Adjoint and Trivial Cohomologies of Nilpotent Complex Lie Algebras of Dimension

Abstract

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

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: ≤7≤7 As to 𝔤_7,0.4(𝜆^′₎≅𝔤_7,0.4(𝜆)⇔𝜆^=±𝜆;𝔤_{7,1.1(𝑖𝜆′)}≅𝔤_{7,1.1(𝑖𝜆)} if and only if 𝜆⇔.𝜆^=𝜆;𝔤_{7,1.2(𝑖𝜆′)}≅𝔤_{7,1.2(𝑖𝜆)}⇔(𝜆^=𝜆or𝜆𝜆^=1);𝔤_{7,1.3(𝑖𝜆′)}≅𝔤_{7,1.3(𝑖𝜆)}⇔𝜆^=𝜆;𝔤_{7,2.1(𝑖𝜆′)}≅𝔤_{7,2.1(𝑖𝜆)}⇔𝜆^= with 𝔤_{7,3.1(𝑖𝜆)},𝔤_{7,3.1(𝑖𝜆′)}≅𝔤_{7,3.1(𝑖𝜆)} any element of the group of transformations of 𝜆^=𝑠(𝜆) which is isomorphic to the symmetric group 𝑠

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 𝜆∶𝐺={𝜆,1/𝜆,1−𝜆,1/(1−𝜆),1−1/𝜆,𝜆/(𝜆−1)}, we give the sequences 𝒮₃. and 𝔤, for, respectively, the spaces of adjoint cocycles and cohomology groups, along with the sequence of Betti numbers, that is, the trivial cohomology (dim𝑍^𝑗(𝔤,𝔤))_{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 (dim𝐻^𝑗(𝔤,𝔤))_{0≤𝑗≤dim𝔤}. 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.

The nonisomorphic 7-dimensional NLAs having the same adjoint cohomology come in 13 pairs, as shown in Table 2. 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.

2. Cohomology Tables

The results appear as in Tables 3, 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.

Table 4 

Cohomology table for NLAs of dimension 7: rank 1.

Table 5 

Cohomology table for NLAs of dimension 7: rank 2. (continues on next page).

Table 6 

Cohomology table for NLAs of dimension 7: rank 3.

Table 7 

Cohomology table for NLAs of dimension 7: rank 4.

Table 8 

Cohomology table for NLAs of dimension 7: rank 5.

Table 9 

Cohomology table for NLAs of dimension 7: rank 6.

Table 10 

Cohomology table for NLAs of dimension 7: rank 7 (Abelian).

Table 11 

Cohomology table for NLAs of dimension 6.

Table 12 

Cohomology table for NLAs of dimension 5.

Table 13 

Cohomology table for NLAs of dimension 𝔤₃×𝐶𝟚

Recall that 𝐶²(abel.)‡■(dim2,rk2) the center of 𝐶(abel.)‡■(dim1,rk1) and 𝑍⁰(𝔤,𝔤)=𝐻⁰(𝔤,𝔤)=𝔠 Recall also the following facts about Poincaré duality (PD) ([16, Theorem 6.10]). For any complex 𝔤,-dimensional Lie algebra 𝑍¹(𝔤,𝔤)=Der(𝔤),𝐻¹(𝔤,𝔤)=Der(𝔤)/ad(𝔤). and any 𝑁-module 𝔤 cohomology and homology are related by the formulae (with 𝔤 the contragredient 𝑉,-module, and 𝑉^∗)𝔤whence0≤𝑘≤𝑁Now, as the algebra 𝐻𝐻_𝑘(𝔤,𝑉)≅𝐻^𝑘𝔤,𝑉^∗^∗,(2.1)_𝑘(𝔤,𝑉)≅𝐻_𝑁−𝑘(𝔤,𝑉⊗_ℂ(^𝑁𝔤)^∗),(2.2) considered in this paper is nilpotent, hence unimodular (i.e., 𝐻^𝑘(𝔤,𝑉)≅𝐻^𝑁−𝑘(𝔤,𝑉^∗⊗_ℂ𝑁𝔤)^∗.(2.3) for all 𝔤), one has Tr(ad𝑋)=0 (the trivial module), and (2.2), (2.3) read, respectively,𝑋∈𝔤⋀^𝑁𝔤≅ℂ

Hence, 𝐻𝐻^𝑘(𝔤,𝑉)≅𝐻_𝑁−𝑘(𝔤,𝑉),(2.4)_𝑘(𝔤,𝑉)≅𝐻^𝑁−𝑘𝔤,𝑉^∗^∗.(2.5) and 𝐻^𝑁(𝔤,𝔤)≅𝐻₀(𝔤,𝔤)≅𝔤/[𝔤,𝔤]≅(𝐻¹(𝔤,ℂ))^∗, being equipped with the coadjoint representation. We say that PD holds true for the cohomology of the 𝐻^𝑁−𝑘(𝔤,𝔤)≅𝐻^𝑘(𝔤,𝔤^∗)^∗,𝔤^∗-module 𝔤 if 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 𝐻^𝑁−𝑘(𝔤,𝑉)≅𝐻^𝑘(𝔤,𝑉). among them, those quadratic NLAs have a ‡;. Recall that a Lie algebra ■ is called quadratic if there exists a nondegenerate symmetric bilinear form 𝔤 on 𝐵 which is invariant, that is, 𝔤 for all 𝐵([𝑥,𝑦],𝑧)+𝐵(𝑦,[𝑥,𝑧])=0 This amounts to the adjoint and coadjoint representations being equivalent, and hence implies PD for the adjoint cohomology. It is known that quadratic structures 𝑥,𝑦,𝑧∈𝔤. on 𝐵 are in one-to-one correspondence with those elements 𝔤 whose super-Poisson bracket 𝐼∈⋀³𝔤^∗ vanishes [17], the correspondence being {𝐼,𝐼} with 𝐵↦𝐼_𝐵, for all 𝐼_𝐵(𝑥,𝑦,𝑧)=𝐵([𝑥,𝑦],𝑧) Recall that the super-Poisson bracket is, for 𝑥,𝑦,𝑧∈𝔤., with 𝜔∈⋀^𝑘𝔤^∗,𝜋∈⋀𝔤^∗,{𝜔,𝜋}=2(−1)^𝑘∑_𝑖,𝑗𝐵(𝑦_𝑖,𝑦_𝑗)(𝑥_𝑖⌋𝜔)∧(𝑥_𝑗⌋𝜋) the (left) interior product, ⌋ the basis of (𝑥_𝑗)_{1≤𝑗≤dim𝔤} (in which the commutation relations are given in [8]) and 𝔤 such that 𝑦_𝑗 with 𝐵(𝑦_𝑗,⋅)=𝜔^𝑗, the dual basis to (𝜔^𝑗)_{1≤𝑗≤dim𝔤}. There are only 6 quadratic non-Abelian NLAs of dimension (𝑥_𝑗)_{1≤𝑗≤dim𝔤} (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 ≤7 Here are Aut𝔤.s and 𝐵s in the basis 𝐼_𝐵(𝜔^𝑗).

3. About the Programs

All computations were made by developing programs with the computer algebra system Reduce. 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 𝔤_7,2.4∶𝐼𝐵_𝐵=𝜔¹⊗𝜔⁷+𝜔⁷⊗𝜔¹+𝜔²⊗𝜔⁶+𝜔⁶⊗𝜔²−𝜔³⊗𝜔⁵+𝜔⁵⊗𝜔³+𝜔⁴⊗𝜔⁴;𝔤_6,3=𝜔¹∧𝜔³∧𝜔⁴−𝜔¹∧𝜔²∧𝜔⁵;×ℂ∶𝐼𝐵_𝐵=𝜔¹⊗𝜔⁶+𝜔⁶⊗𝜔¹−𝜔²⊗𝜔⁵+𝜔⁵⊗𝜔²+𝜔³⊗𝜔⁴+𝜔⁴⊗𝜔³+𝜔⁷⊗𝜔⁷;𝔤_5,4=𝜔¹∧𝜔²∧𝜔³;×ℂ²∶𝐼𝐵_𝐵=𝜔¹⊗𝜔⁵+𝜔⁵⊗𝜔¹−𝜔²⊗𝜔⁴+𝜔⁴⊗𝜔²+𝜔³⊗𝜔³+𝜔⁶⊗𝜔⁶+𝜔⁷⊗𝜔⁷;𝔤_6,3=𝜔¹∧𝜔²∧𝜔³;∶𝐼𝐵_𝐵=𝜔¹⊗𝜔⁶+𝜔⁶⊗𝜔¹−𝜔²⊗𝜔⁵+𝜔⁵⊗𝜔²+𝜔³⊗𝜔⁴+𝜔⁴⊗𝜔³;𝔤_5,4=𝜔¹∧𝜔²∧𝜔³;×ℂ∶𝐼𝐵_𝐵=𝜔¹⊗𝜔⁵+𝜔⁵⊗𝜔¹−𝜔²⊗𝜔⁴+𝜔⁴⊗𝜔²+𝜔³⊗𝜔³+𝜔⁶⊗𝜔⁶;𝔤_5,4=𝜔¹∧𝜔²∧𝜔³;∶𝐼𝐵_𝐵=𝜔¹⊗𝜔⁵+𝜔⁵⊗𝜔¹−𝜔²⊗𝜔⁴+𝜔⁴⊗𝜔²+𝜔³⊗𝜔³;=𝜔¹∧𝜔²∧𝜔³.(2.6) and, when the commutation relations of 𝐻²(𝔤,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 all 𝔤 (𝐻^𝑘(𝔤,ℂ)) 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 0≤𝑘≤𝑁 which does the same, yet offers the option to make use of the computed bases of 2^ and 𝐻^𝑘(𝔤,ℂ) (𝐻^𝑁−𝑘(𝔤,ℂ)) to get the matrix of the bilinear form in PD and modify the basis of 2𝑘≤𝑁 so as to get the dual basis in PD of the basis of 𝐻^𝑁−𝑘(𝔤,ℂ)

As to program 3, harmonic cohomology comes naturally in the following way, which can be formulated for unimodular 𝐻^𝑘(𝔤,ℂ). suppose we already computed a basis of 𝔤∶only for 𝐻^𝑘(𝔤,ℂ); how to deduce by PD a basis of 2𝑘≤𝑁? Let 𝐻^𝑁−𝑘(𝔤,ℂ) (𝜚∶𝐶^𝑘(𝔤,ℂ)=⋀^𝑘𝔤^∗→𝐶_𝑁−𝑘(𝔤,ℂ)=⋀^𝑁−𝑘𝔤) be the isomorphism defined by 0≤𝑘≤𝑁, where 𝜚(𝑓)=Ω⌊𝑓 and Ω=𝑥₁∧⋯∧𝑥_𝑁 denotes the (right) interior product. As ⌊ for all 𝜚(𝑑𝑓)=(−1)^𝑘+1𝜕(𝜚(𝑓)) (𝑓∈𝐶^𝑘(𝔤,ℂ) boundary operator) [18], 𝜕 defines an isomorphism 𝜚 which is actually (up to the factor 𝐻^𝑘(𝔤,ℂ)→𝐻_𝑁−𝑘(𝔤,ℂ),) the one of (2.4) ((−1)^{𝑘(𝑁−𝑘)}). Now, what we look for is an explicit identification algorithm 𝑉=ℂ to be implemented in programs. For any subset 𝐻_𝑁−𝑘(𝔤,ℂ)→𝐻^𝑁−𝑘(𝔤,ℂ) of 𝐼 denote {1,…,𝑁}, for 𝜔^𝐼=𝜔^𝑖1∧⋯∧𝜔^𝑖𝑘 (𝐼={𝑖₁,…,𝑖_𝑘}), 1≤𝑖₁<⋯<𝑖_𝑘≤𝑁,1≤𝑘≤𝑁 and similarly for 𝜔^∅=1, Let 𝑥_𝐼. be the Hermitian scalar product on (⋅|⋅) obtained by decreeing the basis ⋀𝔤^∗=⨁^𝑁𝑘=0𝐶^𝑘(𝔤,ℂ) to be orthonormal. For (𝜔^𝐼) let 1≤𝑘≤𝑁, be the conjugate linear bijective map 𝑧↦𝑔_𝑧 defined by 𝐶_𝑘(𝔤,ℂ)→𝐶^𝑘(𝔤,ℂ) for all (𝑓∣𝑔_𝑧)=𝑓(𝑧) (we set 𝑓∈𝐶^𝑘(𝔤,ℂ),𝑧∈𝐶_𝑘(𝔤,ℂ) for 𝑔₁=1). Then for any subset 𝑘=0 of 𝐼 and {1,…,𝑁},𝑔_𝑥𝐼=𝜔^𝐼 where 𝑔_{𝜚(𝜔𝐼)}=𝜌_𝐼,𝐼^′𝜔^𝐼′, is the complementary subset to I, and 𝐼^ Let 𝜌_𝐼,𝐼^′=(−1)^{𝑁′𝐼,𝐼},𝑁_𝐼,𝐼^′=card{(𝑖,𝑗)∈𝐼×𝐼^;𝑗<𝑖}. be the adjoint of 𝑑^∗ on 𝑑 Then ⋀𝔤^∗. for all 𝑑^∗𝑔_𝑧=𝑔_𝜕𝑧 and 𝑧∈⋀𝔤=⨁^𝑁𝑘=0𝐶_𝑘(𝔤,ℂ).𝑑 are disjoint on 𝑑^∗ in the sense of [19], hence ⋀𝔤^∗ is isomorphic to ker𝑑/im𝑑=⨁^𝑁𝑘=0𝐻^𝑘(𝔤,ℂ) where kerΔ=ker𝑑∩im𝑑^∗,. Then Δ=𝑑𝑑^∗+𝑑^∗𝑑 is the {𝑓∈𝐶^𝑘(𝔤,ℂ);Δ𝑓=0} harmonic cocycle space. It is contained in the 𝑘th cocycle space 𝑘th Each equivalence class of 𝑍^𝑘(𝔤,ℂ). modulo coboundaries contains exactly one harmonic cocycle.

Lemma 3.1. Suppose that 𝑍^𝑘(𝔤,ℂ) is unimodular and let 𝔤. Then 𝑓∈𝑍^𝑘(𝔤,ℂ)(0≤𝑘≤𝑁) if and only if 𝑔_𝜚(𝑓)∈𝑍^𝑁−𝑘(𝔤,ℂ) is harmonic; in that case, 𝑓 is also harmonic.

Proof. It is enough to prove 𝑔_𝜚(𝑓) that is, 𝑑(𝑔_𝜚(𝑓))=(−1)^𝑘𝑔_𝜚(𝑑^∗_𝑓), for all 𝑑Φ𝑓=(−1)^𝑘Φ𝑑^∗𝑓 with 𝑓∈𝐶^𝑘(𝔤,ℂ), the Hodge operator on Φ defined by ⋀𝔤^∗ For Φ𝑓=𝑔_𝜚(𝑓). subset of cardinality 𝐼 of 𝑘 hence {1,…,𝑁},Φ²(𝜔^𝐼)=𝜌_𝐼,𝐼^′Φ(𝜔^𝐼′)=𝜌_𝐼,𝐼^′𝜌_𝐼^′_,𝐼𝜔^𝐼=(−1)^{𝑘(𝑁−𝑘)}𝜔^𝐼, Now, Φ²=⨁^𝑁𝑘=0(−1)^{𝑘(𝑁−𝑘)}×Id_𝐶^𝑘_(𝔤,ℂ). implies successively 𝑓∈𝐶^𝑘(𝔤,ℂ),𝑑^∗Φ𝑓=𝑑^∗(𝑔_𝜚(𝑓))=𝑔_{𝜕𝜚(𝑓)}=(−1)^𝑘+1𝑔_{𝜚(𝑑𝑓)}=(−1)^𝑘+1Φ(𝑑𝑓) and 𝑑^∗Φ²𝑓=(−1)^{𝑁−𝑘+1}Φ𝑑Φ𝑓 which reads Φ𝑑^∗Φ²𝑓=(−1)^{𝑁−𝑘+1}Φ²𝑑Φ𝑓=(−1)^{𝑘(𝑁−𝑘+1)}𝑑Φ𝑓,

Then Φ𝑑^∗𝑓=(−1)^𝑘𝑑Φ𝑓. is isomorphic to the space of harmonic cocycles and the map 𝐻^𝑘(𝔤,ℂ) which assigns to the class of the harmonic cocycle [𝑓]↦[𝑔_𝜚(𝑓)] the class of the harmonic cocycle 𝑓 is a conjugate-linear isomorphism from 𝑔_𝜚(𝑓) onto 𝐻^𝑘(𝔤,ℂ) If 𝐻^𝑁−𝑘(𝔤,ℂ). is a basis for ([𝜓_𝑗]) (𝐻^𝑘(𝔤,ℂ)) consisting of harmonic cocycles, then 2𝑘≤𝑁 is a basis for ([𝑔_{𝜚(𝜓𝑗)}]) consisting of harmonic cocycles. Hence we see that the price to be paid for computing 𝐻^𝑁−𝑘(𝔤,ℂ) and their base only for 𝐻^𝑘(𝔤,ℂ), yet get bases for the whole cohomology, is to go to harmonic cohomology. That was implemented as an option in the variant program 2𝑘≤𝑁: with that option on, the basis of 3^ is computed as explained, then modified into the dual basis in PD of the basis of 𝐻^𝑁−𝑘(𝔤,ℂ) With the option off, no use of PD occurs: harmonic cocycles and bases are computed independently for each 𝐻^𝑘(𝔤,ℂ)(2𝑘≤𝑁). (𝑘).

All programs handle dimensions up to 7 and (if necessary) one continuous parameter 0≤𝑘≤𝑁 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 𝐿. up to 𝐻^𝑘(𝔤,ℂ) 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 𝑘=7 of the Lie algebra gets higher, typically for the computation of the dimensions of trivial cohomology, with 4 GB RAM: 𝑁 second for 1𝑁=7, seconds for 10𝑁=9, minutes for 3 hours for 𝑁=11,3 up to 10 days for 𝑁=13,

The programs are downloadable in the companion archive [20] (programs 1,2,3 are, resp., 𝑁=15.2007.𝚐𝚎𝚗𝚎𝙻𝚙𝚕𝚞𝚜2007.𝚛𝚎𝚍,𝚗𝚌𝚕2007.𝚛𝚎𝚍,𝚗𝚌𝚕𝚑𝚊𝚛, and the variants program𝚛𝚎𝚍, program 2^ are, resp., 3^2007.𝚗𝚌𝚕𝚍2007.𝚛𝚎𝚍,𝚗𝚌𝚕𝚍𝚑𝚊𝚛 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 𝚛𝚎𝚍, 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.