Classification Problems in Lie algebra

Question

My field of interest is Lie group analysis of PDEs, currently I am struggling with techniques for classification of Lie algebra into mutually conjugate classes of 1- and 2-dimensional sub-algebras through adjoint action.

1. I have studied technique of Peter Olver (chapter 3, example 3.12) where successive adjoint action on general element in Lie algebra leads to optimal classification of Lie algebra.
2. The second technique is by given Patera and co-authors through abstract Lie algebra.

I believe that the later technique is far superior to that of Olver, but as I am being not much familiar with abstract Lie algebra, so I don't know how should I proceed for classification of Lie algebra using technique given by Patera and co-authors. As an example I list Lie algebra for well known KdV equation as follow:

$V_{1}=\partial_{t}, V_{2}=\partial_{x}, V_{3}=x\,\partial_{x}+2t\,\partial_{t}, V_{4}=u\,\partial_{u}, V_{5}=\,2t\,\partial_{x}-xu\,\partial_{u}$, $V_{6}=\,4tx\,\partial_{x}+4t^{2}\partial_{t}-(x^{2}+2t)\,u\,\partial_{u}$

Can anybody explain in detail, the classification of above Lie algebra into 1- and 2-dimensional optimal sub-algebras based on technique of Patera and co-authors. It seems to me that, firstly one have to decompose abstract Lie algebra into some sort of direct sums of sub-algebras but I am not sure.

I have studied following articles as well:

1. Group‐invariant solutions and optimal systems for multidimensional hydrodynamics
2. A Note on Optimal Systems for the Heat Equation
3. A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems

Additional Example :

I have asked also this type of question to expert in Lie algebra classification for classification of 1-dimensional Lie algebra using abstract Lie algebra, the question was actually snapshot pdf doc with 8-dimensional Lie algebra which want to classify

The expert responded as

The first thing to do before starting a sub algebra classification is to identify the Lie algebra that you are considering as an abstract Lie algebra in a basis independent way. In your case the structure is very simple

Your algebra L is the direct sum of 4 indecomposable Lie algebras:

 L = L_1 + L_2 + L_3 + L_4

with

L_1 = { V-2-V_1, V_3, V_4} = sl(2,R),  L_2 = {V_6, V_7, V_5} , L_3 }, 

L_3= {V_1+V_2}, L_3 ={{V_8} . 

The one dimensional algebras {V_1+V_2} and {V_8} form a basis for the center of L. As a direct sum of in-decomposable Lie algebras the algebra L has two types of sub-algebras: splitting and non-splitting. Splitting subalgebras are themselves direct sums of subalgebras of the indecomposable components of L. The non splitting subalgebras are obtained from the splitting ones by a procedure called the Goursat twist. A representative list of nontrivial sub-algebras of sl(2,R) is given by the two-dimensional algebra

 {V_1-V_2, V_3} 

and three one dimensional sub-algebras

 {V_3-V_4} =o(2), {V-2-V__1} = o(1,1)

and

V_3 ( a nilpotent element in sl(2,R).

There are 3 two-dimensional mutually non conjugate sub-algebras of s_3,1, namely {V_6,V_5}, {V_6,V_7} and {V_5,V_7}. Mutually non-conjugate one dimensional subalgebras are {V_5},{V_6},{V_7},{V_5+ V_7},{V_5 -V_7}. The last two are conjugate by an outer automorphism (like time reversal or space reflection). These can be used if you are studying an equation invariant under such transformations.

Answer 1

The classification of $6$-dimensional abstract Lie algebras over $\mathbb{R}$ or $\mathbb{C}$ goes as follows. By Levi's theorem we have $L\cong S\ltimes R$, where $R$ denotes the solvable radical of $L$, and $S$ a (semisimple) Levi subalgebra. We have two cases. Either $L$ is solvable, or $L$ is not solvable. In the second case, $S\neq 0$, and we can use the classification of semisimple Lie algebras of dimension $n\le 6$ for the Levi subalgebras. There are only a few cases to consider for $S\neq 0$, e.g., $S=\mathfrak{sl}_2(\mathbb{C})$, or $S=\mathfrak{sl}_2(\mathbb{C})\times \mathfrak{sl}_2(\mathbb{C})$. Over $\mathbb{R}$ we also have to consider $\mathfrak{so}_3(\mathbb{R})$ and direct products. Now it is quite easy to classify all non-solvable Lie algebras of dimension $6$.
For the second case, $L$ is solvable, and we can use a classification of the nilpotent radical $N$ of $R$ to obtain a classification. This is expained here.

Once you have the classification of $6$-dimensional Lie algebras, e.g., the classification by Turkowsky, you can just check about $1$-and $2$-dimensional subalgebras.