Mixing
Second-order term of Baker-Campbell-Hausdorff formula
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Let $G$ be a Lie group with Lie algebra $mathfrak g$. I'm trying to prove the first two terms of the Baker-Campbell-Hausdorff formula for $exp(tX)exp(tY)$ satisfy $$
(exp tX)(exp tY) = expbig(t(X+Y) + frac 1 2 t^2 [X,Y] + t^3 hat Z(t)big)
$$
for some smooth function $hat Z: (-epsilon, epsilon) to mathfrak g$ and some $epsilon > 0$, for every $X, Y in mathfrak g$.
What I've tried: Since $exp : mathfrak g to G$ restricts to a diffeomorphism from a neighborhood of $0 in mathfrakg$ to a neighborhood of $e in G$, the map $phi : (-epsilon, epsilon) to mathfrak g$ defined by
$$
phi(t) = exp^-1(exp tX exp tY)
$$
is smooth for some $epsilon > 0$. So the problem amounts to proving $phi''(0) = [X,Y]$ and applying Taylor's theorem. But I'm not sure how to do this without knowing the derivative of multiplication $G times G to G$ at general points $(g,h) in G times G$. I'm wondering if there's a more clever approach having something to do with the adjoint representations $mathrmAd: G to mathrmGL(mathfrak g)$ and $mathrmAd_* = mathrmad: mathfrak g to mathfrakgl(mathfrakg)$ since $mathrmad(X)Y = [X,Y]$. Any advice would be appreciated.
differential-geometry lie-groups lie-algebras smooth-manifolds
asked Jul 21 at 16:25
D Ford
417212
add a comment |Â
up vote
1
down vote
favorite
Let $G$ be a Lie group with Lie algebra $mathfrak g$. I'm trying to prove the first two terms of the Baker-Campbell-Hausdorff formula for $exp(tX)exp(tY)$ satisfy $$
(exp tX)(exp tY) = expbig(t(X+Y) + frac 1 2 t^2 [X,Y] + t^3 hat Z(t)big)
$$
for some smooth function $hat Z: (-epsilon, epsilon) to mathfrak g$ and some $epsilon > 0$, for every $X, Y in mathfrak g$.
What I've tried: Since $exp : mathfrak g to G$ restricts to a diffeomorphism from a neighborhood of $0 in mathfrakg$ to a neighborhood of $e in G$, the map $phi : (-epsilon, epsilon) to mathfrak g$ defined by
$$
phi(t) = exp^-1(exp tX exp tY)
$$
is smooth for some $epsilon > 0$. So the problem amounts to proving $phi''(0) = [X,Y]$ and applying Taylor's theorem. But I'm not sure how to do this without knowing the derivative of multiplication $G times G to G$ at general points $(g,h) in G times G$. I'm wondering if there's a more clever approach having something to do with the adjoint representations $mathrmAd: G to mathrmGL(mathfrak g)$ and $mathrmAd_* = mathrmad: mathfrak g to mathfrakgl(mathfrakg)$ since $mathrmad(X)Y = [X,Y]$. Any advice would be appreciated.
differential-geometry lie-groups lie-algebras smooth-manifolds
asked Jul 21 at 16:25
D Ford
417212
1
Maybe this is useful? en.wikipedia.org/wiki/Derivative_of_the_exponential_map
– md2perpe
Jul 22 at 20:40
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $G$ be a Lie group with Lie algebra $mathfrak g$. I'm trying to prove the first two terms of the Baker-Campbell-Hausdorff formula for $exp(tX)exp(tY)$ satisfy $$
(exp tX)(exp tY) = expbig(t(X+Y) + frac 1 2 t^2 [X,Y] + t^3 hat Z(t)big)
$$
for some smooth function $hat Z: (-epsilon, epsilon) to mathfrak g$ and some $epsilon > 0$, for every $X, Y in mathfrak g$.
What I've tried: Since $exp : mathfrak g to G$ restricts to a diffeomorphism from a neighborhood of $0 in mathfrakg$ to a neighborhood of $e in G$, the map $phi : (-epsilon, epsilon) to mathfrak g$ defined by
$$
phi(t) = exp^-1(exp tX exp tY)
$$
is smooth for some $epsilon > 0$. So the problem amounts to proving $phi''(0) = [X,Y]$ and applying Taylor's theorem. But I'm not sure how to do this without knowing the derivative of multiplication $G times G to G$ at general points $(g,h) in G times G$. I'm wondering if there's a more clever approach having something to do with the adjoint representations $mathrmAd: G to mathrmGL(mathfrak g)$ and $mathrmAd_* = mathrmad: mathfrak g to mathfrakgl(mathfrakg)$ since $mathrmad(X)Y = [X,Y]$. Any advice would be appreciated.
differential-geometry lie-groups lie-algebras smooth-manifolds
asked Jul 21 at 16:25
D Ford
417212
Let $G$ be a Lie group with Lie algebra $mathfrak g$. I'm trying to prove the first two terms of the Baker-Campbell-Hausdorff formula for $exp(tX)exp(tY)$ satisfy $$
(exp tX)(exp tY) = expbig(t(X+Y) + frac 1 2 t^2 [X,Y] + t^3 hat Z(t)big)
$$
for some smooth function $hat Z: (-epsilon, epsilon) to mathfrak g$ and some $epsilon > 0$, for every $X, Y in mathfrak g$.
What I've tried: Since $exp : mathfrak g to G$ restricts to a diffeomorphism from a neighborhood of $0 in mathfrakg$ to a neighborhood of $e in G$, the map $phi : (-epsilon, epsilon) to mathfrak g$ defined by
$$
phi(t) = exp^-1(exp tX exp tY)
$$
is smooth for some $epsilon > 0$. So the problem amounts to proving $phi''(0) = [X,Y]$ and applying Taylor's theorem. But I'm not sure how to do this without knowing the derivative of multiplication $G times G to G$ at general points $(g,h) in G times G$. I'm wondering if there's a more clever approach having something to do with the adjoint representations $mathrmAd: G to mathrmGL(mathfrak g)$ and $mathrmAd_* = mathrmad: mathfrak g to mathfrakgl(mathfrakg)$ since $mathrmad(X)Y = [X,Y]$. Any advice would be appreciated.
differential-geometry lie-groups lie-algebras smooth-manifolds
asked Jul 21 at 16:25
D Ford
417212
asked Jul 21 at 16:25
D Ford
417212
asked Jul 21 at 16:25
D Ford
417212
asked Jul 21 at 16:25
D Ford
417212
417212
1
Maybe this is useful? en.wikipedia.org/wiki/Derivative_of_the_exponential_map
– md2perpe
Jul 22 at 20:40
add a comment |Â
1
Maybe this is useful? en.wikipedia.org/wiki/Derivative_of_the_exponential_map
– md2perpe
Jul 22 at 20:40
1
1
Maybe this is useful? en.wikipedia.org/wiki/Derivative_of_the_exponential_map
– md2perpe
Jul 22 at 20:40
Maybe this is useful? en.wikipedia.org/wiki/Derivative_of_the_exponential_map
– md2perpe
Jul 22 at 20:40
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2858640%2fsecond-order-term-of-baker-campbell-hausdorff-formula%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
Maybe this is useful? en.wikipedia.org/wiki/Derivative_of_the_exponential_map
– md2perpe
Jul 22 at 20:40