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

Serdica Mathematical Journal (2012)

- Volume: 38, Issue: 1-3, page 273-296
- ISSN: 1310-6600

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topFındık, 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 -

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