Doubt in understading Proof of Matrix lie group and Lie algebra locally homemorphicabout orthogonal complementInvariance under conjugation, equivalent in Lie Group and Lie Algebra?If the Lie algebra is a direct sum, then the Lie group is a direct product?Lie group and Lie algebra automorphismsOrthogonal complement is identity minus outer-product?Lie algebra and Lie Group correspondenceDimension of Lie algebra and Lie groupUnderstanding Lie algebra of matrix Lie groupUnderstanding the proof of the theorem that the exponential map is locally one-to-one at 1 and onto“If $g$ is semisimple, It is not too hard to see that $H^2(g,a)=0$. With a little supplementary argument…”
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Doubt in understading Proof of Matrix lie group and Lie algebra locally homemorphic
about orthogonal complementInvariance under conjugation, equivalent in Lie Group and Lie Algebra?If the Lie algebra is a direct sum, then the Lie group is a direct product?Lie group and Lie algebra automorphismsOrthogonal complement is identity minus outer-product?Lie algebra and Lie Group correspondenceDimension of Lie algebra and Lie groupUnderstanding Lie algebra of matrix Lie groupUnderstanding the proof of the theorem that the exponential map is locally one-to-one at 1 and onto“If $g$ is semisimple, It is not too hard to see that $H^2(g,a)=0$. With a little supplementary argument…”
$begingroup$
I was reading Brian C Hall Lie Group book In that I encountered following proof .
I understand Whole proof. But have one doubt Why Auther take Orthogonal complement into consideration As I think Direct sum also be ok.
I thought hard But I do not understand why Author requires orthogonal complement.
I uploaded photo Because I do not want to miss any details ..
I would be thankful if someone helps me
Any help will be appreciated
real-analysis linear-algebra proof-explanation lie-groups lie-algebras
asked Mar 22 at 7:50
SRJSRJ
1,8961620
$endgroup$
add a comment |
$begingroup$
I was reading Brian C Hall Lie Group book In that I encountered following proof .
I understand Whole proof. But have one doubt Why Auther take Orthogonal complement into consideration As I think Direct sum also be ok.
I thought hard But I do not understand why Author requires orthogonal complement.
I uploaded photo Because I do not want to miss any details ..
I would be thankful if someone helps me
Any help will be appreciated
real-analysis linear-algebra proof-explanation lie-groups lie-algebras
asked Mar 22 at 7:50
SRJSRJ
1,8961620
$endgroup$
add a comment |
$begingroup$
I was reading Brian C Hall Lie Group book In that I encountered following proof .
I understand Whole proof. But have one doubt Why Auther take Orthogonal complement into consideration As I think Direct sum also be ok.
I thought hard But I do not understand why Author requires orthogonal complement.
I uploaded photo Because I do not want to miss any details ..
I would be thankful if someone helps me
Any help will be appreciated
real-analysis linear-algebra proof-explanation lie-groups lie-algebras
asked Mar 22 at 7:50
SRJSRJ
1,8961620
$endgroup$
I was reading Brian C Hall Lie Group book In that I encountered following proof .
I understand Whole proof. But have one doubt Why Auther take Orthogonal complement into consideration As I think Direct sum also be ok.
I thought hard But I do not understand why Author requires orthogonal complement.
I uploaded photo Because I do not want to miss any details ..
I would be thankful if someone helps me
Any help will be appreciated
real-analysis linear-algebra proof-explanation lie-groups lie-algebras
real-analysis linear-algebra proof-explanation lie-groups lie-algebras
asked Mar 22 at 7:50
SRJSRJ
1,8961620
asked Mar 22 at 7:50
SRJSRJ
1,8961620
asked Mar 22 at 7:50
SRJSRJ
1,8961620
asked Mar 22 at 7:50
SRJSRJ
1,8961620
asked Mar 22 at 7:50
SRJSRJ
1,8961620
1,8961620
add a comment |
add a comment |
1 Answer
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$begingroup$
The "orthogonal" part of that isn't important. What's crucial is that it's a complementary space, so that $mathfrakgoplus D$ is exactly the full space $M_n(mathbbC)$. We use this $D$ to extend the exponential map from $mathfrakg$ to the full space - but not as the standard matrix exponential (We only have $e^X+Y=e^Xe^Y$ if $X$ and $Y$ commute). We use the inverse of this extended map, and that's why we needed a complementary space.
Why use the (real) orthogonal complement? Because it's a simple complementary space we can just write down and move on from.
answered Mar 22 at 8:06
jmerryjmerry
17k11633
$endgroup$
add a comment |
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$begingroup$
The "orthogonal" part of that isn't important. What's crucial is that it's a complementary space, so that $mathfrakgoplus D$ is exactly the full space $M_n(mathbbC)$. We use this $D$ to extend the exponential map from $mathfrakg$ to the full space - but not as the standard matrix exponential (We only have $e^X+Y=e^Xe^Y$ if $X$ and $Y$ commute). We use the inverse of this extended map, and that's why we needed a complementary space.
Why use the (real) orthogonal complement? Because it's a simple complementary space we can just write down and move on from.
answered Mar 22 at 8:06
jmerryjmerry
17k11633
$endgroup$
add a comment |
$begingroup$
The "orthogonal" part of that isn't important. What's crucial is that it's a complementary space, so that $mathfrakgoplus D$ is exactly the full space $M_n(mathbbC)$. We use this $D$ to extend the exponential map from $mathfrakg$ to the full space - but not as the standard matrix exponential (We only have $e^X+Y=e^Xe^Y$ if $X$ and $Y$ commute). We use the inverse of this extended map, and that's why we needed a complementary space.
Why use the (real) orthogonal complement? Because it's a simple complementary space we can just write down and move on from.
answered Mar 22 at 8:06
jmerryjmerry
17k11633
$endgroup$
add a comment |
$begingroup$
The "orthogonal" part of that isn't important. What's crucial is that it's a complementary space, so that $mathfrakgoplus D$ is exactly the full space $M_n(mathbbC)$. We use this $D$ to extend the exponential map from $mathfrakg$ to the full space - but not as the standard matrix exponential (We only have $e^X+Y=e^Xe^Y$ if $X$ and $Y$ commute). We use the inverse of this extended map, and that's why we needed a complementary space.
Why use the (real) orthogonal complement? Because it's a simple complementary space we can just write down and move on from.
answered Mar 22 at 8:06
jmerryjmerry
17k11633
$endgroup$
The "orthogonal" part of that isn't important. What's crucial is that it's a complementary space, so that $mathfrakgoplus D$ is exactly the full space $M_n(mathbbC)$. We use this $D$ to extend the exponential map from $mathfrakg$ to the full space - but not as the standard matrix exponential (We only have $e^X+Y=e^Xe^Y$ if $X$ and $Y$ commute). We use the inverse of this extended map, and that's why we needed a complementary space.
Why use the (real) orthogonal complement? Because it's a simple complementary space we can just write down and move on from.
answered Mar 22 at 8:06
jmerryjmerry
17k11633
answered Mar 22 at 8:06
jmerryjmerry
17k11633
answered Mar 22 at 8:06
jmerryjmerry
17k11633
answered Mar 22 at 8:06
jmerryjmerry
17k11633
17k11633
add a comment |
add a comment |
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