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?



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


