11 One-Forms and Deformed de Rham Complex 203The cohomology of the complex (C∗(g,V),d) is calledthe cohomology of the
Lie algebragassociated to the representationρ:g→gl(V).
Letgbe a Lie algebra andω∈g∗is a closed 1-form. Then the complex
(Λ∗(g∗),dω) coincides with the cochain complex of the Lie algebragassociated
with the one-dimensional representationρω:g→K, whereρω(ξ)=ω(ξ),
ξ∈g.
The proof follows from the formula:
(ω∧a)(X 1 ,...,Xq+1)=∑q+1i=1(−1)i+1ω(Xi)(a(X 1 ,...,Xˆi,...,Xq+1)).The cohomologyH∗ω(g) coincides with the Lie algebra cohomology with
trivial coefficients ifω=0.Ifω= 0 the deformed differential dω is not
compatible with the exterior product∧inΛ∗(g)
dω(a∧b)=d(a∧b)+ω∧a∧b=dω(a)∧b+(−1)degaa∧dω(b)and the cohomologyHω∗(g) has no natural multiplicative structure.
LetG/Γ be a compact solvmanifold, whereGis a completely solvable
Lie group and ̃ωis a closed 1-form onG/Γ. From the previous sections it
follows that the cohomologyHω∗ ̃(G/Γ,C) is isomorphic to the Lie algebra
cohomologyHω∗(g) whereω∈g∗is the left-invariant 1-form that represents
the class [ ̃ω]∈H^1 (G/Γ,R).
One can define by means ofωa one-dimensional representationρω:G→
C∗:
ρω(g) = exp∫
γ(e,g)ω,whereγ(e, g) is a path connecting the identityewithg∈G(let us recall that
Gis a simply connected). Asωis the left invariant 1-form then
∫γ(e,g 1 g 2 )ω=∫
γ(e,g 1 )ω+∫
γ(g 1 ,g 1 g 2 )ω=∫
γ(e,g 1 )ω+∫
g− 11 γ(e,g 2 )ωholds on andρω(g 1 g 2 )=ρω(g 1 )ρω(g 2 ).
The representationρωinduces the representation of corresponding Lie al-
gebrag(we denote it by the same symbol):ρω(X)=ω(X).
Letgbe ann-dimensional real completely solvable Lie algebra (or complex
solvable) andb^1 (g) = dimH^1 (g)=k≥1. Then exists a basise^1 ,...,ening∗
such that
de^1 =···=dek=0,dek+s=αk+s∧ek+s+Pk+s(e^1 ,...,ek+s−^1 ),s=1,...,n−k,