Topology in Molecular Biology

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

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   (1982)


6. S. P. Novikov,On the exotic De-Rham cohomology. Perturbation theory as a
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