that the are the only irreducible representations of , which is one of the conclusions of the theorem of the highest weight. Dimension The representations are -dimensional. This follows easiest from counting the dimensions in any concrete realization, such as the one given in representations of and . For a Lie general algebra the Weyl dimension formula, applies, where is the set of positive roots, is the highest weight, and is half the sum of the positive roots. The inner product is that of the Lie algebra invariant under the action of the Weyl group on the Cartan subalgebra. The roots (really elements of are via this inner product identified with elements of For the formula reduces to , where the present notation must be taken into account. The highest weight is . By taking tensor products, the result follows. Faithfulness If a representation of a Lie group is not faithful, then is a nontrivial normal subgroup. There are three relevant cases. is non-discrete and abelian. is non-discrete and non-abelian. is discrete. In this case , where is the center of . In the case of , the first case is excluded since is semi-simple. The second case (and the first case) is excluded because is simple. For the third case, is isomorphic to the quotient But is the center of It follows that the center of is trivial, and this excludes the third case. The conclusion is that every representation and every projective representation for finite-dimensional vector spaces are faithful. By using the fundamental Lie correspondence, the statements and the reasoning above translate directly to Lie algebras with (abelian) nontrivial non-discrete normal subgroups replaced by (one-dimensional) nontrivial ideals in the Lie algebra, and the center of replaced by the center of The center of any semisimple Lie algebra is trivial and is semi-simple and simple, and hence has no non-trivial ideals. A related fact is that if the corresponding representation of is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding representation is not faithful, but is . Non-unitarity The Lie algebra representation is not Hermitian. Accordingly, the corresponding (projective) representation of the group is never unitary. This is due to the non-compactness of the Lorentz group. In fact, a connected simple non-compact Lie group cannot have any nontrivial unitary finite-dimensional representations. There is a topological proof of this. Let , where is