[16] P. Koshlukov, “Weak polynomial identities for the matrix algebra of order two”, J. Algebra, 188 (1997), 610–625 | DOI | MR | Zbl

[17] P. Koshlukov, “Finitely based ideals of weak polynomial identities”, Commun. Algebra, 26 (1998), 3335–3359 | DOI | MR | Zbl

[18] V. Kurlin, “Compressed Drinfeld associators”, J. Algebra, 292 (2005), 184–242 | DOI | MR | Zbl

[19] V. Kurlin, “The Baker–Campbell–Hausdorff formula in the free metabelian Lie algebra”, J. Lie Theory, 17 (2007), 525–538 | MR | Zbl

[20] L. Le Bruyn, Trace Rings of Generic 2 by 2 Matrices, Mem. Amer. Math. Soc., 66, no. 363, 1987 | MR

[21] A. A. Mikhalev, V. Shpilrain, J. Yu, Combinatorial Methods. Free Groups, Polynomials, Free Algebras, CMS Books in Mathematics, Springer, New York, 2004 | DOI | MR

[22] Algebra and Logic, 12 (1973), 47–63 | DOI | MR | Zbl

[23] J.-P. Serre, Lie Algebras and Lie Groups, W. A. Benjamin Inc., New York–Amsterdam, 1965 | MR | Zbl