We study a family of infinite-dimensional algebras that are similar to semisimple Lie algebras as well as symplectic reflection algebras. Infinitesimal Hecke algebras over sl(2) have a triangular decomposition and a nontrivial center, which yields an analogue of Duflo's Theorem (about primitive ideals), as well as a block decomposition of the BGG Category O. These algebras also have a quantized version, with similar representation theory; in particular, Category O has a block decomposition, even though the center is trivial. Finally, we discuss some questions about the higher rank cases. (Joint with A.Tikaradze, and also with W.L.Gan.)