TITLE:
Discussion on the Homology Theory of Lie Algebras
AUTHORS:
Lilong Kang, Yu Wang, Caiyu Du
KEYWORDS:
Lie Algebra, Differential Sequence, Differential Fractional Algebra, Cohomology
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.12 No.7,
July
12,
2024
ABSTRACT: Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent manifolds. So we start with the differential sequence of Lie algebras. The Lie algebra g has the differential sequence
E
0
,
E
1
,⋯,
E
s
⋯
, which leads to the chain complex
E
s
0
→
Δ
s
0
E
s
s
→
Δ
s
1
⋯
→
Δ
s
i
E
s
(
i+1
)s
→
Δ
s
i+1
⋯
of
E
s
by discussing the chain complex
E
1
0
→
Δ
1
0
E
1
1
→
Δ
1
1
⋯
→
Δ
1
r−1
E
1
r
→
Δ
1
r
⋯
of
E
1
and proves that
E
s+1
i
≅
H
i
(
E
s
)=
Ker
Δ
s
i+1
/
Im
Δ
s
i
and therefore
E
s+1
≅H(
E
s
)
by the chain complex of
E
s
(see Theorem 2).