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Let $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main…
In this paper, we study the connectedness of the commuting graph of a general Lie algebra and provide a process to determine whether the commuting graph is connected or not, as well as to compute an upper bound for its diameter. In…
It is well known that a finite-dimensional Lie algebra over a field of characteristic zero is simple exactly when its derivation algebra is simple. In this paper we characterize those Lie algebras of arbitrary dimension over any field that…
Using a classical and quantum mechanical analysis, we show that the magnetic field gives rise to dynamical symmetries of a three-dimensional axially symmetric two-electron quantum dot with a parabolic confinement. These symmetries manifest…
The notion of hidden symmetry algebra used in the context of exactly solvable systems is re-examined from the purely algebraic way, analyzing subspaces of commuting polynomials that generate finite-dimensional quadratic algebras. By…
We study connections between the ring of symmetric functions and the characters of irreducible finite-dimensional representations of quantum affine algebras. We study two families of representations of the symplectic and orthogonal Lie…
A general discussion is presented of the response of a gauge-field system to external magnetic fields, in the light of a theorem due to S. Elitzur. As a result a natural understanding emerges of some recent puzzling results from lattice MC…
The normal form for a system of ode's is constructed from its polynomial symmetries of the linear part of the system, which is assumed to be semi-simple. The symmetries are shown to have a simple structure such as invariant function times…
The conjecture of a hidden $E_{10}$ symmetry of M-theory is supported by the close connection between the dynamics of D=11 supergravity near a spacelike singularity and a truncation of an one-dimensional $\sigma$-model with $E_{10}$…
We present a formalization, in the theorem prover Lean, of the classification of solvable Lie algebras of dimension at most three over arbitrary fields. Lie algebras are algebraic objects which encode infinitesimal symmetries, and as such…
In this work, we reconsider the study of 2D materials involving double lattice structures associated with periodic polygons. In tessellated periodic representation, it appears two periodic polygons of $k$ sides of unequal side lengths at…
$E_8$ is prominent in mathematics and theoretical physics, and is generally viewed as an exceptional symmetry in an eight-dimensional space very different from the space we inhabit; for instance the Lie group $E_8$ features heavily in…
We study an algebraic structure of magical supergravities in three dimensions. We show that if the commutation relations among the generators of the quasi-conformal group in the super-Ehlers decomposition are in a particular form, then one…
We investigate 2-dimensional Viscoelastic equations with a view of Lie groups. In this sense, we answer question of the symmetry classification. We provide the algebra of symmetry and build the optimal system of Lie subalgebras. Reductions…
We study 10D super Yang-Mills theory with the gauge groups $E_6$, $E_7$ and $E_8$. We consider the torus/orbifold compacfitication with magnetic fluxes and Wilson lines. They lead to 4D interesting models with three families of quarks and…
The distribution of superclusters in the Local Supercluster neighbourhood presents such a remarkable periodicity that some kind of network must fit the observed large scale structure. A three dimension chessboard has been suggested$^{1}$.…
This paper studies the canonical symmetric connection $\nabla$ associated to any Lie group $G$. The salient properties of $\nabla$ are stated and proved. The Lie symmetries of the geodesic system of a general linear connection are…
We investigate symplectic nilpotent Lie groups with Lagrangian normal subgroups. We show that there exists a bijection between the isomorphism classes of nilpotent Lie groups with Lagrangian normal subgroups and the isomorphism classes of…
We study the homology and cohomology groups of super Lie algebra of supersymmetries and of super Poincare Lie algebra in various dimensions. We give complete answers for (non-extended) supersymmetry in all dimensions $\leq 11$. For…
We give a new construction of the Lie algebra of type $E_8$, in terms of $3\times3$ matrices, such that the Lie bracket has a natural description as the matrix commutator. This leads to a new interpretation of the Freudenthal-Tits magic…