Some equations and their solutions in free lie algebra

Let L be a free Lie algebras of rank at least 2 over an arbitrary field K. In this paper we study equations of the form [x , u] + [y , v] = 0, where u, v are components of L and x, y are indeterminates. In this case where u and v are free generators of L, we exhibit two series of solutions, we work out the determinates of the homogeneous components of the solution space, and we determine its radical. In the general case we show that the result on free generator coefficients are sufficient to obtain the solution space up to finite codimension. As an application we determine the radical of the bilinear equation [x1 , x2] + [x3 , x4] = 0. Also as an application we give an algorithm for the exhibit the solution of the equation [x , a] + [y , b] = 0, where L = L(a , b) is a free Lie algebra of rank 2.