Killing Form - Connection With Real Forms

Connection With Real Forms

Suppose that g is a semisimple Lie algebra over the field of real numbers. By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries ±1. By Sylvester's law of inertia, the number of positive entries is an invariant of the bilinear form, i.e. it does not depend on the choice of the diagonalizing basis, and is called the index of the Lie algebra g. This is a number between 0 and the dimension of g which is an important invariant of the real Lie algebra. In particular, a real Lie algebra g is called compact if the Killing form is negative definite. It is known that under the Lie correspondence, compact Lie algebras correspond to compact Lie groups.

If g_C is a semisimple Lie algebra over the complex numbers, then there are several non-isomorphic real Lie algebras whose complexification is g_C, which are called its real forms. It turns out that every complex semisimple Lie algebra admits a unique (up to isomorphism) compact real form g. The real forms of a given complex semisimple Lie algebra are frequently labeled by the positive index of inertia of their Killing form.

For example, the complex special linear algebra sl(2, C) has two real forms, the real special linear algebra, denoted sl(2, R), and the special unitary algebra, denoted su(2). The first one is noncompact, the so-called split real form, and its Killing form has signature (2,1). The second one is the compact real form and its Killing form is negative definite, i.e. has signature (0,3). The corresponding Lie groups are the noncompact group SL(2, R) of 2 × 2 real matrices with the unit determinant and the special unitary group SU(2), which is compact.