Tensor Product over quaternions
up vote
1
down vote
favorite
I am calculating a metric using a quaternionic approach, however, I am struggling to simplify (if possible) expressions such as:
$dq,g , otimes , dbar{q}, bar{g}$
where $q$ is a general quaternionic variable and $g$ is a unit quaternion. I know that the tensor product in this case is symmetric since it comes from a coordinate, $q_1$, and using $|dq_1|^2 = dq_1 dbar{q_1} = dbar{q_1}dq_1$ where $q_1 = qg$
Taking $d$: $dq_1 = dq,g+q,dg$ and similarly $dbar{q_1} = dbar{q}, bar{g} +bar{q},dbar{g}$.
Then:
$|dq_1|^2 = dq,g otimes dbar{q}, bar{g} + dq,g , otimes ,bar{q},dbar{g} + q, dg ,otimes ,dbar{q}, bar{g} + q, dg ,otimes , bar{q},dbar{g} $
So my question is whether I can get these tensor products into forms that have, say, $dg , otimes , dbar{g}$ together.
differential-geometry quaternions
asked yesterday
Bunneh
536
add a comment |
up vote
1
down vote
favorite
I am calculating a metric using a quaternionic approach, however, I am struggling to simplify (if possible) expressions such as:
$dq,g , otimes , dbar{q}, bar{g}$
where $q$ is a general quaternionic variable and $g$ is a unit quaternion. I know that the tensor product in this case is symmetric since it comes from a coordinate, $q_1$, and using $|dq_1|^2 = dq_1 dbar{q_1} = dbar{q_1}dq_1$ where $q_1 = qg$
Taking $d$: $dq_1 = dq,g+q,dg$ and similarly $dbar{q_1} = dbar{q}, bar{g} +bar{q},dbar{g}$.
Then:
$|dq_1|^2 = dq,g otimes dbar{q}, bar{g} + dq,g , otimes ,bar{q},dbar{g} + q, dg ,otimes ,dbar{q}, bar{g} + q, dg ,otimes , bar{q},dbar{g} $
So my question is whether I can get these tensor products into forms that have, say, $dg , otimes , dbar{g}$ together.
differential-geometry quaternions
asked yesterday
Bunneh
536
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am calculating a metric using a quaternionic approach, however, I am struggling to simplify (if possible) expressions such as:
$dq,g , otimes , dbar{q}, bar{g}$
where $q$ is a general quaternionic variable and $g$ is a unit quaternion. I know that the tensor product in this case is symmetric since it comes from a coordinate, $q_1$, and using $|dq_1|^2 = dq_1 dbar{q_1} = dbar{q_1}dq_1$ where $q_1 = qg$
Taking $d$: $dq_1 = dq,g+q,dg$ and similarly $dbar{q_1} = dbar{q}, bar{g} +bar{q},dbar{g}$.
Then:
$|dq_1|^2 = dq,g otimes dbar{q}, bar{g} + dq,g , otimes ,bar{q},dbar{g} + q, dg ,otimes ,dbar{q}, bar{g} + q, dg ,otimes , bar{q},dbar{g} $
So my question is whether I can get these tensor products into forms that have, say, $dg , otimes , dbar{g}$ together.
differential-geometry quaternions
asked yesterday
Bunneh
536
I am calculating a metric using a quaternionic approach, however, I am struggling to simplify (if possible) expressions such as:
$dq,g , otimes , dbar{q}, bar{g}$
where $q$ is a general quaternionic variable and $g$ is a unit quaternion. I know that the tensor product in this case is symmetric since it comes from a coordinate, $q_1$, and using $|dq_1|^2 = dq_1 dbar{q_1} = dbar{q_1}dq_1$ where $q_1 = qg$
Taking $d$: $dq_1 = dq,g+q,dg$ and similarly $dbar{q_1} = dbar{q}, bar{g} +bar{q},dbar{g}$.
Then:
$|dq_1|^2 = dq,g otimes dbar{q}, bar{g} + dq,g , otimes ,bar{q},dbar{g} + q, dg ,otimes ,dbar{q}, bar{g} + q, dg ,otimes , bar{q},dbar{g} $
So my question is whether I can get these tensor products into forms that have, say, $dg , otimes , dbar{g}$ together.
differential-geometry quaternions
differential-geometry quaternions
asked yesterday
Bunneh
536
asked yesterday
Bunneh
536
asked yesterday
Bunneh
536
asked yesterday
Bunneh
536
asked yesterday
Bunneh
536
536
add a comment |
add a comment |
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005029%2ftensor-product-over-quaternions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
