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Let $G$ be a Lie group and $V$ be a vector space. Let $\rho_{l} : G \times V \to V$ be a left representation and $\rho_r:V \times G \to V$ be a right representation which commutes with $\rho_l$ in the sense that $\rho_l(g_1 , \rho_r(v , g_2) ) = \rho_r( \rho_l(g_1, v) , g_2)$. Then the group multiplication $(g_1,v_1) \cdot (g_2,v_2) = (g_1 \cdot g_2 \quad,\quad \rho_r(v_1,g_2) + \rho_l(g_1,v_2))$ makes the set $G \times V$ into a Lie group. The identity is $(e_G, e_H)$ and the inverse of $(g,h)$ is $(g^{-1}, - \rho_l[ g^{-1} , \rho_r(h,g^{-1} ) ] )$ and associativity can be verified by hand. This Lie group appears to be some sort of sum of a left semi-direct product with a right semi-direct product. Does it have a name?

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up vote 4 down vote accepted

It is isomorphic to the semidirect product of $G$ with $V$ for the left representation $g\mapsto (v\mapsto g.v.g^{-1})$.

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isomorphism is $(g,v) \mapsto (g,v \cdot g^{-1})$ – hoj201 Jul 28 '13 at 21:19

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