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Let $G=H\ltimes K$ denote a semidirect product Lie group with Lie algebra $\mathfrak g=\mathfrak h \oplus \mathfrak k$, where $\mathfrak k$ is an ideal and $\mathfrak h$ is a subalgebra of the same dimension as $\mathfrak k$. There exist…

We study the affine variety $L_{n}(\mathfrak{g})$ of Lie algebra representations, the collection of all homomorphisms from an arbitrary $n$-dimensional Lie algebra into a fixed real semi-simple Lie algebra $\mathfrak{g}$. Using techniques…

The structure of the observable algebra ${\mathfrak O}_{\Lambda}$ of lattice QCD in the Hamiltonian approach is investigated. As was shown earlier, ${\mathfrak O}_{\Lambda}$ is isomorphic to the tensor product of a gluonic…

Given a group $X$ we study the algebraic structure of the compact right-topological semigroup $\lambda(X)$ consisting of maximal linked systems on $X$. This semigroup contains the semigroup $\beta(X)$ of ultrafilters as a closed…

Let $\mathfrak g$ be a semisimple Lie algebra and $\mathfrak k\subset\mathfrak g$ be a reductive in $\mathfrak g$ subalgebra. A $(\mathfrak g, \mathfrak k)$-module is a $\mathfrak g$-module which after restriction to $\mathfrak k$ becomes a…

A new highly symmetrical model of the compact Lie algebra $\mathfrak{g}^c_2$ is provided as a twisted ring group for the group $\mathbb{Z}_2^3$ and the ring $\mathbb{R}\oplus\mathbb{R}$. The model is self-contained and can be used without…

Let g = Lie(G) be the Lie algebra of a simple algebraic group G over an algebraically closed field of characteristic 0. Let e be a nilpotent element of g and let g_e = Lie(G_e) where G_e stands for the stabiliser of e in G. For g classical,…

For a perfect Lie algebra $\mathfrak{h}$ we classify all Lie algebras containing $\mathfrak{h}$ as a subalgebra of codimension $1$. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product…

Let $\Gamma$ denote a finite, undirected, connected graph, with vertex set $X$. Fix a vertex $x \in X$. Associated with $x$ is a certain subalgebra $T=T(x)$ of ${\rm Mat}_X(\mathbb C)$, called the subconstituent algebra. The algebra $T$ is…

Derived brackets provide a mechanism for generating algebraic structures from graded Lie superalgebras, with applications in Poisson geometry, mathematical physics, and the theory of algebroids. In this paper, we present a complete…

Let $G$ and $\tilde G$ be connected complex reductive Lie groups, $G$ semisimple. Let $\Lambda^+$ be the monoid of dominant weights for a positive root system $\Delta^+$, and let $l(w)$ be the length of a Weyl group element $w$. Let…

Let $\mathfrak g$ be a semisimple Lie algebra, $\vartheta\in {\sf Aut}(\mathfrak g)$ a finite order automorphism, and $\mathfrak g_0$ the subalgebra of fixed points of $\vartheta$. Recently, we noticed that using $\vartheta$ one can…

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical…

Let Gamma be a Q-polynomial distance-regular graph with vertex set X, diameter D geq 3 and adjacency matrix A. Fix x in X and let A*=A*(x) be the corresponding dual adjacency matrix. Recall that the Terwilliger algebra T=T(x) is the…

In this article studies questions about the existence of left-invariant K\"{a}hler and semi-para-K\"{a}hler structures on six-dimensional unsolvable Lie groups whose Lie algebras are semidirect products. According to the classification…

For the $p$-dimensional filiform Lie algebra ${\mathfrak m}_2(p)$ over a field ${\mathbb F}$ of prime characteristic $p\ge 5$ with nonzero Lie brackets $[e_1,e_i] = e_{i+1}$ for $1<i<p$ and $[e_2,e_i]=e_{i+2}$ for $2<i<p-1$, we show that…

We will introduce a notion of normal subshifts. A subshift $(\Lambda,\sigma)$ is said to be normal if it satisfies a certain synchronizing property called $\lambda$-synchronizing and is infinite as a set. We have lots of purely infinite…

Let ${\mathbb{TL}_n^{\! \mathbb Q}} $ be the rational Temperley-Lieb algebra, with loop parameter $ 2 $. In the first part of the paper we study the seminormal idempotents $ E_{ \mathfrak{t}} $ for ${\mathbb{TL}_n^{\! \mathbb Q}}$ for $…

Given a scalar parameter $q$, the $q$-deformed Heisenberg algebra $\mathcal{H}(q)$ is the unital associative algebra with two generators $A,B$ that satisfy the $q$-deformed commutation relation $AB-qBA= I$, where $I$ is the multiplicative…

We introduce a linear algebraic object called a bidiagonal pair. Roughly speaking, a bidiagonal pair is a pair of diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the…