in . It can be shown that the Lie map of is a Lie algebra isomorphism: The map is also onto. Thus , since it is simply connected, is the universal covering group of , isomorphic to the group of above. Non-surjectiveness of exponential mapping for SL(2, C) The exponential mapping is not onto. The matrix is in but there is no such that . In general, if is an element of a connected Lie group with Lie algebra then, by , The matrix can be written Realization of representations of and and their Lie algebras The complex linear representations of and are more straightforward to obtain than the representations. They can be (and usually are) written down from scratch. The holomorphic group representations (meaning the corresponding Lie algebra representation is complex linear) are related to the complex linear Lie algebra representations by exponentiation. The real linear representations of are exactly the -representations. They can be exponentiated too. The -representations are complex linear and are (isomorphic to) the highest weight-representations. These are usually indexed with only one integer (but half-integers are used here). The mathematics convention is used in this section for convenience. Lie algebra elements differ by a factor of and there is no factor of in the exponential mapping compared to the physics convention used elsewhere. Let the basis of be This choice of basis, and the notation, is standard in the mathematical literature. Complex linear representations The irreducible holomorphic -dimensional representations can be realized on the space of homogeneous polynomial of degree in 2 variables the elements of which are The action of is given by The associated -action is, using and the definition above, for the basis elements of With a choice of basis for , these representations become matrix Lie algebras. Real linear representations The -representations are realized on a space of polynomials in homogeneous of degree in and homogeneous of degree in The representations are given by By employing again it is found that In particular for the basis elements, Properties of the (m, n) representations The representations, defined above via (as restrictions to the real form ) of tensor products of irreducible complex linear representations and of are irreducible, and they are the only irreducible representations. Irreducibility follows from the unitarian trick and that a representation of is irreducible if and only if , where are irreducible representations of . Uniqueness follows from