Complexification of a Lie algebra

2020 Mathematics Subject Classification: Primary: 17B [MSN][ZBL]

The complexification of a Lie algebra $\def\fg{ {\mathfrak g}}\fg$ over $\R$ is a complex Lie algebra $\fg_\C$ that is the tensor product of the algebra $\fg$ with the complex field $\C$ over the field of real numbers $\R$:

$$\fg_\C=\fg\otimes_\R \C.$$ Thus, the complexification of the Lie algebra $\fg$ is obtained from $\fg$ by extending the field of scalars from $\R$ to $\C$. As elements of the algebra $\fg_\C$ one can consider pairs $(u,v)$, $u,v\in\fg$; the operations in $\fg_\C$ are then defined by the formulas:

$$(u_1,v_1)+(u_2,v_2)=(u_1+u_2,v_1+v_2),$$

$$\def\a{\alpha}\a+i\def\b{\beta}\b = (\a u - \b v,\a v + \b u)\ \textrm{ for any } \a,\b\in \R,$$

$$[(u_1,v_1),(u_2,v_2)] = [u_1,u_2] - [v_1,v_2].[v_1,u_2]+[u_1,v_2]).$$ The algebra $\fg_\C$ is also called the complex hull of the Lie algebra $\fg$.

Certain important properties of an algebra are preserved under complexification: $\fg_\C$ is nilpotent, solvable or semi-simple if and only if $\fg$ has this property. However, simplicity of $\fg$ does not, in general, imply that of $\fg_\C$.

The notion of the complexification of a Lie algebra is closely related to that of a real form of a complex Lie algebra (cf. Form of an (algebraic) structure). A real Lie subalgebra $\def\ff{ {\mathfrak f}}\ff$ of a complex Lie algebra $\def\fh{ {\mathfrak h}}\fh$ is called a real form of $\fh$ if each element $x\in\fh$ is uniquely representable in the form $x=u+iv$, where $u,v\in\ff$. The complexification of $\ff$ is naturally isomorphic to $\fh$. Not every complex Lie algebra has a real form. On the other hand, a given complex Lie algebra may, in general, have several non-isomorphic real forms. Thus, the Lie algebra of all real matrices of order $n$ and the Lie algebra of all anti-Hermitian matrices of order $n$ are non-isomorphic real forms of the Lie algebra of all complex matrices of order $n$ (which also has other real forms).

References

[Ga]	F. Gantmakher, "On the classification of real simple Lie groups" Mat. Sb., 5 : 2 (1939) pp. 217–250
[Na]	M.A. Naimark, "Theory of group representations", Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[Zh]	D.P. Zhelobenko, "Compact Lie groups and their representations", Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013