A Lie group G is a group G that also has the structure of a smooth manifold. Further, the group operations of multiplication G×G → G and inversion G → G must be smooth. An example is S¹ considered as unit modulus complex numbers under multiplication. Real and complex matrix groups are further examples. The Lie correspondence connects Lie groups and Lie algebras, and there is a rich representation theory of compact Lie groups largely due to Weyl. See also topological group.