Complexification

Complexification of a Real Lie Algebra

The complexification of a finite-dimensional real vector space V, as denoted by V_c, is the space of formal linear combinations v₁+ iv₂ , with v₁, v₂ ∈ V. 

Let g be a finite-dimensional real Lie algebra and g_c its complexification (as a real vector space). Then, the bracket operation on g has a unique extension to g_c which makes g_c into a complex Lie algebra. The complex Lie algebra g_c is called the complexification of the real Lie algebra g. 

Isomorphisms of Complex Lie Algebra

The Lie algebra gl(n;C), sl(n;C), so(n;C), and sp(n;C) are complex Lie algebras. In addition, there are also following isomorphisms of complex Lie algebras:

gl(n;R)_c  ≅ gl(n;C)

u(n)_c  ≅ gl(n;C)

su(n)_c  ≅ sl(n;C)

sl(n;R)_c  ≅ sl(n;C)

so(n)_c  ≅ so(n;C)

sp(n;R)_c  ≅ sp(n;C)

sp(n)_c  ≅ sp(n;C)

(u(n) is the space of all nxn complex skew-self-adjoint matrices. If X is any nxn complex matrix, then X = (X-X^*)/2 + i(X+X^*)/2i. Thus, X can be written as a skew matrix plus i times a skew matrix. Every X in gl(n;C) can be written uniquely as X₁ + iX₂, with X₁ and X₂ in u(n). It follows that u(n)_c  ≅ gl(n;C). If X has trace zero, then so do X₁ and X₂ , which has su(n)_c  ≅ sl(n;C))

Note that u(n)_c  ≅ gl(n;R)_c  ≅ gl(n;C). However, u(n)_c is not isomorphic to gl(n;R), except when n = 1. The real algebra u(n) and gl(n;R) are called real forms of the complex Lie algebra gl(n;C). 

In physics, we do not always clearly distinguish a matrix Lie algebra and its Lie algebra, or between a real Lie algebra and its complexification. For example, some references in the literature to SU(2) actually refer to the complexified Lie algebra sl(2;C). 

Representation and Complexification

Let g be a real Lie algebra and g_c its complexification. Then, every finite-dimensional complex representation π of g has a unique extension to a complex-linear representations of g_c , also denoted as π and given by π(X+iY) = π(X) + iπ(Y) for all X, Y ∈ g. Furthermore, π is irreducible as a representation of g_c  if and only if it is irreducible as a representation of g.

If π is a complex representation of the real Lie algebra g, acting on the complex vector space V. Then, saying that π is irreducible means that there is no nontrivial invariant complex subspace W ⊂ V. Even though g is a real Lie algebra, when considering complex representations of g, we are interested only in complex invariant subspaces.