# Graded Symmetry Groups: Plane and Simple

@article{Roelfs2021GradedSG, title={Graded Symmetry Groups: Plane and Simple}, author={Martin Roelfs and Steven De Keninck}, journal={ArXiv}, year={2021}, volume={abs/2107.03771} }

The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the classic matrix Lie algebra approach, while retaining information about the number of reflections in a given transformation. This imposes a graded structure on Lie groups, not evident in their matrix representation. By embracing this graded structure, the…

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## References

SHOWING 1-10 OF 35 REFERENCES

### Lie-groups as Spin groups.

- Mathematics
- 1993

It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of geometric algebra is available…

### Geometric invariant decomposition of SU(3)

- Mathematics
- 2021

A novel invariant decomposition of diagonalizable n × n matrices into n commuting matrices is presented. This decomposition is subsequently used to split the fundamental representation of su(3) Lie…

### Clifford algebra of points, lines and planes

- MathematicsRobotica
- 2000

The Clifford algebra for the group of rigid body motions is described and three examples of the use of the algebra are given: a computer graphics problem on the visibility of the apparent crossing of a pair of lines, an assembly problem concerning a double peg-in-hole assembly, and a problem from computer vision on finding epipolar lines in a stereo vision system.

### Geometric Algebra for Physicists

- Mathematics, Physics
- 2003

Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with…

### Lie Groups, Lie Algebras, and Representations: An Elementary Introduction

- Mathematics
- 2004

Preface Part I General Theory 1 Matrix Lie Groups 1.1 Definition of a Matrix Lie Group 1.2 Examples of Matrix Lie Groups 1.3 Compactness 1.4 Connectedness 1.5 Simple Connectedness 1.6 Homomorphisms…

### Geometry, Kinematics, and Rigid Body Mechanics in Cayley-Klein Geometries

- Mathematics
- 2011

The 19 century witnessed a dramatic freeing of the human intellect from its naive assumptions. This development is particularly striking in mathematics. New geometries were discovered which…

### Square Root and Logarithm of Rotors in 3D Conformal Geometric Algebra Using Polar Decomposition

- MathematicsGuide to Geometric Algebra in Practice
- 2011

This chapter gives explicit formulas for the square root and the logarithm of rotors in 3D CGA and classifies the types of conformal transformations and their orbits.

### Unitary representations of the inhomogeneous Lorentz group

- Physics
- 1964

SummaryA realization of the unitary representation [m, s] of the inhomogeneous Lorentz group is derived, in the case of nonzero mass, which has simple transformation properties yet has no superfluous…

### Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics

- Mathematics
- 1984

1 / Geometric Algebra.- 1-1. Axioms, Definitions and Identities.- 1-2. Vector Spaces, Pseudoscalars and Projections.- 1-3. Frames and Matrices.- 1-4. Alternating Forms and Determinants.- 1-5.…

### Geometric algebra for computer science - an object-oriented approach to geometry

- Computer ScienceThe Morgan Kaufmann series in computer graphics
- 2007

An introduction to Geometric Algebra that will give a strong grasp of its relationship to linear algebra and its significance for 3D programming of geometry in graphics, vision, and robotics is found.