# Adjoint and Trivial Cohomologies of Nilpotent Complex Lie Algebras of Dimension≤7

Authors: L. Magnin
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

We present tables for adjoint and trivial cohomologies of complex nilpotent Lie
algebras of dimension≤7. Attention is paid to quadratic Lie algebras, Poincaré duality,
and harmonic cocycles.


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## Body

## 1. Introduction

There are several classifications of the complex
nilpotent Lie algebras (NLA) of dimension≤7 (see [1–7], and the comparisons in [3, 5, 8]). In dimension ≤6, there are also
historical notations which go back to Dixmier [9] and Vergne [10]. In dimension 7, out of the
several classifications, Carles' classification
([2] with a couple
corrections as in [8]) is of particular interest, as it is based on weight
systems, with notations taking into account the rank and the labeling of the
weight systems, in the spirit of the historical notations. We label the NLAs of
dimension ≤7 according to the historical notations in
dimensions ≤6 and according to Carles' notations in dimension
7. We refer to [8]
for commutation relations (they are given in a basis that diagonalizes a
maximal torus, i.e., a maximal Abelian subalgebra of the derivation algebra
consisting of semisimple elements) and discussion of the classification, as
well as for comparison with other classifications. Let us simply recall here
that there are 98 (nonequivalent) weight systems for complex 7-dimensional
indecomposable NLAs: 1 in rank 0, 24 in rank 1, 45 in rank 2, 24 in rank 3, and
4 in rank 4. The indecomposable NLAs of dimension 7 are almost classified by
their weight system with a few exceptions, and one gets 123 nonisomorphic
indecomposable NLAs of dimension 7, of which 6 continuous 1-parameter series
(each continuous series counts as one algebra). We also recall the isomorphisms
for the continuous series: 𝔤7,0.4(λ′)≅𝔤7,0.4(λ)⇔λ′=±λ;𝔤7,1.1(iλ′)≅𝔤7,1.1(iλ)⇔λ′=λ;𝔤7,1.2(iλ′)≅𝔤7,1.2(iλ)⇔(λ′=λorλλ′=1);𝔤7,1.3(iλ′)≅𝔤7,1.3(iλ)⇔λ′=λ;𝔤7,2.1(iλ′)≅𝔤7,2.1(iλ)⇔λ′=λ.   As to 𝔤7,3.1(iλ),𝔤7,3.1(iλ′)≅𝔤7,3.1(iλ) if and only if λ′=s(λ) with s any element of the group of transformations of λ:G={λ,1/λ,1−λ,1/(1−λ),1−1/λ,λ/(λ−1)}, which is isomorphic to the symmetric group 𝒮3.Now, given some 7-dimensional NLA, it is not always
easy to match it (up to isomorphism) to an algebra of the list. For that
purpose, however, the adjoint cohomology is very effective. Adjoint and trivial
cohomologies for all complex 7-dimensional indecomposable NLAs, along with
their weight systems under the action of the maximal torus have been computed
in [11] (see also
[12], and for trivial
cohomology [13]).
(Beside the Abelian case, there is a couple special
instances in which there are formulae valid in any dimension: for standard
filiform [11] and
Heisenberg Lie algebras [14, 15].)However, on one hand that work is unpublished, and on
the other hand, when identifying a particular NLA, one has to look up quickly
some particular cohomology sequence. Hence there is a point in publishing a
handy list of cohomology for all NLAs of dimension≤7. In the present paper, we write down such a
list. For each NLA 𝔤, we give the sequences (dimZj(𝔤,𝔤))0≤j≤dim𝔤 and (dimHj(𝔤,𝔤))0≤j≤dim𝔤 for, respectively, the spaces of adjoint
cocycles and cohomology groups, along with the sequence of Betti numbers, that
is, the trivial cohomology (dimHj(𝔤,ℂ))0≤j≤dim𝔤.
We also pay attention to quadratic Lie algebras, Poincaré duality and harmonic
cocycles.Nonisomorphic NLAs of dimension≤6 have different adjoint cohomologies, even
though their trivial cohomologies may be equal. In dimension 7, up to 14
nonisomorphic NLAs (of which 2 continuous series) may share the same trivial
cohomology (e.g., (1,2,3,4,4,3,2,1)), hence the trivial cohomology is
ineffective in separating nonisomorphic Lie algebras; it does not refine the
classification by weight systems of the algebras either. However, the adjoint
cohomology does separate all but 13 pairs of NLAs, and refines the
classification by weight systems of the algebras, with only 4 exceptions. For
any continuous series, the adjoint cohomology is the same for all but some singular
values of the parameter at which gaps occur. The singular values for the 6
continuous series are listed in Table 1. Throughout the paper, we denote ω=exp(2iπ/3).
For continuous series, the term generic will refer to the values of the
parameter which are not singular.Table 1
Singular values for continuous series of 7-dimensional NLAs.
AlgebraSingular values𝔤7,0.4(λ):No singular value𝔤7,1.1(iλ):λ=1,λ=3,λ∈{−ω,−ω2}𝔤7,1.2(iλ)(λ≠−1)λ=1,λ∈{−ω,−ω2}𝔤7,1.3(iλ)λ=1;𝔤7,2.1(iλ)λ=0,λ=12,λ=2,λ∈{−ω,−ω2}𝔤7,3.1(iλ)λ=0,1,λ=−1,2,12,λ∈{−ω,−ω2}The nonisomorphic 7-dimensional NLAs having the same
adjoint cohomology come in 13 pairs, as shown in Table2. Adjoint cohomology
refines the classification by weight systems of the NLAs, except for the 4
pairs # 2,3,4,8 in Table 2. In each of those 4 pairs, the weight system on the
cohomology is identical for the 2 components [11].Table 2
Pairs of
7-dimensional NLAs having same adjoint cohomology.
#PairAdjoint cohomology1(𝔤7,0.4(λ),𝔤7,0.5)(1,4,9,15,16,12,7,2)2(𝔤7,1.1(iλ)(λ=−ω),𝔤7,1.1(iλ)(λ=−ω2))(1,4,9,14,16,12,6,2)3(𝔤7,1.2(iλ)(generic),𝔤7,1.2(ii))(1,6,16,27,30,23,12,3)4(𝔤7,1.3(iλ)(generic),𝔤7,1.3(iii))(1,7,19,30,32,23,11,3)5(𝔤7,1.20,𝔤7,1.21)(1,5,12,17,17,15,10,3)6(𝔤7,2.1(iλ)(generic),𝔤7,1.3(ii))(1,8,20,31,33,24,12,3)7(𝔤7,2.1(iλ)(λ=0),𝔤7,2.11)(2,9,21,33,34,24,12,3)8(𝔤7,2.1(iλ)(λ=−ω),𝔤7,2.1(iλ)(λ=−ω2))(1,8,20,31,34,25,12,3)9(𝔤7,2.26,𝔤7,3.4)(2,8,19,32,36,26,12,3)10(𝔤7,2.31,𝔤7,2.41)(1,7,18,29,32,24,12,3)11(𝔤7,2.34,𝔤7,2.35)(2,7,18,29,33,25,11,3)12(𝔤7,2.36,𝔤6,12×ℂ)(2,11,29,47,51,38,18,4)13(𝔤6,11×ℂ,𝔤4×𝔤3)(2,12,31,48,51,38,18,4)


## 2. Cohomology Tables

The results
appear as in Tables3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. For continuous series, the places where gaps occur
for the singular values are underlined.Table 3
Cohomology table for NLAs of dimension 7: rank 0.
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbers𝔤7,0.1(1,10,48,114,147,109,44,7)(1,4,9,15,16,11,6,2)(1,2,3,4,4,3,2,1)𝔤7,0.2(1,10,49,113,147,109,44,7)(1,4,10,15,15,11,6,2)(1,2,3,4,4,3,2,1)𝔤7,0.3(1,11,50,116,149,110,44,7)(1,5,12,19,20,14,7,2)(1,2,4,6,6,4,2,1)𝔤7,0.4(λ)(1,10,48,114,147,110,44,7)(1,4,9,15,16,12,7,2)(1,2,3,4,4,3,2,1)𝔤7,0.5(1,10,48,114,147,110,44,7)(1,4,9,15,16,12,7,2)(1,2,3,4,4,3,2,1)𝔤7,0.6(1,10,48,114,148,110,44,7)(1,4,9,15,17,13,7,2)(1,2,3,5,5,3,2,1)𝔤7,0.7(1,10,49,115,148,110,44,7)(1,4,10,17,18,13,7,2)(1,2,3,4,4,3,2,1)𝔤7,0.8(1,10,50,114,148,112,45,7)(1,4,11,17,17,15,10,3)(1,3,4,4,4,4,3,1)Table 4
Cohomology table for NLAs of dimension 7: rank 1.
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbers𝔤7,1.01(i)(1,11,51,113,148,112,45,7)(1,5,13,17,16,15,10,3)(1,3,4,4,4,4,3,1)𝔤7,1.01(ii)(1,12,53,118,152,113,45,7)(1,6,16,24,25,20,11,3)(1,3,5,7,7,5,3,1)𝔤7,1.02(1,11,49,114,148,110,44,7)(1,5,11,16,17,13,7,2)(1,2,3,5,5,3,2,1)𝔤7,1.03(1,11,49,115,148,110,44,7)(1,5,11,17,18,13,7,2)(1,2,3,4,4,3,2,1)𝔤7,1.1(iλ) generic(1,10,48,113,147,109,44,7)(1,4,9,14,15,11,6,2)(1,2,3,4,4,3,2,1)𝔤7,1.1(iλ)λ=1(1,11̲,49̲,113,147,109,44,7)(1,5̲,11̲,15̲,15,11,6,2)unchanged𝔤7,1.1(iλ)λ=3(1,10,48,114̲,147,110̲,44,7)(1,4,9,15̲,16̲,
12,7,2)(1,2,4̲,5̲,5̲,4̲,2,1)𝔤7,1.1(iλ)λ∈{−ω,−ω2}(1,10,48,113,148̲,109,44,7)(1,4,9,14,16̲,12̲,6,2)(1,2,3,5̲,5̲,3,2,1)𝔤7,1.1(ii)(1,11,48,114,147,109,44,7)(1,5,10,15,16,11,6,2)(1,2,3,4,4,3,2,1)𝔤7,1.1(iii)(2,10,48,113,147,109,44,7)(2,5,9,14,15,11,6,2)(1,2,3,4,4,3,2,1)𝔤7,1.1(iv)(1,11,48,114,149,110,44,7)(1,5,10,15,18,14,7,2)(1,2,3,6,6,3,2,1)𝔤7,1.1(v)(1,10,50,113,148,112,45,7)(1,4,11,16,16,15,10,3)(1,3,4,4,4,4,3,1)𝔤7,1.1(vi)(1,11,50,115,149,112,45,7)(1,5,12,18,19,16,10,3)(1,3,4,4,4,4,3,1)𝔤7,1.2(iλ) (λ≠−1) generic(1,12,53,121,154,114,45,7)(1,6,16,27,30,23,12,3)(1,3,6,7,7,6,3,1)𝔤7,1.2(iλ)λ∈{1,−ω,−ω2}(1,12,53,122̲,155̲,114,45,7)(1,6,16,28̲,32̲,24̲,12,3)(1,3,6,8̲,8̲,6,3,1)𝔤7,1.2(ii)(1,12,53,121,154,114,45,7)(1,6,16,27,30,23,12,3)(1,3,6,7,7,6,3,1)𝔤7,1.2(iii)(2,12,53,122,154,114,45,7)(2,7,16,28,31,23,12,3)(1,3,6,7,7,6,3,1)𝔤7,1.2(iv)(1,12,53,122,154,114,45,7)(1,6,16,28,31,23,12,3)(1,3,6,7,7,6,3,1)𝔤7,1.3(iλ) generic(1,13,55,122,155,113,45,7)(1,7,19,30,32,23,11,3)(1,3,5,7,7,5,3,1)𝔤7,1.3(iλ)λ=1(1,13,55,122,156̲,113,45,7)(1,7,19,30,33̲,24̲,11,3)(1,3,5,8̲,8̲,5,3,1)𝔤7,1.3(ii)(1,14,55,123,155,114,45,7)(1,8,20,31,33,24,12,3)(1,3,6,8,8,6,3,1)𝔤7,1.3(iii)(1,13,55,122,155,113,45,7)(1,7,19,30,32,23,11,3)(1,3,5,7,7,5,3,1)𝔤7,1.3(iv)(2,13,57,123,157,113,45,7)(2,8,21,33,35,25,11,3)(1,3,5,9,9,5,3,1)𝔤7,1.3(v)(1,13,56,123,155,117,46,7)(1,7,20,32,33,27,16,4)(1,4,6,7,7,6,4,1)𝔤7,1.4(1,12,50,116,149,110,44,7)(1,6,13,19,20,14,7,2)(1,2,4,6,6,4,2,1)𝔤7,1.5‡(2,11,49,115,147,109,44,7)(2,6,11,17,17,11,6,2)(1,2,3,4,4,3,2,1)𝔤7,1.6(1,12,52,118,149,110,44,7)(1,6,15,23,22,14,7,2)(1,2,4,6,6,4,2,1)𝔤7,1.7(2,15,59,128,159,115,45,7)(2,10,25,40,42,29,13,3)(1,3,7,11,11,7,3,1)𝔤7,1.8(1,11,53,120,152,113,45,7)(1,5,15,26,27,20,11,3)(1,3,5,6,6,5,3,1)𝔤7,1.9(2,14,58,124,156,114,45,7)(2,9,23,35,35,25,12,3)(1,3,6,9,9,6,3,1)𝔤7,1.10(1,11,50,116,150,111,44,7)(1,5,12,19,21,16,8,2)(1,2,4,7,7,4,2,1)𝔤7,1.11(1,11,52,117,151,113,45,7)(1,5,14,22,23,19,11,3)(1,3,5,6,6,5,3,1)𝔤7,1.12(1,12,53,120,153,113,45,7)(1,6,16,26,28,21,11,3)(1,3,5,7,7,5,3,1)𝔤7,1.13(2,12,51,117,150,111,44,7)(2,7,14,21,22,16,8,2)(1,2,4,7,7,4,2,1)𝔤7,1.14(2,11,49,115,148,110,44,7)(2,6,11,17,18,13,7,2)(1,2,3,4,4,3,2,1)𝔤7,1.15(1,13,54,120,153,113,45,7)(1,7,18,27,28,21,11,3)(1,3,5,7,7,5,3,1)𝔤7,1.16(2,15,59,128,158,115,45,7)(2,10,25,40,41,28,13,3)(1,3,7,10,10,7,3,1)𝔤7,1.17(1,11,48,114,147,110,44,7)(1,5,10,15,16,12,7,2)(1,2,3,4,4,3,2,1)𝔤7,1.18(2,13,56,123,155,114,45,7)(2,8,20,32,33,24,12,3)(1,3,6,8,8,6,3,1)𝔤7,1.19(2,11,55,120,157,113,45,7)(2,6,17,28,32,25,11,3)(1,3,5,9,9,5,3,1)𝔤7,1.20(1,11,50,114,148,112,45,7)(1,5,12,17,17,15,10,3)(1,3,4,4,4,4,3,1)𝔤7,1.21(1,11,50,114,148,112,45,7)(1,5,12,17,17,15,10,3)(1,3,4,4,4,4,3,1)Table 5
Cohomology
table for NLAs of dimension 7: rank 2. (continues on next page).
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbers𝔤7,2.1(iλ) generic(1,14,55,123,155,114,45,7)(1,8,20,31,33,24,12,3)(1,3,6,8,8,6,3,1)𝔤7,2.1(iλ)λ=0(2̲,14,56̲,124̲,155,114,45,7)(2̲,9̲,21̲,33̲,34̲,24,12,3)unchanged𝔤7,2.1(iλ)λ=12(1,15̲,55,123,155,114,45,7)(1,9̲,21̲,31,33,24,12,3)unchanged𝔤7,2.1(iλ)λ=2(1,14,55,125̲,157̲,115̲,45,7)(1,8,20,33̲,37̲,27̲,13̲,3)(1,3,7̲,9̲,9̲,7̲,3,1)𝔤7,2.1(iλ)λ∈{−ω,−ω2}(1,14,55,123,156̲,114,45,7)(1,8,20,31,34̲,25̲,12,3)(1,3,6,9̲,9̲,6,3,1)𝔤7,2.1(ii)(1,14,56,123,155,114,45,7)(1,8,21,32,33,24,12,3)(1,3,6,8,8,6,3,1)𝔤7,2.1(iii)(1,14,58,125,159,118,46,7)(1,8,23,36,39,32,17,4)(1,4,7,8,8,7,4,1)𝔤7,2.1(iv)(1,14,58,124,157,118,46,7)(1,8,23,35,36,30,17,4)(1,4,7,8,8,7,4,1)𝔤7,2.1(v)(2,14,57,124,157,114,45,7)(2,9,22,34,36,26,12,3)(1,3,6,10,10,6,3,1)𝔤7,2.2(1,15,59,130,159,116,45,7)(1,9,25,42,44,30,14,3)(1,3,7,9,9,7,3,1)𝔤7,2.3(1,13,53,119,149,110,44,7)(1,7,17,25,23,14,7,2)(1,2,4,6,6,4,2,1)𝔤7,2.4‡▪(2,12,49,115,147,110,44,7)(2,7,12,17,17,12,7,2)(1,2,3,4,4,3,2,1)𝔤7,2.5(1,12,49,114,149,110,44,7)(1,6,12,16,18,14,7,2)(1,2,3,6,6,3,2,1)𝔤7,2.6(2,12,49,115,148,110,44,7)(2,7,12,17,18,13,7,2)(1,2,3,4,4,3,2,1)𝔤7,2.7‡(2,13,52,120,150,111,44,7)(2,8,16,25,25,16,8,2)(1,2,4,7,7,4,2,1)𝔤7,2.8(2,13,51,119,152,112,44,7)(2,8,15,23,26,19,9,2)(1,2,5,9,9,5,2,1)𝔤7,2.9(2,12,50,118,151,112,44,7)(2,7,13,21,24,18,9,2)(1,2,5,8,8,5,2,1)𝔤7,2.10(1,12,51,115,149,112,45,7)(1,6,14,19,19,16,10,3)(1,3,4,4,4,4,3,1)𝔤7,2.11(2,14,56,124,155,114,45,7)(2,9,21,33,34,24,12,3)(1,3,6,8,8,6,3,1)𝔤7,2.12(2,14,57,123,157,113,45,7)(2,9,22,33,35,25,11,3)(1,3,5,9,9,5,3,1)𝔤7,2.13(1,12,50,114,148,112,45,7)(1,6,13,17,17,15,10,3)(1,3,4,4,4,4,3,1)𝔤7,2.14(1,12,51,113,148,112,45,7)(1,6,14,17,16,15,10,3)(1,3,4,4,4,4,3,1)𝔤7,2.15(1,13,54,119,152,113,45,7)(1,7,18,26,26,20,11,3)(1,3,5,7,7,5,3,1)𝔤7,2.16(1,14,55,120,153,113,45,7)(1,8,20,28,28,21,11,3)(1,3,5,7,7,5,3,1)𝔤7,2.17(2,13,53,122,154,114,45,7)(2,8,17,28,31,23,12,3)(1,3,6,7,7,6,3,1)𝔤7,2.18(2,15,58,125,157,115,45,7)(2,10,24,36,37,27,13,3)(1,3,6,10,10,6,3,1)𝔤7,2.19(2,15,58,125,156,114,45,7)(2,10,24,36,36,25,12,3)(1,3,6,9,9,6,3,1)𝔤7,2.20(2,16,59,128,158,115,45,7)(2,11,26,40,41,28,13,3)(1,3,7,10,10,7,3,1)𝔤7,2.21(2,16,60,129,158,115,45,7)(2,11,27,42,42,28,13,3)(1,3,7,10,10,7,3,1)𝔤7,2.22(2,14,56,125,156,115,45,7)(2,9,21,34,36,26,13,3)(1,3,7,9,9,7,3,1)𝔤7,2.23(1,13,58,123,156,118,46,7)(1,7,22,34,34,29,17,4)(1,4,7,7,7,7,4,1)𝔤7,2.24(2,13,54,123,155,114,45,7)(2,8,18,30,33,24,12,3)(1,3,6,8,8,6,3,1)𝔤7,2.25(1,14,56,123,155,117,46,7)(1,8,21,32,33,27,16,4)(1,4,6,7,7,6,4,1)𝔤7,2.26(2,13,55,124,157,114,45,7)(2,8,19,32,36,26,12,3)(1,3,6,10,10,6,3,1)𝔤7,2.27(2,17,65,132,164,119,46,7)(2,12,33,50,51,38,18,4)(1,4,8,11,11,8,4,1)𝔤7,2.28(1,16,63,131,163,119,46,7)(1,10,30,47,49,37,18,4)(1,4,8,10,10,8,4,1)𝔤7,2.29(2,14,59,126,159,117,46,7)(2,9,24,38,40,31,16,4)(1,4,6,9,9,6,4,1)𝔤7,2.30(1,15,56,125,156,117,46,7)(1,9,22,34,36,28,16,4)(1,4,6,8,8,6,4,1)𝔤7,2.31(1,13,54,122,155,114,45,7)(1,7,18,29,32,24,12,3)(1,3,6,8,8,6,3,1)𝔤7,2.32(1,14,55,122,156,113,45,7)(1,8,20,30,33,24,11,3)(1,3,5,8,8,5,3,1)𝔤7,2.33(1,12,53,120,153,113,45,7)(1,6,16,26,28,21,11,3)(1,3,5,7,7,5,3,1)𝔤7,2.34(2,12,55,121,157,113,45,7)(2,7,18,29,33,25,11,3)(1,3,5,9,9,5,3,1)𝔤7,2.35(2,12,55,121,157,113,45,7)(2,7,18,29,33,25,11,3)(1,3,5,9,9,5,3,1)𝔤7,2.36(2,16,62,132,164,119,46,7)(2,11,29,47,51,38,18,4)(1,4,8,11,11,8,4,1)𝔤7,2.37(1,13,53,121,154,114,45,7)(1,7,17,27,30,23,12,3)(1,3,6,7,7,6,3,1)𝔤7,2.38(2,16,65,131,164,119,46,7)(2,11,32,49,50,38,18,4)(1,4,8,11,11,8,4,1)𝔤7,2.39(2,17,63,131,163,115,45,7)(2,12,31,47,49,33,13,3)(1,3,7,13,13,7,3,1)𝔤7,2.40(3,16,63,132,159,115,45,7)(3,12,30,48,46,29,13,3)(1,3,7,11,11,7,3,1)𝔤7,2.41(1,13,54,122,155,114,45,7)(1,7,18,29,32,24,12,3)(1,3,6,8,8,6,3,1)𝔤7,2.42(2,14,58,126,159,114,45,7)(2,9,23,37,40,28,12,3)(1,3,6,10,10,6,3,1)𝔤7,2.43(2,16,60,129,159,115,45,7)(2,11,27,42,43,29,13,3)(1,3,7,11,11,7,3,1)𝔤7,2.44(2,16,60,130,161,116,45,7)(2,11,27,43,46,32,14,3)(1,3,7,11,11,7,3,1)𝔤7,2.45(2,17,65,134,165,119,46,7)(2,12,33,52,54,39,18,4)(1,4,8,12,12,8,4,1)𝔤6,12×ℂ(2,16,62,132,164,119,46,7)(2,11,29,47,51,38,18,4)(1,4,8,11,11,8,4,1)𝔤6,17×ℂ(2,14,55,122,154,113,45,7)(2,9,20,30,31,22,11,3)(1,3,5,7,7,5,3,1)𝔤6,19×ℂ(2,13,53,120,153,113,45,7)(2,8,17,26,28,21,11,3)(1,3,5,7,7,5,3,1)𝔤6,20×ℂ(2,12,50,115,149,112,45,7)(2,7,13,18,19,16,10,3)(1,3,4,4,4,4,3,1)Table 6
Cohomology table for NLAs of dimension 7: rank 3.
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbers𝔤7,3.1(iλ) generic(1,15,59,130,159,116,45,7)(1,9,25,42,44,30,14,3)(1,3,7,9,9,7,3,1)𝔤7,3.1(iλ)λ=0,1(2̲,15,59,130,161̲,116,45,7)(2̲,10̲,25,42,46̲,32̲,14,3)(1,3,7,11̲,11̲,7,3,1)𝔤7,3.1(iλ)λ=−1,2,12(1,17̲,59,131̲,159,117̲,45,7)(1,11̲,27̲,43̲,45̲,31̲,15̲,3)(1,3,8̲,10̲,10̲,8̲,3,1)𝔤7,3.1(iλ)λ∈{−ω,−ω2}(1,15,59,130,160̲,116,45,7)(1,9,25,42,45̲,31̲,14,3)(1,3,7,10̲,10̲,7,3,1)𝔤7,3.1(iii)(1,15,63,130,161,119,46,7)(1,9,29,46,46,35,18,4)(1,4,8,9,9,8,4,1)𝔤7,3.2(2,17,62,130,158,115,45,7)(2,12,30,45,43,28,13,3)(1,3,7,10,10,7,3,1)𝔤7,3.3(2,15,56,126,156,115,45,7)(2,10,22,35,37,26,13,3)(1,3,7,9,9,7,3,1)𝔤7,3.4(2,13,55,124,157,114,45,7)(2,8,19,32,36,26,12,3)(1,3,6,10,10,6,3,1)𝔤7,3.5(2,14,56,125,157,114,45,7)(2,9,21,34,37,26,12,3)(1,3,6,10,10,6,3,1)𝔤7,3.6(3,18,64,134,163,117,45,7)(3,14,33,51,52,35,15,3)(1,3,8,14,14,8,3,1)𝔤7,3.7(2,15,61,131,164,119,46,7)(2,10,27,45,50,38,18,4)(1,4,8,11,11,8,4,1)𝔤7,3.8(2,19,68,139,169,120,46,7)(2,14,38,60,63,44,19,4)(1,4,9,14,14,9,4,1)𝔤7,3.9(2,18,65,137,168,120,46,7)(2,13,34,55,60,43,19,4)(1,4,9,13,13,9,4,1)𝔤7,3.10(1,15,58,124,157,118,46,7)(1,9,24,35,36,30,17,4)(1,4,7,8,8,7,4,1)𝔤7,3.11(2,18,65,136,166,119,46,7)(2,13,34,54,57,40,18,4)(1,4,8,13,13,8,4,1)𝔤7,3.12(3,19,66,147,172,122,46,7)(3,15,36,66,74,49,21,4)(1,4,11,14,14,11,4,1)𝔤7,3.13(2,16,59,127,159,117,46,7)(2,11,26,39,41,31,16,4)(1,4,6,9,9,6,4,1)𝔤7,3.14(2,18,65,135,166,120,46,7)(2,13,34,53,56,41,19,4)(1,4,9,12,12,9,4,1)𝔤7,3.15(2,17,63,133,164,119,46,7)(2,12,31,49,52,38,18,4)(1,4,8,11,11,8,4,1)𝔤7,3.16(1,14,58,123,157,118,46,7)(1,8,23,34,35,30,17,4)(1,4,7,8,8,7,4,1)𝔤7,3.17(1,16,56,127,157,117,46,7)(1,10,23,36,39,29,16,4)(1,4,6,9,9,6,4,1)𝔤7,3.18(1,19,70,135,170,125,47,7)(1,13,40,58,60,50,25,5)(1,5,10,11,11,10,5,1)𝔤7,3.19(2,19,70,148,171,124,47,7)(2,14,40,71,74,50,24,5)(1,5,9,15,15,9,5,1)𝔤7,3.20(2,19,66,133,163,115,45,7)(2,14,36,52,51,33,13,3)(1,3,7,13,13,7,3,1)𝔤7,3.21(2,15,58,127,159,114,45,7)(2,10,24,38,41,28,12,3)(1,3,6,10,10,6,3,1)𝔤7,3.22(2,15,57,124,157,114,45,7)(2,10,23,34,36,26,12,3)(1,3,6,10,10,6,3,1)𝔤7,3.23(3,17,63,132,159,116,45,7)(3,13,31,48,46,30,14,3)(1,3,7,11,11,7,3,1)𝔤7,3.24(3,22,75,150,175,122,46,7)(3,18,48,78,80,52,21,4)(1,4,11,17,17,11,4,1)𝔤6,5×ℂ(3,18,65,135,165,119,46,7)(3,14,34,53,55,39,18,4)(1,4,8,11,11,8,4,1)𝔤6,7×ℂ(3,20,70,142,170,120,46,7)(3,16,41,65,67,45,19,4)(1,4,9,14,14,9,4,1)𝔤6,8×ℂ(3,19,66,135,165,119,46,7)(3,15,36,54,55,39,18,4)(1,4,8,11,11,8,4,1)𝔤6,10×ℂ(2,17,64,134,165,119,46,7)(2,12,32,51,54,39,18,4)(1,4,8,11,11,8,4,1)𝔤6,11×ℂ(2,17,63,132,164,119,46,7)(2,12,31,48,51,38,18,4)(1,4,8,11,11,8,4,1)𝔤6,13×ℂ(2,15,58,125,159,118,46,7)(2,10,24,36,39,32,17,4)(1,4,7,8,8,7,4,1)𝔤6,14×ℂ(3,16,58,126,158,115,45,7)(3,12,25,37,39,28,13,3)(1,3,6,10,10,6,3,1)𝔤6,15×ℂ(2,14,56,124,157,115,45,7)(2,9,21,33,36,27,13,3)(1,3,6,10,10,6,3,1)𝔤6,16×ℂ(2,15,57,123,154,113,45,7)(2,10,23,33,32,22,11,3)(1,3,5,7,7,5,3,1)𝔤6,18×ℂ(2,13,51,115,149,112,45,7)(2,8,15,19,19,16,10,3)(1,3,4,4,4,4,3,1)𝔤5,6×ℂ2(3,18,63,132,164,119,46,7)(3,14,32,48,51,38,18,4)(1,4,8,11,11,8,4,1)Table 7
Cohomology table for NLAs of dimension 7: rank 4.
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbers𝔤7,4.1(3,20,71,149,174,122,46,7)(3,16,42,73,78,51,21,4)(1,4,11,16,16,11,4,1)𝔤7,4.2(3,25,78,157,178,123,46,7)(3,21,54,88,90,56,22,4)(1,4,12,18,18,12,4,1)𝔤7,4.3(2,21,72,150,175,125,47,7)(2,16,44,75,80,55,25,5)(1,5,10,16,16,10,5,1)𝔤7,4.4(1,28,91,140,189,134,48,7)(1,22,70,84,84,78,35,6)(1,6,14,14,14,14,6,1)𝔤6,1×ℂ(3,24,81,157,184,127,47,7)(3,20,56,91,96,66,27,5)(1,5,12,18,18,12,5,1)𝔤6,2×ℂ(2,20,70,138,171,125,47,7)(2,15,41,61,64,51,25,5)(1,5,10,12,12,10,5,1)𝔤6,3×ℂ‡▪(4,25,80,153,178,123,46,7)(4,22,56,86,86,56,22,4)(1,4,11,20,20,11,4,1)𝔤6,4×ℂ(3,19,68,140,169,120,46,7)(3,15,38,61,64,44,19,4)(1,4,9,14,14,9,4,1)𝔤6,6×ℂ(3,21,72,143,170,120,46,7)(3,17,44,68,68,45,19,4)(1,4,9,14,14,9,4,1)𝔤6,9×ℂ(2,16,63,132,163,119,46,7)(2,11,30,48,50,37,18,4)(1,4,8,11,11,8,4,1)𝔤5,3×ℂ2(3,22,75,149,179,126,47,7)(3,18,48,77,83,60,26,5)(1,5,11,15,15,11,5,1)𝔤5,4×ℂ2‡▪(4,22,68,135,166,120,46,7)(4,19,41,56,56,41,19,4)(1,4,8,11,11,8,4,1)𝔤5,5×ℂ2(3,19,65,133,164,119,46,7)(3,15,35,51,52,38,18,4)(1,4,8,11,11,8,4,1)𝔤4×𝔤3(2,17,63,132,164,119,46,7)(2,12,31,48,51,38,18,4)(1,4,8,11,11,8,4,1)Table 8
Cohomology table for NLAs of dimension 7: rank 5.
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbers𝔤5,1×ℂ2(3,29,91,166,194,134,48,7)(3,25,71,110,115,83,35,6)(1,6,14,19,19,14,6,1)𝔤5,2×ℂ2(4,27,89,168,190,128,47,7)(4,24,67,110,113,73,28,5)(1,5,13,21,21,13,5,1)𝔤4×ℂ3(4,25,78,150,179,126,47,7)(4,22,54,81,84,60,26,5)(1,5,11,15,15,11,5,1)Table 9
Cohomology table for NLAs of dimension 7: rank 6.
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbers𝔤3×ℂ4(5,34,106,191,209,136,48,7)(5,32,91,150,155,100,37,6)(1,6,16,25,25,16,6,1)Table 10
Cohomology table for NLAs of dimension 7: rank 7
(Abelian).
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbersℂ7‡▪(7,49,147,245,245,147,49,7)(7,49,147,245,245,147,49,7)(1,7,21,35,35,21,7,1)Table 11
Cohomology table for NLAs of dimension 6.
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbersRank 1𝔤6,12(1,11,40,71,68,33,6)(1,6,15,21,19,11,3)(1,3,5,6,5,3,1)𝔤6,17(1,10,36,67,64,32,6)(1,5,10,13,11,6,2)(1,2,3,4,3,2,1)𝔤6,19(1,9,35,66,64,32,6)(1,4,8,11,10,6,2)(1,2,3,4,3,2,1)𝔤6,20(1,8,34,64,63,32,6)(1,3,6,8,7,5,2)(1,2,2,2,2,2,1)Rank 2𝔤6,5(2,12,42,72,68,33,6)(2,8,18,24,20,11,3)(1,3,5,6,5,3,1)𝔤6,7(2,14,44,75,69,33,6)(2,10,22,29,24,12,3)(1,3,6,8,6,3,1)𝔤6,8(2,13,42,72,68,33,6)(2,9,19,24,20,11,3)(1,3,5,6,5,3,1)𝔤6,10(1,12,41,72,68,33,6)(1,7,17,23,20,11,3)(1,3,5,6,5,3,1)𝔤6,11(1,12,40,71,68,33,6)(1,7,16,21,19,11,3)(1,3,5,6,5,3,1)𝔤6,13(1,10,38,68,67,33,6)(1,5,12,16,15,10,3)(1,3,4,4,4,3,1)𝔤6,14(2,11,37,68,66,32,6)(2,7,12,15,14,8,2)(1,2,4,6,4,2,1)𝔤6,15(1,10,36,67,66,32,6)(1,5,10,13,13,8,2)(1,2,4,6,4,2,1)𝔤6,16(1,11,37,67,64,32,6)(1,6,12,14,11,6,2)(1,2,3,4,3,2,1)𝔤6,18(1,9,34,64,63,32,6)(1,4,7,8,7,5,2)(1,2,2,2,2,2,1)𝔤5,6×ℂ(2,12,40,71,68,33,6)(2,8,16,21,19,11,3)(1,3,5,6,5,3,1)Rank 3𝔤6,1(2,17,50,82,74,34,6)(2,13,31,42,36,18,4)(1,4,8,10,8,4,1)𝔤6,2(1,14,44,73,72,34,6)(1,9,22,27,25,16,4)(1,4,6,6,6,4,1)𝔤6,3‡▪(3,18,48,78,72,33,6)(3,15,30,36,30,15,3)(1,3,8,12,8,3,1)𝔤6,4(2,13,43,74,69,33,6)(2,9,20,27,23,12,3)(1,3,6,8,6,3,1)𝔤6,6(2,15,45,75,69,33,6)(2,11,24,30,24,12,3)(1,3,6,8,6,3,1)𝔤6,9(1,11,41,70,68,33,6)(1,6,16,21,18,11,3)(1,3,5,6,5,3,1)𝔤5,3×C(2,15,47,79,76,34,6)(2,11,26,36,32,17,4)(1,4,7,8,7,4,1)𝔤5,4×C‡▪(3,15,42,72,69,33,6)(3,12,21,24,21,12,3)(1,3,5,6,5,3,1)𝔤5,5×C(2,13,41,71,68,33,6)(2,9,18,22,19,11,3)(1,3,5,6,5,3,1)Rank 4𝔤5,1×C(2,21,55,86,79,35,6)(2,17,40,51,45,24,5)(1,5,9,10,9,5,1)𝔤5,2×C(3,19,55,86,75,34,6)(3,16,38,51,41,19,4)(1,4,9,12,9,4,1)𝔤4×C2(3,17,48,79,73,34,6)(3,14,29,37,32,17,4)(1,4,7,8,7,4,1)𝔤3×𝔤3(2,16,50,82,74,34,6)(2,12,30,42,36,18,4)(1,4,8,10,8,4,1)Rank 5𝔤3×C3(4,24,65,97,81,35,6)(4,22,53,72,58,26,5)(1,5,11,14,11,5,1)Rank 6 (abelian)C6‡▪(6,36,90,120,90,36,6)(6,36,90,120,90,36,6)(1,6,15,20,15,6,1)Table 12
Cohomology table for NLAs of dimension 5.
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbersRank 1𝔤5,6(1,8,24,34,22,5)(1,4,7,8,6,2)(1,2,3,3,2,1)Rank 2𝔤5,3(1,10,28,37,23,5)(1,6,13,15,10,3)(1,3,4,4,3,1)𝔤5,4‡▪(2,10,24,35,22,5)(2,7,9,9,7,2)(1,2,3,3,2,1)𝔤5,5(1,9,24,34,22,5)(1,5,8,8,6,2)(1,2,3,3,2,1)Rank 3𝔤5,1(1,15,30,41,24,5)(1,11,20,21,15,4)(1,4,5,5,4,1)𝔤5,2(2,13,31,39,23,5)(2,10,19,20,12,3)(1,3,6,6,3,1)𝔤4×C(2,11,28,37,23,5)(2,8,14,15,10,3)(1,3,4,4,3,1)Rank 4𝔤3×C𝟚(3,16,37,43,24,5)(3,14,28,30,17,4)(1,4,7,7,4,1)Rank 5C5‡▪(5,25,50,50,25,5)(5,25,50,50,25,5)(1,5,10,10,5,1)Table 13
Cohomology table for NLAs of dimension≤4.
AlgebraAdjoint cocyclesAdjoint cohomologyBetti numbers𝔤4(dim4,rk2)(1,7,15,14,4)(1,4,6,5,2)(1,2,2,2,1)𝔤3×C(dim4,rk3)(2,10,19,15,4)(2,8,13,10,3)(1,3,4,3,1)C4(abel.)‡▪(dim4,rk4)(4,16,24,16,4)(4,16,24,16,4)(1,4,6,4,1)𝔤3(dim3,rk2)(1,6,8,3)(1,4,5,2)(1,2,2,1)C3(abel.)‡▪(dim3,rk3)(3,9,9,3)(3,9,9,3)(1,3,3,1)C2(abel.)‡▪(dim2,rk2)(2,4,2)(2,4,2)(1,2,1)C(abel.)‡▪(dim1,rk1)(1,1)(1,1)(1,1)Recall thatZ0(𝔤,𝔤)=H0(𝔤,𝔤)=𝔠 the center of 𝔤, and Z1(𝔤,𝔤)=Der(𝔤),H1(𝔤,𝔤)=Der(𝔤)/ad(𝔤).  Recall also the following facts about Poincaré
duality (PD) ([16, Theorem 6.10]). For any complex N-dimensional Lie algebra 𝔤 and any 𝔤-module V, cohomology and homology are related by the
formulae (with V* the contragredient 𝔤-module, and 0≤k≤N)(2.1)Hk(𝔤,V)≅Hk(𝔤,V*)*,(2.2)Hk(𝔤,V)≅HN−k(𝔤,V⊗ℂ(⋀N𝔤)*),whence(2.3)Hk(𝔤,V)≅HN−k(𝔤,V*⊗ℂ⋀N𝔤)*.Now, as the algebra 𝔤 considered in this paper is nilpotent, hence
unimodular (i.e., Tr(adX)=0 for all X∈𝔤), one has ⋀N𝔤≅ℂ (the trivial module), and (2.2), (2.3) read,
respectively,(2.4)Hk(𝔤,V)≅HN−k(𝔤,V),(2.5)Hk(𝔤,V)≅HN−k(𝔤,V*)*.Hence,HN(𝔤,𝔤)≅H0(𝔤,𝔤)≅𝔤/[𝔤,𝔤]≅(H1(𝔤,ℂ))*, and HN−k(𝔤,𝔤)≅Hk(𝔤,𝔤*)*,𝔤* being equipped with the coadjoint
representation. We say that PD holds true for the cohomology of the 𝔤-module V if HN−k(𝔤,V)≅Hk(𝔤,V). PD holds true for the trivial cohomology.
However, it does not hold true in general for the adjoint cohomology. The NLAs
satisfying Poincaré duality for the adjoint cohomology are signalled with a ‡; among them, those quadratic NLAs have a ▪.
Recall that a Lie algebra 𝔤 is called quadratic if there exists a
nondegenerate symmetric bilinear form B on 𝔤 which is invariant, that is, B([x,y],z)+B(y,[x,z])=0 for all x,y,z∈𝔤. This amounts to the adjoint and coadjoint
representations being equivalent, and hence implies PD for the adjoint
cohomology. It is known that quadratic structures B on 𝔤 are in one-to-one correspondence with those
elements I∈⋀3𝔤* whose super-Poisson bracket {I,I} vanishes [17], the correspondence being B↦IB, with IB(x,y,z)=B([x,y],z) for all x,y,z∈𝔤. Recall that the super-Poisson bracket is, for ω∈⋀k𝔤*,π∈⋀𝔤*,{ω,π}=2(−1)k∑i,jB(yi,yj)(xi⌋ω)∧(xj⌋π), with ⌋ the (left) interior product, (xj)1≤j≤dim𝔤 the basis of 𝔤 (in which the commutation relations are given
in [8]) and yj such that B(yj,⋅)=ωj, with (ωj)1≤j≤dim𝔤 the dual basis to (xj)1≤j≤dim𝔤. There are only 6 quadratic non-Abelian NLAs
of dimension ≤7 (only one indecomposable in each dimension
5,6,7). Each of them has only one quadratic structure, up to equivalence under the
natural action of Aut𝔤. Here are Bs and IBs in the basis (ωj).(2.6)𝔤7,2.4:B=ω1⊗ω7+ω7⊗ω1+ω2⊗ω6+ω6⊗ω2−(ω3⊗ω5+ω5⊗ω3)+ω4⊗ω4;IB=ω1∧ω3∧ω4−ω1∧ω2∧ω5;𝔤6,3×ℂ:B=ω1⊗ω6+ω6⊗ω1−(ω2⊗ω5+ω5⊗ω2)+ω3⊗ω4+ω4⊗ω3+ω7⊗ω7;IB=ω1∧ω2∧ω3;𝔤5,4×ℂ2:B=ω1⊗ω5+ω5⊗ω1−(ω2⊗ω4+ω4⊗ω2)+ω3⊗ω3+ω6⊗ω6+ω7⊗ω7;IB=ω1∧ω2∧ω3;𝔤6,3:B=ω1⊗ω6+ω6⊗ω1−(ω2⊗ω5+ω5⊗ω2)+ω3⊗ω4+ω4⊗ω3;IB=ω1∧ω2∧ω3;𝔤5,4×ℂ:B=ω1⊗ω5+ω5⊗ω1−(ω2⊗ω4+ω4⊗ω2)+ω3⊗ω3+ω6⊗ω6;IB=ω1∧ω2∧ω3;𝔤5,4:B=ω1⊗ω5+ω5⊗ω1−(ω2⊗ω4+ω4⊗ω2)+ω3⊗ω3;IB=ω1∧ω2∧ω3.


## 3. About the Programs

All computations were made by developing programs with
the computer algebra systemReduce. The adjoint cohomologies have been
computed by program 1. Trivial cohomologies were computed twice: by program 2,
and by program 3 which computes via harmonic cocycles. Actually, those programs
do more than simply compute the dimensions of the cohomology: program 1
computes a basis for H2(𝔤,𝔤) and, when the commutation relations of 𝔤 are given in a basis that diagonalizes a
maximal torus, characters of the adjoint cohomology under the action of the
maximal torus; for trivial cohomology, the programs compute characters and
bases of the eigenspaces under the action of the maximal torus.Program 2 computes allHk(𝔤,ℂ) (0≤k≤N) and their respective bases and characters
independently, making no use of PD. Then PD shows up as a result. There is also
a variant program 2′ which does the same, yet offers the option to make use of the computed
bases of Hk(𝔤,ℂ) and HN−k(𝔤,ℂ) (2k≤N) to get the matrix of the bilinear form in PD
and modify the basis of HN−k(𝔤,ℂ) so as to get the dual basis in PD of the basis
of Hk(𝔤,ℂ).As to program 3, harmonic cohomology comes naturally
in the following way, which can be formulated for unimodular𝔤: suppose we already computed a basis of Hk(𝔤,ℂ)only for 2k≤N;
how to deduce by PD a basis of HN−k(𝔤,ℂ)?
Let ϱ:Ck(𝔤,ℂ)=⋀k𝔤*→CN−k(𝔤,ℂ)=⋀N−k𝔤 (0≤k≤N) be the isomorphism defined by ϱ(f)=Ω⌊f, where Ω=x1∧⋯∧xN and ⌊ denotes the (right) interior product. As ϱ(df)=(−1)k+1∂(ϱ(f)) for all f∈Ck(𝔤,ℂ) (∂ boundary operator) [18], ϱ defines an isomorphism Hk(𝔤,ℂ)→HN−k(𝔤,ℂ), which is actually (up to the factor (−1)k(N−k)) the one of (2.4) (V=ℂ). Now, what we look for is an explicit
identification algorithm HN−k(𝔤,ℂ)→HN−k(𝔤,ℂ) to be implemented in programs. For any subset I of {1,…,N}, denote ωI=ωi1∧⋯∧ωik for I={i1,…,ik} (1≤i1<⋯<ik≤N,1≤k≤N), ω∅=1, and similarly for xI. Let (⋅|⋅)
be the Hermitian
scalar product on ⋀𝔤*=⨁k=0NCk(𝔤,ℂ) obtained by decreeing the basis (ωI) to be orthonormal. For 1≤k≤N,
let z↦gz
be the conjugate
linear bijective map Ck(𝔤,ℂ)→Ck(𝔤,ℂ) defined by (f∣gz)=f(z) for all f∈Ck(𝔤,ℂ),z∈Ck(𝔤,ℂ) (we set g1=1 for k=0). Then for any subset I of {1,…,N},gxI=ωI and gϱ(ωI)=ρI,I′ωI′, where I′ is the complementary subset to I, and ρI,I′=(−1)NI,I′,NI,I′=card{(i,j)∈I×I′;j<i}. Let d* be the adjoint of d on ⋀𝔤*. Then d*gz=g∂z for all z∈⋀𝔤=⨁k=0NCk(𝔤,ℂ).d and d* are disjoint on ⋀𝔤* in the sense of [19], hence kerd/imd=⨁k=0NHk(𝔤,ℂ) is isomorphic to kerΔ=kerd∩imd*, where Δ=dd*+d*d.
Then {f∈Ck(𝔤,ℂ);Δf=0} is the kth harmonic cocycle space. It is contained in the kth cocycle space Zk(𝔤,ℂ). Each equivalence class of Zk(𝔤,ℂ) modulo coboundaries contains exactly one
harmonic cocycle.Lemma 3.1.
Suppose
that𝔤
is unimodular and
let f∈Zk(𝔤,ℂ)(0≤k≤N). Then gϱ(f)∈ZN−k(𝔤,ℂ) if and only if f is harmonic; in that case, gϱ(f) is also harmonic.Proof.
It is enough to proved(gϱ(f))=(−1)kgϱ(d*f), that is, dΦf=(−1)kΦd*f for all f∈Ck(𝔤,ℂ), with Φ the Hodge operator on ⋀𝔤* defined by Φf=gϱ(f). For I subset of cardinality k of {1,…,N},Φ2(ωI)=ρI,I′Φ(ωI′)=ρI,I′ρI′,IωI=(−1)k(N−k)ωI, hence Φ2=⨁k=0N(−1)k(N−k)×IdCk(𝔤,ℂ). Now, f∈Ck(𝔤,ℂ),d*Φf=d*(gϱ(f))=g∂ϱ(f)=(−1)k+1gϱ(df)=(−1)k+1Φ(df) implies successively d*Φ2f=(−1)N−k+1ΦdΦf and Φd*Φ2f=(−1)N−k+1Φ2dΦf=(−1)k(N−k+1)dΦf, which reads Φd*f=(−1)kdΦf.ThenHk(𝔤,ℂ) is isomorphic to the space of harmonic
cocycles and the map [f]↦[gϱ(f)] which assigns to the class of the harmonic
cocycle f the class of the harmonic cocycle gϱ(f) is a conjugate-linear isomorphism from Hk(𝔤,ℂ) onto HN−k(𝔤,ℂ). If ([ψj]) is a basis for Hk(𝔤,ℂ) (2k≤N) consisting of harmonic cocycles, then ([gϱ(ψj)]) is a basis for HN−k(𝔤,ℂ) consisting of harmonic cocycles. Hence we see
that the price to be paid for computing Hk(𝔤,ℂ) and their base only for 2k≤N,
yet get bases for the whole cohomology, is to go to harmonic cohomology. That
was implemented as an option in the variant program 3′: with that option on, the basis of HN−k(𝔤,ℂ) is computed as explained, then modified into
the dual basis in PD of the basis of Hk(𝔤,ℂ)(2k≤N).   With the option off, no use of PD occurs:
harmonic cocycles and bases are computed independently for each k (0≤k≤N).All programs handle dimensions up to 7 and (if
necessary) one continuous parameterL. Though they are meant for nilpotent Lie
algebras whose commutation relations are given in a basis that diagonalizes a
maximal torus, they can be directly applied to any Lie algebra of dimension ≤7 as well, not necessarily nilpotent nor
unimodular (except for the variants), giving explicit calculation of
cohomology: in that case, all material involving weights has simply to be
skipped. As to trivial cohomology, note that program 2 which computes Hk(𝔤,ℂ) up to k=7 makes it possible, thanks to PD, to write down
the dimensions of trivial cohomology for unimodular Lie algebras of dimensions
up to 15. In the same way, program 3 can handle higher dimensions. However,
restrictions may come on one hand from the amount of dynamic storage space
available, and on the other from the running time, which increases steeply as
the dimension N of the Lie algebra gets higher, typically for
the computation of the dimensions of trivial cohomology, with 4 GB RAM: 1 second for N=7,10 seconds for N=9,3 minutes for N=11,3 hours for N=13, up to 10 days for N=15.The programs are downloadable in the companion archive
[20] (programs 1,2,3
are, resp., geneLplus2007.red,ncl2007.red,nclhar2007.red, and the variants
program2′, program 3′ are, resp., ncld2007.red,ncldhar2007.red in [20]) hence we will not enter
technicalities about procedures here. Let us simply mention the following
concerning the continuous parameter. In the presence of the continuous
parameter L,
cocycles equations depend on certain unknowns and on L. They are linear with respect to the unknowns.
The programs define an algorithm which solves the equations over the rational
function field generated by the parameter, while keeping track of the divisions
that have been done. If the parameter is a zero of one of the polynomials by
which a division occurred, it may very well not be a singular value: one has to
compute again the cohomology for all such values.



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*Source: 805305-2008-12-01.xml*