TY - JOUR

T1 - Extensions and automorphisms of Lie algebras

AU - Bardakov, Valeriy G.

AU - Singh, Mahender

PY - 2017/9/1

Y1 - 2017/9/1

N2 - Let 0 -> A -> L -> B -> 0 be a short exact sequence of Lie algebras over a field F, where A is abelian. We show that the obstruction for a pair of automorphisms in Aut(A) x Aut(B) to be induced by an automorphism in Aut(L) lies in the Lie algebra cohomology H-2(B; A). As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in Aut (L-n,2((1))) x Aut (L-n,2(ab)) to be induced by an automorphism in Aut (L-n,L-2 ), where (L-n,L-2) is a free nilpotent Lie algebra of rank n and step 2.

AB - Let 0 -> A -> L -> B -> 0 be a short exact sequence of Lie algebras over a field F, where A is abelian. We show that the obstruction for a pair of automorphisms in Aut(A) x Aut(B) to be induced by an automorphism in Aut(L) lies in the Lie algebra cohomology H-2(B; A). As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in Aut (L-n,2((1))) x Aut (L-n,2(ab)) to be induced by an automorphism in Aut (L-n,L-2 ), where (L-n,L-2) is a free nilpotent Lie algebra of rank n and step 2.

KW - Automorphism of Lie algebra

KW - cohomology of Lie algebra

KW - extension of Lie algebras

KW - free nilpotent Lie algebra

UR - http://www.scopus.com/inward/record.url?scp=84990188135&partnerID=8YFLogxK

U2 - 10.1142/S0219498817501626

DO - 10.1142/S0219498817501626

M3 - Article

AN - SCOPUS:84990188135

VL - 16

JO - Journal of Algebra and its Applications

JF - Journal of Algebra and its Applications

SN - 0219-4988

IS - 9

M1 - 1750162

ER -