Journal of Generalized Lie Theory and Applications

The study offers the entire classification of automorphic Lie algebras based on sln(C), where sln(C) contains no trivial summands, the poles are in any of the exceptional G-orbits in C and the symmetry group G acts on sln(C) through inner automorphisms. The analysis of the algebras within the framework of traditional invariant theory is a crucial aspect of the categorization. This offers both a strong computational tool and raises new concerns from an algebraic standpoint (such as structure theory) that indicate to other uses for these algebras outside of the realm of integrable systems. The study demonstrates, in particular, that the class of automorphic Lie algebras connected to TOY groups (tetrahedral, octahedral and icosahedral groups) depend solely on the automorphic functions of the group, making them group independent Lie algebras. This may be proven by generalising the classical idea to the case of Lie algebras over a polynomial ring and creating a Chevalley normal form for these algebras.

Numerous variations of physical issues, such as those in fluid dynamics, solid mechanics, plasma physics, quantum field theory, as well as in mathematics and engineering, include nonlinear partial differential equations. Systems of nonlinear partial differential equations have also been shown to appear in chemical and biological applications. The analytical analysis of a fully generalised (3+1)-dimensional nonlinear potential Yu- Toda-Sasa-Fukuyama equation, with applications in physics and engineering, is presented in this article. In contrast to earlier study on the problem that has already been done, the extended form of the potential Yu-Toda-Sasa-Fukuyama equation is investigated in greater detail in this paper, leading to the achievement of many novel solutions that are of interest. The nonlinear partial differential equation is fundamentally reduced to an integrable form by the use of the Lie group theory, allowing for direct integration of the outcome.

In this paper we study the Lie symmetries of the canonical connection on Lie groups for the special case when the Lie algebra has a codimension two abelian nilradical. In this particular case, we have only one algebra in dimension four, namely A_4,12 and three algebras in dimension five; namely . For each of these algebras we investigate and classify the symmetry algebra associated with its geodesic equations.