Related papers: Erlangen Program at Large-0: Starting with the gro…

The powerful group theoretical formalism of potential algebras is extended to non-Hermitian Hamiltonians with real eigenvalues by complexifying so(2,1), thereby getting the complex algebra sl(2,\C) or $A_1$. This leads to new types of both…

We use the shear construction to construct and classify a wide range of two-step solvable Lie groups admitting a left-invariant SKT structure. We reduce this to a specification of SKT shear data on Abelian Lie algebras, and which then is…

In the past few years, an action of $\mathrm{PGL}_2(\mathbb F_q)$ on the set of irreducible polynomials in $\mathbb F_q[x]$ has been introduced and many questions have been discussed, such as the characterization and number of invariant…

Let G be a two generator subgroup of PSL(2,C). The Jorgensen number J(G) of G is defined by J(G)=inf{ |tr^2 A-4|+|tr[A,B]-2| ; G=<A,B>}. If G is a non-elementary Kleinian group, then J(G) >= 1. This inequality is called Jorgensen's…

We give the construction of weighted Lagranngiann Grassmannians$wLGr(3,6)$ and weighted partial $A_3$ flag variety $wFl_{1,3}$ coming from the symplectic Lie group $Sp(6,\mathbb C)$ and the general linear group $GL(4,\mathbb C)$…

We discuss algebraic and analytic structure of rational Lax operators. With algebraic reductions of Lax equations we associate a reduction group - a group of twisted automorphisms of the corresponding infinite dimensional Lie algebra. We…

In this paper we introduce the systematic study of invariant functions and equivariant mappings defined on Minkowski space under the action of the Lorentz group. We adapt some known results from the orthogonal group acting on the Euclidean…

We develop and describe continuous and discrete transforms of class functions on a compact semisimple, but not simple, Lie group $G$ as their expansions into series of special functions that are invariant under the action of the even…

Let V be a finite-dimensional superspace and G a simple (or a ``close'' to simple) matrix Lie superalgebra, i.e., a Lie subsuperalgebra in GL(V). Under the classical invariant theory for G we mean the description of G-invariant elements of…

Using the dilaton scalar and axion pseudoscalar fields we construct a number of scalars and differential forms which are symmetric under the $\mathbf{Z}_2$-subgroup of the group $SL(2, \mathbf{R})$. These invariants enable us to establish…

We describe the full group of isometries of each left invariant Riemannian metric on the simply connected unimodular nilpotent or solvable $(R)$-type Lie groups of dimension four.

We combine the coordinate method and Erlangen program in the framework of noncommutative geometry through an investigation of symmetries of noncommutative coordinate algebras. As the model we use the coherent states construction and the…

This is an informal announcement of results to be described and proved in detail in a paper to appear. We give various results on the structure of approximate subgroups in linear groups such as $\SL_n(k)$. For example, generalising a result…

We determine fundamental systems of invariants for complex solvable rigid Lie algebras having nonsplit nilradicals of characteristic sequence $(3,1,..,1)$, these algebras being the natural followers of solvable algebras having Heisenberg…

Deformed generalized gauge groups, whch were created from physical considerations and made it possible to clarify some long-standing problems in physics, such as the problem of motion and the problem of the energy of the gravitational…

Let $G\subset\GL(V)$ be a complex reductive group. Let $G'$ denote $\{\phi\in\GL(V)\mid p\circ\phi=p\text{for all} p\in\C[V]^G\}$. We show that, in general, $G'=G$. In case $G$ is the adjoint group of a simple Lie algebra $\lieg$, we show…

The systematic study of CR manifolds originated in two pioneering 1932 papers of \'Elie Cartan. In the first, Cartan classifies all homogeneous CR 3-manifolds, the most well-known case of which is a one-parameter family of left-invariant CR…

In this paper, we resolve a conjecture of Green and Liebeck [Disc. Math., 343 (8):117119, 2019] on codes in $PGL(2,q)$. To be specific, we show that: if $D$ is a dihedral subgroup of order $2(q+1)$ in $G=PGL(2,q)$, and $A=\{g\in G: g^{q+1}=…

We compute all signatures of $PSL_2(\mathbb{F}_7)$, and $PSL_2(\mathbb{F}_{11})$ which classify all orientation preserving actions of the groups $PSL_2(\mathbb{F}_7)$, and $PSL_2(\mathbb{F}_{11})$ on compact, connected, orientable surfaces…

Symmetry group of Lie algebras and superalgebras constructed from (\epsilon,\delta) Freudenthal- Kantor triple systems has been studied. Especially, for a special (\epsilon,\epsilon) Freudenthal- Kantor triple, it is SL(2) group. Also,…