In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and . The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the matrices used to represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension .