11 One-Forms and Deformed de Rham Complex 207
wherenis an integer. The corresponding Lie algebrag 2 has the following
basis:
e 1 =
⎛
⎜
⎜
⎝
02 π 00
− 2 π 000
0001
0000
⎞
⎟
⎟
⎠,e^2 =
⎛
⎜
⎜
⎝
0001
0000
0000
0000
⎞
⎟
⎟
⎠,e^3 =
⎛
⎜
⎜
⎝
0000
0001
0000
0000
⎞
⎟
⎟
⎠,
and the following structure relations:
[e 1 ,e 2 ]=− 2 πe 3 , [e 1 ,e 3 ]=2πe 2 , [e 2 ,e 3 ]=0.
As the eigenvalues of ad(e 1 ) are equal to 0,± 2 πi the Lie groupG 2 is not
completely solvable.
The left-invariant 1-forms
e^1 =dz, e^2 =cos2πzdx−sin 2πzdy, e^3 =sin2πzdx+cos 2πzdy (11.33)
are the dual basis toe 1 ,e 2 ,e 3 and
de^1 =0, de^2 =− 2 πe^1 ∧e^3 , de^3 =2πe^1 ∧e^2. (11.34)
The cohomologyH∗(g 2 ) is spanned by the cohomology classes of:
e^1 ,e^2 ∧e^3 ,e^1 ∧e^2 ∧e^3.
But
dimH^1 (g 2 )=1=dimH^1 (G 2 /Γ 2 ,R)=3.
This example shows that, generally speaking, Hattori’s theorem does not
hold for non-completely solvable Lie groupsG, but the inclusion of left-
invariant differential formsψ:Λ∗(g∗)→Λ∗(G/Γ) always induces the in-
jectionψ∗in cohomology.
References
- S.P. Novikov, Soviet Math. Dokl. 24 , 222–226, (1981)
- S. P. Novikov, Russ. Math. Surveys 37 (5), 1–56, (1982)
- S. P. Novikov, Soviet Math. Dokl. 33 (5), 551–555, (1986)
- A.V. Pazhitnov, Soviet Math. Dokl. 35 , 1–2, (1987)
- E. Witten,Supersymmetry and Morse theory, J. Differential Geom. 17 , 661–692,
(1982) - S. P. Novikov,On the exotic De-Rham cohomology. Perturbation theory as a
spectral sequence, arXiv:math-ph/0201019 - L. Alaniya, Russ. Math. Surveys 54 (5) 1019–1020 (1999)
- K. Nomizu, Ann. Math. 59 , 531–538, (1954)
- D.V. Millionshchikov, Russ. Math. Surveys 57 (4) 813–814, (2002)
- D.V. Millionshchikov, Math. Notes (in Russian) 77(1–2), 61–71 (2005)