Classification of 8-dimensional Nilsolitons by Symbolic Computation

Abstract

In this paper, we develop an algorithm to classify 8 dimensional nilsolitons with simple nilsoliton derivation. We restrict our classifications to the nilsolitons corresponding to singular Gram matrix with nullity 1-3. This work can be considered as a continuation paper to our previous study where we introduced a procedure to classify algebras in dimension 8 that admit simple derivations and singular Gram matrices U. Having the singular Gram matrices, there exists infinitely many solutions to Uv = [1](m), where the solutions are exactly the squares of the structure constants. Also, the structure constants have to satisfy the Jacobi identity for the algebra to be a Lie algebra. In our previous work, we did not introduce a procedure to create and solve the Jacobi identity(s). In this study, we take care of this issue by using computer algorithms for each index set. Thus, we complete classification of all 8 dimensional in-decomposable nilsolitons with the nullity of corresponding Gram matrix is in the set {0,1,2,3} . We provide several examples to illustrate the algorithm. For the implementation process of the newly introduced algorithm, we use MATLAB R2020b.