group element in the standard representation. For the Lorentz group, the -representation is projective when is a half-integer. See . For a projective representation of , it holds that since any loop in traversed twice, due to the double connectedness, is contractible to a point, so that its homotopy class is that of a constant map. It follows that is a double-valued function. It is not possible to consistently choose a sign to obtain a continuous representation of all of , but this is possible locally around any point. The covering group SL(2, C) Consider as a real Lie algebra with basis where the sigmas are the Pauli matrices. From the relations is obtained which are exactly on the form of the -dimensional version of the commutation relations for (see conventions and Lie algebra bases below). Thus, the map , , extended by linearity is an isomorphism. Since is simply connected, it is the universal covering group of . A geometric view Let be a path from to , denote its homotopy class by and let be the set of all such homotopy classes. Define the set and endow it with the multiplication operation where is the path multiplication of and : With this multiplication, becomes a group isomorphic to the universal covering group of . Since each has two elements, by the above construction, there is a 2:1 covering map . According to covering group theory, the Lie algebras and of are all isomorphic. The covering map is simply given by . An algebraic view For an algebraic view of the universal covering group, let act on the set of all Hermitian matrices by the operation The action on is linear. An element of may be written in the form The map is a group homomorphism into Thus is a 4-dimensional representation of . Its kernel must in particular take the identity matrix to itself, and therefore . Thus for in the kernel so, by Schur's lemma, is a multiple of the identity, which must be since . The space is mapped to Minkowski space , via The action of on preserves determinants. The induced representation of on via the above isomorphism, given by preserves the Lorentz inner product since This means that belongs to the full Lorentz group . By the main theorem of connectedness, since is connected, its image under in is connected, and hence is contained