Outer Automorphisms of Lie Algebras related with Generic 2×2 Matrices

Fındık, Şehmus. "Outer Automorphisms of Lie Algebras related with Generic 2×2 Matrices." Serdica Mathematical Journal 38.1-3 (2012): 273-296. <http://eudml.org/doc/288267>.

@article{Fındık2012,
abstract = {2010 Mathematics Subject Classification: 17B01, 17B30, 17B40, 16R30.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. 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 outer automorphisms of the completion L^ of L with respect to the formal power series topology and of the related associative algebra W^. As a consequence we obtain similar results for the automorphisms of the relatively free algebra F2/F2^(c+1) = F2(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)) and for W/W^(c+1). We show that such automorphisms are Z2-graded, i.e., they map the linear combinations of elements of odd, respectively even degree to linear combinations of the same kind.},
author = {Fındık, Şehmus},
journal = {Serdica Mathematical Journal},
keywords = {Free Lie Algebras; Generic Matrices; Inner Automorphisms; Outer Automorphisms},
language = {eng},
number = {1-3},
pages = {273-296},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Outer Automorphisms of Lie Algebras related with Generic 2×2 Matrices},
url = {http://eudml.org/doc/288267},
volume = {38},
year = {2012},
}

TY - JOUR
AU - Fındık, Şehmus
TI - Outer Automorphisms of Lie Algebras related with Generic 2×2 Matrices
JO - Serdica Mathematical Journal
PY - 2012
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 38
IS - 1-3
SP - 273
EP - 296
AB - 2010 Mathematics Subject Classification: 17B01, 17B30, 17B40, 16R30.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. 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 outer automorphisms of the completion L^ of L with respect to the formal power series topology and of the related associative algebra W^. As a consequence we obtain similar results for the automorphisms of the relatively free algebra F2/F2^(c+1) = F2(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)) and for W/W^(c+1). We show that such automorphisms are Z2-graded, i.e., they map the linear combinations of elements of odd, respectively even degree to linear combinations of the same kind.
LA - eng
KW - Free Lie Algebras; Generic Matrices; Inner Automorphisms; Outer Automorphisms
UR - http://eudml.org/doc/288267
ER -