Geodesic
Outer Automorphisms of Lie Algebras related with Generic 2×2 Matrices
Serdica Mathematical Journal, Tome 38 (2012) no. 1-3, pp. 273-296.

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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.
Keywords: Free Lie Algebras, Generic Matrices, Inner Automorphisms, Outer Automorphisms 
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     title = {Outer {Automorphisms} of {Lie} {Algebras} related with {Generic} 2{\texttimes}2 {Matrices}},
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Fındık, Şehmus. Outer Automorphisms of Lie Algebras related with Generic 2×2 Matrices. Serdica Mathematical Journal, Tome 38 (2012) no. 1-3, pp. 273-296. http://geodesic.mathdoc.fr/item/SMJ2_2012_38_1-3_a13/