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Starting with Lie's classification of finite-dimensional transitive Lie algebras of vector fields on $\mathbb C^2$ we construct Lie algebras of vector fields on the bundle $\mathbb C^2 \times \mathbb C$ by lifting the Lie algebras from the…

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group $G$, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a…

We develop a Lie algebraic approach to systematically calculate the evolution operator of the generalized two-dimensional quadratic Hamiltonian with time-dependent coefficients. Although the development of the Lie algebraic approach…

A method of classification of integrable equations on quad-graphs is discussed based on algebraic ideas. We assign a Lie ring to the equation and study the function describing the dimensions of linear spaces spanned by multiple commutators…

We implement two algorithms in MATHEMATICA for classifying automorphisms of lower-dimensional non-commutative Lie algebras. The first algorithm is a brute-force approach whereas the second is an evolutionary strategy. These algorithms are…

Derivations extend the concept of differentiation from functions to algebraic structures as linear operators satisfying the Leibniz rule. In Lie algebras, derivations form a Lie algebra via the commutator bracket of linear endomorphisms.…

We consider the Lie-algebraic notion of commutant in the setting of Poisson algebra. This provides a framework for deforming Hamiltonian differential equations. By taking a subalgebra of the algebra of integrals, and considering the set of…

Let $\mathbb{K}$ be a field, $R$ be an associative and commutative $\mathbb{K}$-algebra and $L$ be a Lie algebra over $\mathbb{K}$. We give some descriptions of injections from $L$ to Lie algebra of $\mathbb{K}$-derivations of $R$ in the…

We describe the reduction procedure for a symplectic Lie algebroid by a Lie subalgebroid and a symmetry Lie group. Moreover, given an invariant Hamiltonian function we obtain the corresponding reduced Hamiltonian dynamics. Several examples…

We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant…

We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four…

This paper introduces Lie groups in degenerate geometric (Clifford) algebras that preserve four fundamental subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations. We prove that…

We propose a perturbation algorithm for Hamiltonian systems on a Lie algebra $\mathbb{V}$, so that it can be applied to non-canonical Hamiltonian systems. Given a Hamiltonian system that preserves a subalgebra $\mathbb{B}$ of $\mathbb{V}$,…

Decomposition classes provide a way of partitioning the Lie algebras of an algebraic group into equivalence classes based on the Jordan decomposition. In this paper, we investigate the decomposition classes of the Lie algebras of connected…

We develop the Lie theory of Lie-admissible algebras whose product is enriched with higher operations modeled on directed graphs with a view to apply it to the deformation theories controlled by this kind of Lie algebras. We produce…

We consider the 2-generated free metabelian associative and Lie algebras over the complex field and the invariants of the dihedral groups of finite order acting on these algebras. In the associative case we find a finite set of generators…

A Lie system is the non-autonomous system of differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie…

We study the boundary-localized Lie algebra generated by the rank-one perturbation \(T = U + \varepsilon E\) of the unilateral shift on \(\ell^2(\mathbb{Z}_{\ge\ 0})\). While the polynomial algebra \(\langle T \rangle\) is abelian, the…

We study a Lie algebra $\mathcal A_{a_1,\ldots,a_{n-1}}$ of deformed skew-symmetric $n \times n$ matrices endowed with a Lie bracket given by a choice of deformed symmetric matrix. The deformations are parametrized by a sequence of real…

Automorphic Lie Algebras arise in the context of reduction groups introduced in the late 1970s in the field of integrable systems. They are subalgebras of Lie algebras over a ring of rational functions, defined by invariance under the…