1928. The Lie algebra The irreducible complex linear representations of the complexification, of the Lie algebra of the Lorentz group are to be found. A convenient basis for is given by the three generators of rotations and the three generators of boosts. They are explicitly given in conventions and Lie algebra bases. The Lie algebra is complexified, and the basis is changed to the components of its two ideals The components of and separately satisfy the commutation relations of the Lie algebra and, moreover, they commute with each other, where are indices which each take values , and is the three-dimensional Levi-Civita symbol. Let and denote the complex linear span of and respectively. One has the isomorphisms where is the complexification of The utility of these isomorphisms comes from the fact that all irreducible representations of , and hence all irreducible complex linear representations of are known. The irreducible complex linear representation of is isomorphic to one of the highest weight representations. These are explicitly given in complex linear representations of The unitarian trick The Lie algebra is the Lie algebra of It contains the compact subgroup with Lie algebra The latter is a compact real form of Thus from the first statement of the unitarian trick, representations of are in one-to-one correspondence with holomorphic representations of By compactness, the Peter–Weyl theorem applies to , and hence orthonormality of irreducible characters may be appealed to. The irreducible unitary representations of are precisely the tensor products of irreducible unitary representations of . By appeal to simple connectedness, the second statement of the unitarian trick is applied. The objects in the following list are in one-to-one correspondence: Holomorphic representations of Smooth representations of Real linear representations of Complex linear representations of Tensor products of representations appear at the Lie algebra level as either of where is the identity operator. Here, the latter interpretation, which follows from , is intended. The highest weight representations of are indexed by for . (The highest weights are actually , but the notation here is adapted to that of ) The tensor products of two such complex linear factors then form the irreducible complex linear representations of Finally, the -linear representations of the real forms of the far left, , and the far right, in are obtained from the -linear representations of characterized in the previous paragraph. The (μ, ν)-representations of sl(2, C) The complex linear representations