Topology of SO(3) For Kids
We present a simple visual description of the topology of the space of three-dimensional rotations, requiring just intuition, imagination and no advanced math.
math.HO↗
arXiv subjects
Orlin Stoytchev
Publications and source records attributed to Orlin Stoytchev.
A Class of Efficient Presentations of Finite Simple Groups
We exhibit a new presentation of the (equilateral) Von Dyck groups $D(2,3,n), \ n\ge 3$, in terms of two generators of order $n$ satisfying three relations, one of which is Artin's braid relation. By dropping the relation which fixes the order of the generators we obtain the universal covering groups of the corresponding Von Dyck groups. In the cases $n=3,\, 4,\,5$, these are respectively the double covers of the finite rotational tetrahedral, octahedral and icosahedral groups. When $n\ge 6$ we obtain infinite covers of the corresponding infinite Von Dyck groups. The interesting cases arise for $n\ge 7$ when these groups act as discrete groups of isometries of the hyperbolic plane. Imposing a suitable third relation we obtain a host of (efficient) presentations of finite simple Chevalley groups of type $A_1$ as well as the sporadic Janko group $J_2$.
math.GR↗
From Foucault's Pendulum to the Gauss--Bonnet Theorem
We present a self-contained proof of the Gauss-Bonnet theorem for two-dimensional surfaces embedded in $R^3$ using just classical vector calculus. The exposition should be accessible to advanced undergraduate and non-expert graduate students. It may be viewed as an illustration and exercise in multivariate calculus and a motivation to go deeper into the fields of geometry and topology.
math.HO↗
The Fundamental Group of $SO(n)$ Via Quotients of Braid Groups
We describe an algebraic proof of the well-known topological fact that $π_1(SO(n)) \cong Z/2Z$. The fundamental group of $SO(n)$ appears in our approach as the center of a certain finite group defined by generators and relations. The latter is a factor group of the braid group $B_n$, obtained by imposing one additional relation and turns out to be a nontrivial central extension by $Z/2Z$ of the corresponding group of rotational symmetries of the hyperoctahedron in dimension $n$.
math.HO↗
A Note on Topologically-Trivial Braids
We give a simple characterization of braids that can be unplaited keeping separately their upper ends and their lower ends tied together
math.GT↗
Modular Conjugation and the Implementation of Supersymmetry
Any Z_2-graded C*-dynamical system with a self-adjoint graded-KMS functional on it can be represented (canonically) as a Z_2-graded algebra of bounded operators on a Z_2-graded Hilbert space, so that the grading of the latter is compatible with the functional. The modular conjugation operator plays a crucial role in this reconstruction. The results are generalized to the case of an unbounded graded-KMS functional having as dense domain the union of a net of C*-subalgebras. It is shown that the modulus of such an unbounded graded-KMS functional is KMS.
math-ph↗
Touching the $Z_2$ in Three-Dimensional Rotations
A simple and self-contained proof is presented of the well-known fact that the fundamental group of SO(3) is $Z_2$, using a relationship between closed paths in SO(3) and braids.
math.HO↗
Irreducible Representations of a Class of Current Algebras of Etingof and Frenkel
A class of representations is described for the central extensions, found by Etingof and Frenkel, of current algebras over Riemann surfaces. Their irreducibility is proved. The possibility/impossibility to obtain integrable representations within that class is discussed briefly.
math.RT↗