Complexification (Lie Group)

In the study of Lie groups, the complexification, or universal complexification, of a real Lie group is a complex Lie group together with a continuous homomorphism from the original group into it, chosen so that every continuous homomorphism from the original group into any other complex Lie group factors uniquely through this complex analytic homomorphism. Such complexifications always exist and are unique up to isomorphism, and the Lie algebra of the complexification is obtained from the complexification of the original group's own Lie algebra. For a compact Lie group, the complexification can be built as the Chevalley complexification, named after Claude Chevalley, using the Hopf algebra generated by the matrix coefficients of the group's finite-dimensional unitary representations; every element of this complexification can be written as a unitary element times the exponential of a skew-adjoint element, and the resulting complexification is itself a complex algebraic group. A standard example is the complexification of the compact group SU(2), which is the complex group SL(2, C). This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/