The complex linear representations of the complexification of s l ( 2 , C ) , s l ( 2 , C ) C , obtained via isomorphisms in (A1), stand in one-to-one correspondence with the real linear representations of s l ( 2 , C ) . The set of all real linear irreducible representations of s l ( 2 , C ) are thus indexed by a pair (μ, ν). The complex linear ones, corresponding precisely to the complexification of the real linear s u ( 2 ) representations, are of the form (μ, 0), while the conjugate linear ones are the (0, ν). All others are real linear only. The linearity properties follow from the canonical injection, the far right in (A1), of s l ( 2 , C ) into its complexification. Representations on the form (ν, ν) or (μ, ν) ⊕ (ν, μ) are given by real matrices (the latter are not irreducible). Explicitly, the real linear (μ, ν) -representations of s l ( 2 , C ) are φ φ μ μ , ν ν ( X ) = ( φ φ μ μ ⊗ ⊗ φ φ ν ν ¯ ¯ ) ( X ) = φ φ μ μ ( X ) ⊗ ⊗ Id ν ν + 1 + Id μ μ + 1 ⊗ ⊗ φ φ ν ν ( X ) ¯ ¯ , X ∈ ∈ s l ( 2 , C ) where φ φ μ μ , μ μ = 0 , 1 2 , 1 , 3 2 , … … are the complex linear irreducible representations of s l ( 2 , C ) and φ φ ν ν ¯ ¯ , ν ν = 0 , 1 2 , 1 , 3 2 , … … their complex conjugate representations. (The labeling is usually in the mathematics literature 0, 1, 2,..., but half-integers are chosen here to conform with the labeling for the s o ( 3 , 1 ) Lie algebra.) Here the tensor product is interpreted in the former sense of (A0). These representations are concretely realized below.