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In this note we present explicit and elementary formulas for the correspondence between the group of special Lorentz transformation $SO^+(3,1)$, on the one hand, and its spin group $SL(2,\mathbb{C})$, on the other hand. Although we will not…
We describe geometrically the classical and quantum inhomogeneous groups $G_0=(SL(2, \BbbC)\triangleright \BbbC^2)$ and $G_1=(SL(2, \BbbC)\triangleright \BbbC^2)\triangleright \BbbC$ by studying explicitly their shape algebras as a spaces…
We present a family of non-abelian groups for which the hidden subgroup problem can be solved efficiently on a quantum computer.
We give the complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. This classifications recovers other known classification results in the…
Special generic maps are higher dimensional versions of Morse functions with exactly two singular points, characterizing spheres topologically except 4-dimensional cases and 4-dimensional standard spheres. The class of such maps also…
We develop practical techniques to compute with arithmetic groups $H\leq \mathrm{SL}(n,\mathbb{Q})$ for $n>2$. Our approach relies on constructing a principal congruence subgroup in $H$. Problems solved include testing membership in $H$,…
The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space, de Sitter 3-space, and Minkowski motion group is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a…
In this paper, we establish a complete structural description of flat Lorentzian Lie groups, i.e., Lie groups endowed with a flat left invariant Lorentzian metric, thereby resolving a long-standing open problem in the theory of…
We discuss the basic properties of Lie groupoids, Lie algebroids and Lie pseudo-groups in view of applying these techniques to the analysis of Jordan-H\"older resolutions and, subsequently, to the integration of partial differential…
In this paper, we introduce and study five families of Lie groups in degenerate Clifford geometric algebras. These Lie groups preserve the even and odd subspaces and some other subspaces under the adjoint representation and the twisted…
The group action methods have been playing an important role in recent studies about the configuration problems inside a compact set $E$ in Euclidean spaces with given Hausdorff dimension. In this paper, we further explore the group action…
In this article, we develop the geometry of canonical stratifications of the spaces $\mathcal{M}_{0,n}$ and prepare ground for studying the action of the Galois group $Gal (\overline{\mathbb{Q}} /\mathbb{Q})$ upon strata. We define and…
We show a Prime Geodesic Theorem for the group SL3(Z), counting those geodesics whose lifts lie in the split Cartan subgroup. This is the first arithmetic Prime Geodesic Theorem of higher rank for a non-cocompact group.
We present a mathematical model for a physical theory that is compatible with Einstein's Special Relativity Theory. Our model consists of three pseudo-complex dimensions, representing three real dimensions of space, dual to what could be…
Recent work has constructed neural networks that are equivariant to continuous symmetry groups such as 2D and 3D rotations. This is accomplished using explicit Lie group representations to derive the equivariant kernels and nonlinearities.…
Duality groups of Abelian gauge theories on four manifolds and their reduction to two dimensions are considered. The duality groups include elements that relate different space-times in addition to relating different gauge-coupling…
Lie transformation groups containing a one-dimensional subgroup acting cyclically on a manifold are considered. The structure of the group is found to be considerably restricted by the existence of a one-dimensional subgroup whose orbits…
In this paper we consider the problem of computing the difference Galois groups of order three equations for a large class of difference operators including the shift operator (Case S), the $q$-difference operator (Case Q), the Mahler…
We construct exceptional field theory for the duality group SL(3)$\times$SL(2). The theory is defined on a space with 8 `external' coordinates and 6 `internal' coordinates in the $(3,2)$ fundamental representation, leading to a…
A remarkable number of different numerical algorithms can be understood and analyzed using the concepts of symmetric spaces and Lie triple systems, which are well known in differential geometry from the study of spaces of constant curvature…