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The center of nilpotent Lie algebra and the last abelian term of the derived series
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Let $L$ be a finite-dimensional nilpotent lie algebra. Consider the last non-trivial term of the derived series $A:=[L^n,L^n]$. Is it true that $A$ is equal to the center $Z$ of $L$?
lie-algebras
asked Jul 19 at 5:50
Ronald
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Let $L$ be a finite-dimensional nilpotent lie algebra. Consider the last non-trivial term of the derived series $A:=[L^n,L^n]$. Is it true that $A$ is equal to the center $Z$ of $L$?
lie-algebras
asked Jul 19 at 5:50
Ronald
1,5991821
1
(a) Every metabelian Lie algebra of nilpotency length $ge 3$ is a counterexample. (b) There are two nonabelian complex 4-dimensional nilpotent Lie algebras (up to isomorphism) and both are counterexamples (one is covered by Aristide's answer and the other is covered by (a)).
– YCor
Jul 19 at 20:23
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up vote
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down vote
favorite
Let $L$ be a finite-dimensional nilpotent lie algebra. Consider the last non-trivial term of the derived series $A:=[L^n,L^n]$. Is it true that $A$ is equal to the center $Z$ of $L$?
lie-algebras
asked Jul 19 at 5:50
Ronald
1,5991821
Let $L$ be a finite-dimensional nilpotent lie algebra. Consider the last non-trivial term of the derived series $A:=[L^n,L^n]$. Is it true that $A$ is equal to the center $Z$ of $L$?
lie-algebras
asked Jul 19 at 5:50
Ronald
1,5991821
asked Jul 19 at 5:50
Ronald
1,5991821
asked Jul 19 at 5:50
Ronald
1,5991821
asked Jul 19 at 5:50
Ronald
1,5991821
1,5991821
1
(a) Every metabelian Lie algebra of nilpotency length $ge 3$ is a counterexample. (b) There are two nonabelian complex 4-dimensional nilpotent Lie algebras (up to isomorphism) and both are counterexamples (one is covered by Aristide's answer and the other is covered by (a)).
– YCor
Jul 19 at 20:23
add a comment |Â
1
(a) Every metabelian Lie algebra of nilpotency length $ge 3$ is a counterexample. (b) There are two nonabelian complex 4-dimensional nilpotent Lie algebras (up to isomorphism) and both are counterexamples (one is covered by Aristide's answer and the other is covered by (a)).
– YCor
Jul 19 at 20:23
1
1
(a) Every metabelian Lie algebra of nilpotency length $ge 3$ is a counterexample. (b) There are two nonabelian complex 4-dimensional nilpotent Lie algebras (up to isomorphism) and both are counterexamples (one is covered by Aristide's answer and the other is covered by (a)).
– YCor
Jul 19 at 20:23
(a) Every metabelian Lie algebra of nilpotency length $ge 3$ is a counterexample. (b) There are two nonabelian complex 4-dimensional nilpotent Lie algebras (up to isomorphism) and both are counterexamples (one is covered by Aristide's answer and the other is covered by (a)).
– YCor
Jul 19 at 20:23
add a comment |Â
1 Answer
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No, consider any nilpotent algebra $L$ which is not commutative. Let $M$ be a commutative algebra. Suppose that $L^nneq 0$ and $L^n+1=0$. The center of $Loplus M$ is $Z(L) oplus M$, and$(Loplus M)^n=L^n$, $(Loplus M)^n+1=0$.
answered Jul 19 at 6:50
Tsemo Aristide
51.2k11243
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1 Answer
1
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oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
No, consider any nilpotent algebra $L$ which is not commutative. Let $M$ be a commutative algebra. Suppose that $L^nneq 0$ and $L^n+1=0$. The center of $Loplus M$ is $Z(L) oplus M$, and$(Loplus M)^n=L^n$, $(Loplus M)^n+1=0$.
answered Jul 19 at 6:50
Tsemo Aristide
51.2k11243
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2
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No, consider any nilpotent algebra $L$ which is not commutative. Let $M$ be a commutative algebra. Suppose that $L^nneq 0$ and $L^n+1=0$. The center of $Loplus M$ is $Z(L) oplus M$, and$(Loplus M)^n=L^n$, $(Loplus M)^n+1=0$.
answered Jul 19 at 6:50
Tsemo Aristide
51.2k11243
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
No, consider any nilpotent algebra $L$ which is not commutative. Let $M$ be a commutative algebra. Suppose that $L^nneq 0$ and $L^n+1=0$. The center of $Loplus M$ is $Z(L) oplus M$, and$(Loplus M)^n=L^n$, $(Loplus M)^n+1=0$.
answered Jul 19 at 6:50
Tsemo Aristide
51.2k11243
No, consider any nilpotent algebra $L$ which is not commutative. Let $M$ be a commutative algebra. Suppose that $L^nneq 0$ and $L^n+1=0$. The center of $Loplus M$ is $Z(L) oplus M$, and$(Loplus M)^n=L^n$, $(Loplus M)^n+1=0$.
answered Jul 19 at 6:50
Tsemo Aristide
51.2k11243
answered Jul 19 at 6:50
Tsemo Aristide
51.2k11243
answered Jul 19 at 6:50
Tsemo Aristide
51.2k11243
answered Jul 19 at 6:50
Tsemo Aristide
51.2k11243
51.2k11243
add a comment |Â
add a comment |Â
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1
(a) Every metabelian Lie algebra of nilpotency length $ge 3$ is a counterexample. (b) There are two nonabelian complex 4-dimensional nilpotent Lie algebras (up to isomorphism) and both are counterexamples (one is covered by Aristide's answer and the other is covered by (a)).
– YCor
Jul 19 at 20:23