Topology in Molecular Biology

202 D.V. Millionschikov

Let us return to our examples:

1. The cohomology classesH∗(Tn,R) are represented by invariant forms

dxi^1 ∧···∧dxiq, 1 ≤i 1 <···<iq≤n, q=1,...,n.

2.H∗(H 3 /Γ 3 ,R) is spanned by the cohomology classes of the following left-

invariant forms:

dx,dy,dy∧dz,dx∧(dz−xdy),dx∧dy∧dz.

3.H∗(G 1 /Γ 1 ,R) is spanned by the cohomology classes of:

e^1 =dz, e^2 ∧e^3 =dx∧dy, e^1 ∧e^2 ∧e^3 =dx∧dy∧dz.

11.7 Deformed Differential and Lie Algebra Cohomology

From the definition of Lie algebra cohomology it follows thatH^1 (g)isthe
dual space tog/[g,g]:

1.b^1 (g) = dimH^1 (g)≥2 for a nilpotent Lie algebrag(Dixmier’s theorem
[15]);
2.b^1 (g)≥1 for a solvable Lie algebrag;
3.b^1 (g) = 0 for a semi-simple Lie algebrag.
Consider a Lie algebragwith a non-trivialH^1 (g). Letω∈g∗,dω=0.
One can define:

1. A new deformed differential dωinΛ∗(g∗) by the formula

dω(a)=da+ω∧a.

2. A one-dimensional representation

ρω:g→K,ρω(ξ)=ω(ξ),ξ∈g.

Now we recall the definition of Lie algebra cohomology associated with a
representation. Letgbe a Lie algebra andρ:g→gl(V) its linear representa-
tion. We denote byCq(g,V) the space ofq-linear alternating mappings ofg
intoV. Then one can consider an algebraic complex:

V=C^0 (g,V)
d
−−−−→C^1 (g,V)
d
−−−−→C^2 (g,V)
d
−−−−→C^3 (g,V)
d
−−−−→···

where the differential d is defined by:

(df)(X 1 ,...,Xq+1)=

q∑+1

i=1

(−1)i+1ρ(Xi)(f(X 1 ,...,Xˆi,...,Xq+1))

+

∑

1 ≤i<j≤q+1

(−1)i+j−^1 f([Xi,Xj],X 1 ,...,Xˆi,...,Xˆj,...,Xq+1).(11.25)