TITLE:
The de Rham Cohomology of Compact Lie Groups
AUTHORS:
Yingling Liu, Yu Wang
KEYWORDS:
Lie Group, Lie Algebra, De Rham Cohomology, Lie Algebra Cohomology, Invariant Differential Forms
JOURNAL NAME:
Open Access Library Journal,
Vol.13 No.1,
January
28,
2026
ABSTRACT: Let
G
be a compact connected Lie group with Lie algebra
g
. We prove that the de Rham cohomology
H
dR
•
(
G
)
is isomorphic to the Lie algebra cohomology
H
•
(
g,ℝ
)
, and that both are isomorphic to the space
(
(
Λ
•
g
)
*
)
g
of
g
-invariant forms on the exterior algebra. In other words,
H
•
(
g,ℝ
)≅
H
dR
•
(
G
)≅
(
(
Λ
•
g
)
*
)
g
.
This result reduces the geometric problem of computing
H
dR
•
(
G
)
to the algebraic problem. We use the above isomorphism to calculate the de Rham cohomology of several classical groups
SO(
3
)
,
SO(
4
)
,
SO(
5
)
,
SO(
6
)
, and
SU(
3
)
. We provide detailed computations for
SO(
3
)
. For the higher-dimensional cases
SO(
4
)
,
SO(
5
)
,
SO(
6
)
, and
SU(
3
)
, we utilize MATLAB to compute the de Rham cohomology groups efficiently.