Notes for wClips-120314/0:11:34: Difference between revisions
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It is great to know the relationship with Lie bialgebras, as it is extremely relevant for the study of ${\mathcal A}^v$. Yet it is also good to know that $I{\mathfrak g}$ has a much simpler definition, that avoids some of the complexity. Namely, $I{\mathfrak g}$ is the semi-direct product ${\mathfrak g}^\star\rtimes{\mathfrak g}$, where ${\mathfrak g}$ acts on its dual ${\mathfrak g}^\star$ using the coadjoint action. The metric on $I{\mathfrak g}$ need not ever be explicitly used, yet it is the metric associated with the a norm on $I{\mathfrak g}$, which is simply the contraction map of ${\mathfrak g}^\star$ and ${\mathfrak g}$. |
It is great to know the relationship with Lie bialgebras, as it is extremely relevant for the study of ${\mathcal A}^v$. Yet it is also good to know that $I{\mathfrak g}$ has a much simpler definition, that avoids some of the complexity. Namely, $I{\mathfrak g}$ is the semi-direct product ${\mathfrak g}^\star\rtimes{\mathfrak g}$, where ${\mathfrak g}$ acts on its dual ${\mathfrak g}^\star$ using the coadjoint action. The metric on $I{\mathfrak g}$ need not ever be explicitly used, yet it is the metric associated with the a norm on $I{\mathfrak g}$, which is simply the contraction map of ${\mathfrak g}^\star$ and ${\mathfrak g}$. This definition appears roughly starting at minute 44:00 of this video. |
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Revision as of 19:30, 19 March 2012
It is great to know the relationship with Lie bialgebras, as it is extremely relevant for the study of ${\mathcal A}^v$. Yet it is also good to know that $I{\mathfrak g}$ has a much simpler definition, that avoids some of the complexity. Namely, $I{\mathfrak g}$ is the semi-direct product ${\mathfrak g}^\star\rtimes{\mathfrak g}$, where ${\mathfrak g}$ acts on its dual ${\mathfrak g}^\star$ using the coadjoint action. The metric on $I{\mathfrak g}$ need not ever be explicitly used, yet it is the metric associated with the a norm on $I{\mathfrak g}$, which is simply the contraction map of ${\mathfrak g}^\star$ and ${\mathfrak g}$. This definition appears roughly starting at minute 44:00 of this video.
