The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on Cartans method of moving frames.
The invariants of solvable triangular Lie algebras with one nilindependent diagonal element are studied exhaustively. Bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on Cartans method of moving frames and the special technique developed for triangular and related algebras in [J. Phys. A: Math. Theor. 40 (2007), 7557-7572]. The conjecture of Tremblay and Winternitz [J. Phys. A: Math. Gen. 34 (2001), 9085-9099] on the number and form of elements in the bases is completed and proved.
We show that all results of Yasar and Ozer [Comput. Math. Appl. 59 (2010), 3203-3210] on symmetries and conservation laws of a nonconservative Fokker-Planck equation can be easily derived from results existing in the literature by means of using the fact that this equation is reduced to the linear heat equation by a simple point transformation. Moreover nonclassical symmetries and local and potential conservation laws of the equation under consideration are exhaustively described in the same way as well as infinite series of potential symmetry algebras of arbitrary potential orders are constructed.
A purely algebraic algorithm for computation of invariants (generalized Casimir operators) of Lie algebras by means of moving frames is discussed. Results on the application of the method to computation of invariants of low-dimensional Lie algebras and series of solvable Lie algebras restricted only by a required structure of the nilradical are reviewed.
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants (generalized Casimir operators) are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40, 113; math-ph/0606045], is used to determine the invariants. A conjecture of [J. Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent invariants and their form, is corroborated.
