Title:

Linear equations over free Lie algebras

In this thesis, we study equations of the form $[x_1,u_1]+[x_2, u_2]+\ldots+[x_k,u_k]=0$ over a free Lie algebra $L$, where $k>1$ and the coefficients $u_1, u_2, \ldots,u_k$ belong to $L$. The starting point of this research is a paper [22], in which the authors embarked on a systematic study of very concrete linear equations over free Lie algebras. They focused on the given equations in the case where $k=2$. We generalise and develop a number of the results on equations with two variables to equations with an arbitrary number of indeterminates. Most of the results refer to the case where the coefficients coincide with the free generators of $L$. Throughout our research, we study some features of the solution space of these equations such as the homogenous structure and the fine homogenous structure. The main achievement in this work is that we give a detailed description of the solution space. Then we obtain explicit bases for some specific fine homogeneous components of the solution space, in particular, we give a basis for the "multilinear'' fine homogenous component. Moreover, we generalise earlier results on commutator calculus using the "language'' of free Lie algebras and apply them to determine the radical and the coordinate algebra of the solution space of the given equations.
