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Let $\frak{g}$ be a complex Lie algebra $\frak{g}$, and $R:\frak{g} \to $End$(V)$, a representation for some finite dimensional complex vector space $V$. As is well-known, we can construct from $R$ its dual representation $$ R^*:{\frak g} \to \text{End}(V^*), ~~~~~~~~ g \mapsto \big(R^*(g)(f):v \mapsto - f(R(g)v)\big), $$ and its conjugate representation $$ \overline{R}:{\frak g} \to \text{End}(\overline{V}), ~~~~~~~~ g \mapsto \big(\overline{R}(g):\overline{v} \mapsto \overline{R(g)v}\big), $$ where the ``over-bar'' means complex conjugation.

I would like to ask, for which type of Lie algebra representations are the dual and conjugate representations isomorphic? As a (very) simple example, take the Lie algebra $\frak{u}_1$. Unless I am mistaken, all its finite-dimensional modules have isomorphic duals and conjugates.

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migrated from mathoverflow.net Aug 2 '14 at 12:12

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Three quick comments: 1) I assume from the context that your Lie algebra is over $\mathbb{C}$. 2) To make $R^∗$ a Lie algebra representation, you need to add a negative sign on the right side. 3) You need to re-examine your definition of "conjugate representation". Examples would clarify what is going on. –  Jim Humphreys Jul 31 '14 at 14:07
Check it once and for all for the Lie algebra $End(V)$ itself. I'm voting to close as not research-level. –  Allen Knutson Jul 31 '14 at 15:21
What kind of complex Lie algebra is $\mathfrak{u}_1$ ? –  abx Jul 31 '14 at 16:16

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