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Let $G$ be an algebraic group (say, connected). Given a rep. $\rho:G\to GL(V)$ there is a dual rep. $\rho^{\vee}:G\to GL(V^{\vee})$ defined by $\rho^{\vee}(g)\phi =\phi\circ \rho(g^{-1})$. My question is when the adjoint representation of $G$ is isomorphic to its own dual?

In characteristic $0$, If $G$ is semi-simple than its lie algebra is also semi-simple and therefore the killing form which is always bi-linear, symmetric and $G$-invariant is also non-degenerate and therefore induces an isomorphism between the adjoint representation and its dual. What about the other cases?

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Have you tried MathOverflow? – WetSavannaAnimal aka Rod Vance Sep 29 '14 at 2:21

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