Reflection Groups in Geometry Essay Example | Topics and Well Written Essays - 4000 words

Reflection Groups in Geometry - Essay Example A reflection group is a distinct group produced by multiple reflections of a finite-dimensional (Euclidean) space. Weyl groups of simple Lie algebras and symmetry groups of regular polytypes are examples of finite reflection groups while infinite groups comprise the Weyl groups of infinite-dimensional Kac–Moody algebras and the triangle groups similar to ordinary tessellations of the hyperbolic plane and Euclidean plane. With regard to symmetry, discrete isometry groups of broad Riemannian manifolds that are formed by reflections are grouped into classes leading to hyperbolic reflection groups (corresponding to hyperbolic space), affine (corresponding to Euclidean space) and finite reflection groups (then-sphere). Coxeter groups are reflection groups that are finitely generated. Unlike reflection groups, Coxeter groups are abstract groups that have a certain structure generated by reflections. An investigation of the topology and geometry of reflection groups will help us comp rehend the theoretic properties of the group. The concept of reflection in a Euclidean space and the hypothesis of discrete groups of motions resulting from reflections has its origin in the study of space polyhedral and plane regular polygons that goes back to early mathematics. In the present day, reflection groups are common in many areas of mathematical research, and geometers encounter them as special convex polytopes or discrete groups of isometries of Riemannian spaces with even curvature. On the other hand, an algebraist encounters reflection groups in group theory, particularly in the representation theory, Coxeter groups and invariant theory. Other areas of mathematics where they may be encountered include the theory of arrangements of hyperplanes, a theory of combinations and permutation, a theory of modular forms and quadratic forms, low-dimensional topology, singularity theory, and the theory of hyperbolic real and complex manifolds (Yau 1986).  Ã‚