Complexification of a Lie algebra
over 
The complex Lie algebra 
that is the tensor product of the algebra 
with the complex field 
over the field of real numbers 
:
 |
Thus, the complexification of the Lie algebra 
is obtained from 
by extending the field of scalars from 
to 
. As elements of the algebra 
one can consider pairs 
, 
; the operations in 
are then defined by the formulas:
 |
 |
 |
The algebra 
is also called the complex hull of the Lie algebra 
.
Certain important properties of an algebra are preserved under complexification: 
is nilpotent, solvable or semi-simple if and only if 
has this property. However, simplicity of 
does not, in general, imply that of 
.
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 
of a complex Lie algebra 
is called a real form of 
if each element 
is uniquely representable in the form 
, where 
. The complexification of 
is naturally isomorphic to 
. 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 
and the Lie algebra of all anti-Hermitian matrices of order 
are non-isomorphic real forms of the Lie algebra of all complex matrices of order 
(which also has other real forms).
References
| [1] | M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) |
| [2] | D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) |
| [3] | F. Gantmakher, "On the classification of real simple Lie groups" Mat. Sb. , 5 : 2 (1939) pp. 217–250 |
Complexification of a Lie algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complexification_of_a_Lie_algebra&oldid=30727