Minimal Non-Abelian Groups to Lie Groups/Algebras

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A group is called minimal non-abelian if it is non-abelian and all proper subgroups are abelian.

Does this notion also exist with Lie groups or algebras? As an example, consider the Lie algebra defined by the generators $\{e_1,e_2\}, [e_1,e_2]=e_2$. Do you have a paper at hand?

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These Lie algebras were called semiabelian by some authors. Some old papers are On the structure of simple-semiabelian Lie-algebras by Farnsteiner and On simple semiabelian $p$-adic lie algebras by myself.

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[Math] Torsion for Lie algebras and Lie groups

I don't know the answer to the actual question but here is a situation which should be similar but simpler. Consider an integral polynomial group law $G$, i.e., a group scheme structure on the affine space over $\mathbb Z$. We then have a torsion free finitely generated nilpotent group $G(\mathbb Z)$ and an integral Lie algebra $\mathfrak g_{\mathbb Z}$. We also have a compact manifold $X:=G(\mathbb R)/\mathbb(\mathbb Z)$ and as $G(\mathbb R)$ is contractible we have that the cohomology of $X$ is equal to the cohomology of $G(\mathbb Z)$ (as $X$ is a $K(G(\mathbb Z),1)$). We also have that $G(\mathbb Q)$ is the Malcev completion of $G(\mathbb Z)$ and (consequently) we have that the inclusion $G(\mathbb Z)\hookrightarrow G(\mathbb Q)$ induces an isomorphism on cohomology with rational coefficients. For more obvious reasons (the Chevalley-Eilenberg complexes are isomorphic) $\mathfrak g_{\mathbb Z}\hookrightarrow \mathfrak g_{\mathbb Q}$ induces an isomorphism in cohomology.

Furthermore, $\mathfrak g_{\mathbb Q}$ is the Lie algebra that correspond to the torsion free, divisible nilpotent group $G(\mathbb Q)$ under the Malcev correspondence (i.e., the Campbell-Baker-Hausdorff formula) which results in an isomorphism between the Lie algebra cohomology (with rational coefficients) of $\mathfrak g_{\mathbb Q}$ and the group cohomology (again with rational coefficients) of $G(\mathbb Q)$. Combining everything we get an isomorphism between the cohomology of $\mathfrak g_{\mathbb Z}$ and of $G(\mathbb Z)$ (equal to that of $X$) both with rational coefficients. If I remember correctly Larry Lambe computed examples where the torsion of the integral cohomology was different (perhaps even for strictly upper triangular $4\times4$-matrices).

We can approach the comparison somewhat differently: To begin with, there is a third cohomology involved, the cohomology of the group scheme (over $\mathbb Z$) $G$. We can put all three of these cohomologies on an equal footing. We may consider the subcomplex of the standard complex for computing group cohomology consisting of polynomial cochains $G(\mathbb Z)\hookrightarrow \mathbb Z$, i.e., as $G$ as a scheme is just affine space $G(\mathbb Z)=\mathbb Z^m$ for some $m$ and we demand that the cochain $\mathbb Z^{mn}\hookrightarrow \mathbb Z$ be given by a polynomial. This is clearly a subcomplex and computes the group scheme cohomology of $G$. We may consider another subcomplex consisting of the functions given by polynomials with rational coefficients (still inducing a map $G(\mathbb Z)\hookrightarrow \mathbb Z$). It again is a subcomplex which has been shown to compute the cohomology of the group $G(\mathbb Z)$ (i.e., the inclusion into all cochains is a quasi-isomorphism). Note that such functions are sums of products of binomal polynomials in the projection functions (of $\mathbb Z^{mn}$ to its factors). The third complex is not quite a subcomplex consisting it does of rational valued functions that can be written as sums of products of divided powers of the projection functions. I have never completely checked all the details but am pretty certain that this complex computes the Lie algebra cohomology of $\mathfrak g_{\mathbb Z}$.

As polynomials with integer coefficients are both polynomials with rational coefficients and divided power polynomials we get an explicit map from group scheme cohomology to both group cohomology of integral points and to the integral Lie algebra cohomology. Tensoring with $\mathbb Q$ induces isomorphisms of complexes giving an explicit realisation of the rational isomorphisms. Note however that the group scheme cohomology is definitely distinct from the Lie algebra cohomology and the group cohomology already in the case of the additive group scheme: The $2$-cocycle $((x+y)^p-x^p-y^p)/p$, $p$ a prime, is polynomial which is not a polynomial coboundary. It is however the coboundary of the cochain $(x^p-x)/p$ which is integer-valued as well as of the divided power cochain $x^p/p$.

[Math] Examples of finite dimensional non simple non abelian Lie algebras

A nice example to play around with is the Lie algebra of upper triangular matrices. It is solvable, so has plenty of ideals and things like that.

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[Math] Torsion for Lie algebras and Lie groups

[Math] Examples of finite dimensional non simple non abelian Lie algebras

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