Tensor Product over quaternions











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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







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    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







    share|cite|improve this question
























      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







      share|cite|improve this question













      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






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