Application of superalgebra homology groups to distinguish Engel-like structures

For a long time, tangent bundle of a manifold and the direct sum of multi-vector fields is a prototype of $\mathbb{Z}$-graded superalgebra with the Schouten bracket, and the grading is defined by $a-1$ for $a$-multivector fields. Recently, we noticed the direct sum of differential forms of a manifold has $\mathbb{Z}$-graded superalgebra structure with a super bracket $[A,B]=(-1)^{a}d(A\wedge B)$ for $a$-form $A$, and the grading of $a$-form is given by $-a-1$. By Lie derivation, we also see that $\sum\limits_{p=0}^{n}\Lambda^{p}\mathrm{T}^{*}(M)\oplus\mathrm{T}(M)$ has $\mathbb{Z}$-graded Lie superalgebra structure, which is a “super” superalgebra of the two superalgebra described above.

For a given $\mathbb{Z}$-graded Lie superalgebra, there is a notion of the weighted (co)chain complex and (co)homology groups. In general, those objects are infinite dimensional and hard to understand the entire properties. If we restrict the manifold above to a finite dimensional Lie group, and multivector fields and differential forms to (left) invariant fields and forms, then (co)chain spaces become finite dimensional. Still studying $\mathbb{Z}$-graded Lie super algebra, which is a generalization of Lie algebra, we encounter complicated manipulation of sign changes depending on each degree. In this note, we introduce our trial of using Maple software to gather possible equations of “Engel like” Lie algebra structures by Jacobi identify, then simply solve them and get six possibilities. Getting weighted chain complex and manipulate its homology groups, we claim that those six possibilities are not isomorphic as Lie algebras in generic.

$\mathbb{Z}$-graded Lie super algebra (which is a generalization of Lie algebra), super bracket, the Schouten bracket, chain spaces, the boundary operator, the homology groups, the Betti numbers, and the Euler number.

For a 4-dimensional manifold $M$, if rank 2 distribution $D$ satisfies $D[2]:=D+[D,D]$ has rank 3 and $D[3]:=D[2]+[D[2],D[2]]$ has rank 4, then the pair $(M,D)$ is called an Engel structure. Here, we apply the above notion for 4-dimensional Lie algebra $\mathfrak{g}$. The pair $(\mathfrak{g},D)$ is a Engel-type if $D$ is a 2-dimensional subspace of $\mathfrak{g}$, and $\textrm{dim}D[2]=3$ where $D[2]:=D+[D,D]$, and $\textrm{dim}D[3]=4$ where $D[3]:=D[2]+[D[2],D[2]]$. From the dimensional restriction, we see a specific basis $y_{1},y_{2}$ so that $[y_{1},y_{2}]=y_{3}\in[D,D]$. We choose $y_{4}$ by $[y_{1},y_{3}]=y_{4}$. Thus,

We let know Maple2021 about the relations above and check Jacobi identities by our maple script Engel-try-1.mpl.

We show how Maple works well.

As you expect, in the original Engel-try-1.mpl if we declare save at the almost end of the file, we get an output file Engel-try-1-out4.txt. Preparing the general form of Lie bracket candidate of Engel-type, we gather Jacobi conditions myEqn. Solving them, we have 6 cases, and denote them as ourSolInd[i] and ourSolDep[i] for $i=1,\ldots,6$. As you see, the independent variable $C_{1,4,4}$ of the second and third cases is assumed to be non-zero. ourSolDep[2] looks rather complicated comparing the others. Our main purpose of this note is to distinguish those 6 cases are Lie algebra isomorphic or not by watching their homology groups in superalgebra sense. Our strategy is simple, that is, if two Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ are isomorphic as Lie algebras, their superalgebras are isomorphic, and so weighted homology groups coincide for each primary weight and the superalgebra type. Contrary, if the corresponding homology groups are not equal for some same type of superalgebras and for some primary weight, then the original Lie algebras are not isomorphic.

We have six types after solving the Jacobi identity conditions. In the type 2 and 3, some coefficients are fractions of denominator $C_{1,4,4}$, this means it is non-zero. Looking at those six types, the type 2 seems to be more complicated.

We have a super homology group theory for superalgebra, especially derived from Lie algebras which come from Engel like structure as we got above. We ask Maple to manipulate superalgebra homology groups with the primary weight 0, 1, or 2. In the following, we show the outputs by Maple for the six types in ”generic”. In the weight 0 case, the super homology group is just the Lie algebra homology group and it is natural to treat 0-th chain space $\mathfrak{g}^{0}=\mathbb{R}$, and modify the output by Maple and we conclude the Euler number is 0 for the weight 0 case, too.

Then we have the proposition below.

We have superalgebras on the exterior algebra of differential forms of manifold . We restrict our attention to the invariant forms of Lie groups, and have the super homology groups. We refer to [8] about precise definitions, but in this note, the grading for forms is sharp, namely, the grade of $p$-form is $-(p+1)$. On the contrary, the grade of $p$-multi vector field is $p-1$. Given a (primary) weight $wt$, if $wt$ is non-negative, then the chain space is empty. Otherwise we have chain space bounded to $|wt|$, which is 1-dimensional, consists of $\displaystyle\underbrace{1\bigtriangleup\cdots\bigtriangleup 1}_{|wt|}$, which is a kernel element.

We concentrate for the weight $-5$ here. We follow the definition of $m$ -th chain space. It consists of $\displaystyle\mathfrak{h}_{-1}^{a_{1}}\wedge\mathfrak{h}_{-2}^{a_{2}}\wedge\mathfrak{h}_{-3}^{a_{3}}\wedge\mathfrak{h}_{-4}^{a_{4}}\wedge\mathfrak{h}_{-5}^{a_{5}}$ where $\displaystyle a_{1}+\cdots+a_{5}=m$ and $\displaystyle(-1)a_{1}+\cdots+(-5)a_{5}=-5$. Directly, we see that $m<=5$ and $a_{i}$ comes from the Young diagrams of area 5 and length $m$. Thus the common table becomes as seen on the right end. When $m=5$, the chain space is $\mathfrak{h}_{-1}^{5}$ and 1-dimensional. When $m=4$, the chain space is $\mathfrak{h}_{-1}^{3}\bigtriangleup\mathfrak{h}_{-2}$ and 4-dimensional. When $m=3$, the chain space is $(\mathfrak{h}_{-1}\bigtriangleup\mathfrak{h}_{-2}^{2})\oplus(\mathfrak{h}_{-1}^{2}\bigtriangleup\mathfrak{h}_{-3})$ and 12-dimensional. When $m=2$, the chain space is $(\mathfrak{h}_{-2}\bigtriangleup\mathfrak{h}_{-3})\oplus(\mathfrak{h}_{-1}\bigtriangleup\mathfrak{h}_{-4})$ and 28-dimensional. When $m=1$, the chain space is $\mathfrak{h}_{-5}=\verb|&^|(z_{1},z_{2},z_{3},z_{4})$ and 1-dimensional. By manipulating weight $(-5)$ super homology groups below, we classify 6 types into 3 as Type1, Type2 – Type4, and Type5 – Type 6, which is not new in computation of the previous section.

There is a notion of characteristic foliations in Engel theory, which is rank 1 distribution $\mathfrak{L}$ of Engel distribution $D$ satisfying $[\mathfrak{L},D^{2}]\subset D^{2}$ where $D^{2}:=D+[D,D]$ (, and $D^{3}:=D^{2}+[D^{2},D^{2}]$). In our six cases, they are given by $ay_{1}+by_{2}$ for mySol[3], and $C_{2,3,4}y_{1}-y_{2}$ for the others, up to constant. It will be interesting to study their contributions to super homology discussion.

So far, we studied Engel-like structures first fix dimensional conditions and next check Lie algebra structure. As [4] points out, there is a complete classification list of 4-dimensional Lie algebra by [10]. Since we unfortunately do not have access to the original paper [10], we use the list in [4].