Graded Symmetry Groups: Plane and Simple

Plane Symmetry Groups

This paper discusses plane symmetry groups, also known as planar crystallographic groups or wallpaper groups. The seventeen unique plane symmetry groups describe the symmetries found in two-dimensional patterns such as those found on weaving patterns, the work of the artist M.C. Escher, and of course wallpaper. We shall discuss the fundamental components and properties of plane symmetry groups.

Graded Algebras, Anti-involutions, Simple Groups and Symmetric Spaces

Graded Simple Lie Algebras and Graded Simple Representations

For any finitely generated abelian group Q, we reduce the problem of classification of Q-graded simple Lie algebras over an algebraically closed field of “good” characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional Q-graded simple Lie algebras over any algebraically closed field of characteristi...

Visual Discrimination of the 17 Plane Symmetry Groups

Within most of the 17 plane symmetry groups, individual symmetry operations act in multiple, nonequivalent ways. This, and the fact that many groups can be realized on the basis of different unit cells and generating regions, poses difficulties for visual discrimination and identification. Because of inherent confounds, only few of the groups can be studied by traditional experimental methodolo...

Graded Quantum Groups

Starting from a Hopf algebra endowed with an action of a group π by Hopf automorphisms, we construct (by a “twisted” double method) a quasitriangular Hopf π-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular Hopf π-coalgebras for any finite group π and for infinite groups π such as GLn(k). In particular, we define the graded quantum groups, which are Hopf π-coalg...