Related papers: Classification of real and complex 3-qutrit states
We consider the problem of classifying oriented cycles of linear mappings $F^p\to F^q\to\dots\to F^r\to F^p$ over a field $F$ of complex or real numbers up to homeomorphisms in the spaces $F^p,F^q,\dots,F^r$. We reduce it to the problem of…
We study the GIT-quotient of the Cartesian power of projective space modulo the projective orthogonal group. A classical isomorphism of this group with the Inversive group of birational transformations of the projective space of one…
Motivated by relating the representation theory of the split real and $p$-adic forms of a connected reductive algebraic group $G$, we describe a subset of $2^r$ orbits on the complex flag variety for a certain symmetric subgroup. (Here $r$…
We review known real forms of the quantum orthogonal groups SO_q(N). New *-conjugations are then introduced and we contruct all real forms of quantum orthogonal groups. We thus give an RTT formulation of the *-conjugations on SO_q(N) that…
We classify finite groups $G$ in $\mathrm{PGL}_{4}(\mathbb{C})$ such that $\mathbb{P}^3$ is $G$-birationally rigid.
Loop Quantum Gravity (LQG) is one of the leading approaches to unify quantum physics and General Relativity (GR). The Hilbert space of LQG is spanned by spin-networks which describe the local geometry of quantum space-time. Simulation of…
A complete classification of the multivalued coset groups of order $3$ is given. The proof is based on the classification of rank $3$ groups having regular normal subgroups.
We present the classification of coadjoint orbits of the unitriangular group $UT(7, K)$. We also describe subregular orbits of $UT(n, K)$ for an arbitrary $n$.
In order to define the complex exceptional Lie groups $ {F_4}^C, {E_6}^C, {E_7}^C, {E_8}^C $ and these compact real forms $ F_4,E_6,E_7,E_8 $, we usually use the Cayley algebra $ \mathfrak{C} $. In the present article, we consider replacing…
A rotation in a Euclidean space V is an orthogonal map on V which acts locally as a plane rotation with some fixed angle. We give a classification of all pairs of rotations in finite-dimensional Euclidean space, up to simultaneous…
The aim of this paper is to numerically investigate the orbital dynamics of the circular planar restricted problem of five bodies. By numerically integrating several large sets of initial conditions of orbits we classify them into three…
We study the orbit of $\mathbb{R}$ under the Bianchi group $\operatorname{PSL}_2(\mathcal{O}_K)$, where $K$ is an imaginary quadratic field. The orbit, called a Schmidt arrangement $\mathcal{S}_K$, is a geometric realisation, as an…
Some aspects of phase transitions can be more conveniently studied in the orbit space of the action of the symmetry group. After a brief review of the fundamental ideas of this approach, I shall concentrate on the mathematical aspect and…
In the three-dimensional projective space PG(3,q) over the finite field F_q with q elements, we consider the normal rational curve known as a twisted cubic and the projectivity group G_q that fixes it. For q = 2, 3, 4, we solve the open…
We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and extend it to a complete…
Supergroups are defined in the framework of $\dZ_2$-graded Clifford algebras over the fields of real and complex numbers, respectively. It is shown that cyclic structures of complex and real supergroups are defined by Brauer-Wall groups…
The relational complexity of a subgroup $G$ of $\mathrm{Sym}(\Omega)$ is a measure of the way in which the orbits of $G$ on $\Omega^k$ for various $k$ determine the original action of $G$. Very few precise values of relational complexity…
We numerically investigate the orbital dynamics of a spacecraft, or a comet, or an asteroid in the Pluto-Charon system in a scattering region around Charon using the planar circular restricted three-body problem. The test particle can move…
We consider quotients of the group algebra of the $3$-string braid group $B_3$ by $p$-th order generic polynomial relations on the elementary braids. In cases $p=2,3,4,5$ these quotient algebras are finite dimensional. We give…
In this paper, we give explicit equations for homogeneous spaces corresponding to a rational isogeny of degree $3$. An explicit set of elliptic curves with elements of order $3$ in their Tate-Shafarevich group is constructed. Combining this…