Simple couter-example of preservation of Jordan-Chevalley decomposition
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I was going through chapter 9 of Fulton and Harris, as I am teaching the course, and ran across the example of the Lie algebra $mathbb{C}$, with the following representation:
begin{equation}
rho: t mapsto begin{pmatrix} t & t \ 0 & 0 end{pmatrix}
end{equation}
The book presents this as an example where the matrix is ``neither diagonalizable nor nilpotent'', and where neither the semisimple and nilpotent parts will be in the image of $rho$.
The point they want to showcase is how the semisimplicity of the Lie algebra is crucial.
I'm having trouble with this example, as the matrix in question is already semisimple as it stands.
Can anyone `fix' this example, or show me another neat representation of $mathbb{C}$ where J.C. decomposition is not preserved?
representation-theory
asked Nov 16 at 11:51
Andrés Collinucci
111
add a comment |
up vote
2
down vote
favorite
I was going through chapter 9 of Fulton and Harris, as I am teaching the course, and ran across the example of the Lie algebra $mathbb{C}$, with the following representation:
begin{equation}
rho: t mapsto begin{pmatrix} t & t \ 0 & 0 end{pmatrix}
end{equation}
The book presents this as an example where the matrix is ``neither diagonalizable nor nilpotent'', and where neither the semisimple and nilpotent parts will be in the image of $rho$.
The point they want to showcase is how the semisimplicity of the Lie algebra is crucial.
I'm having trouble with this example, as the matrix in question is already semisimple as it stands.
Can anyone `fix' this example, or show me another neat representation of $mathbb{C}$ where J.C. decomposition is not preserved?
representation-theory
asked Nov 16 at 11:51
Andrés Collinucci
111
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I was going through chapter 9 of Fulton and Harris, as I am teaching the course, and ran across the example of the Lie algebra $mathbb{C}$, with the following representation:
begin{equation}
rho: t mapsto begin{pmatrix} t & t \ 0 & 0 end{pmatrix}
end{equation}
The book presents this as an example where the matrix is ``neither diagonalizable nor nilpotent'', and where neither the semisimple and nilpotent parts will be in the image of $rho$.
The point they want to showcase is how the semisimplicity of the Lie algebra is crucial.
I'm having trouble with this example, as the matrix in question is already semisimple as it stands.
Can anyone `fix' this example, or show me another neat representation of $mathbb{C}$ where J.C. decomposition is not preserved?
representation-theory
asked Nov 16 at 11:51
Andrés Collinucci
111
I was going through chapter 9 of Fulton and Harris, as I am teaching the course, and ran across the example of the Lie algebra $mathbb{C}$, with the following representation:
begin{equation}
rho: t mapsto begin{pmatrix} t & t \ 0 & 0 end{pmatrix}
end{equation}
The book presents this as an example where the matrix is ``neither diagonalizable nor nilpotent'', and where neither the semisimple and nilpotent parts will be in the image of $rho$.
The point they want to showcase is how the semisimplicity of the Lie algebra is crucial.
I'm having trouble with this example, as the matrix in question is already semisimple as it stands.
Can anyone `fix' this example, or show me another neat representation of $mathbb{C}$ where J.C. decomposition is not preserved?
representation-theory
representation-theory
asked Nov 16 at 11:51
Andrés Collinucci
111
asked Nov 16 at 11:51
Andrés Collinucci
111
asked Nov 16 at 11:51
Andrés Collinucci
111
asked Nov 16 at 11:51
Andrés Collinucci
111
asked Nov 16 at 11:51
Andrés Collinucci
111
111
add a comment |
add a comment |
1 Answer
1
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oldest
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up vote
1
down vote
Sure, just take the representation $$t mapsto left( begin{matrix} t & t \ 0 & t end{matrix} right)=left( begin{matrix} t & 0 \ 0 & t end{matrix} right)+left( begin{matrix} 0 & t \ 0 & 0 end{matrix} right).$$
answered Nov 16 at 14:37
Stephen
10.4k12237
Thanks! Nice and simple.
– Andrés Collinucci
Nov 17 at 18:15
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Sure, just take the representation $$t mapsto left( begin{matrix} t & t \ 0 & t end{matrix} right)=left( begin{matrix} t & 0 \ 0 & t end{matrix} right)+left( begin{matrix} 0 & t \ 0 & 0 end{matrix} right).$$
answered Nov 16 at 14:37
Stephen
10.4k12237
Thanks! Nice and simple.
– Andrés Collinucci
Nov 17 at 18:15
add a comment |
up vote
1
down vote
Sure, just take the representation $$t mapsto left( begin{matrix} t & t \ 0 & t end{matrix} right)=left( begin{matrix} t & 0 \ 0 & t end{matrix} right)+left( begin{matrix} 0 & t \ 0 & 0 end{matrix} right).$$
answered Nov 16 at 14:37
Stephen
10.4k12237
Thanks! Nice and simple.
– Andrés Collinucci
Nov 17 at 18:15
add a comment |
up vote
1
down vote
up vote
1
down vote
Sure, just take the representation $$t mapsto left( begin{matrix} t & t \ 0 & t end{matrix} right)=left( begin{matrix} t & 0 \ 0 & t end{matrix} right)+left( begin{matrix} 0 & t \ 0 & 0 end{matrix} right).$$
answered Nov 16 at 14:37
Stephen
10.4k12237
Sure, just take the representation $$t mapsto left( begin{matrix} t & t \ 0 & t end{matrix} right)=left( begin{matrix} t & 0 \ 0 & t end{matrix} right)+left( begin{matrix} 0 & t \ 0 & 0 end{matrix} right).$$
answered Nov 16 at 14:37
Stephen
10.4k12237
answered Nov 16 at 14:37
Stephen
10.4k12237
answered Nov 16 at 14:37
Stephen
10.4k12237
answered Nov 16 at 14:37
Stephen
10.4k12237
10.4k12237
Thanks! Nice and simple.
– Andrés Collinucci
Nov 17 at 18:15
add a comment |
Thanks! Nice and simple.
– Andrés Collinucci
Nov 17 at 18:15
Thanks! Nice and simple.
– Andrés Collinucci
Nov 17 at 18:15
Thanks! Nice and simple.
– Andrés Collinucci
Nov 17 at 18:15
add a comment |
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