Graded Symmetry Groups: Plane and Simple MARTIN ROELFS∗, KU Leuven, Belgium STEVEN DE KENINCK∗, University of Amsterdam, The Netherlands The symmetries described by Pin groups are the result of combining a finite “Every two dimensional 3-reflection can be decom- number of discrete reflections in (hyper)planes. The current work shows how posed as a translation along a line followed or preceded an analysis using geometric algebra provides a picture complementary to by a reflection in the same line.” that of the classic matrix Lie algebra approach, while retaining information Key to understanding theorem1 and its proof is a graded and very about the number of reflections in a given transformation. This imposes a graded structure on Lie groups, not evident in their matrix representation. By geometric perspective on symmetry groups, which we take some embracing this graded structure, the invariant decomposition theorem was time to explain in section2. Following Hamilton, we build all isomet- proven: any composition of : linearly independent reflections can be decom- ric transformations by composing reflections, while devoting extra posed into d:/2e commuting factors, each of which is the product of at most attention to the graded structure this imposes. The factors of any two reflections. This generalizes a conjecture by M. Riesz, and has e.g.the isometry are reflections and bireflections: a pair of reflections that Mozzi-Chasles’ theorem as its 3D Euclidean special case. To demonstrate its form either a rotation, translation, or hyperbolic rotation (boost). utility, we briefly discuss various examples such as Lorentz transformations, We will not only prove theorem1, but also provide an analytical Wigner rotations, and screw transformations. The invariant decomposition : solution for the decomposition of : reflections into d 2 e commuting also directly leads to closed form formulas for the exponential and logarith- simple orthogonal factors. Having such a geometrically inspired de- mic function for all Spin groups, and identifies element of geometry such composition makes many algebraic operations, such as computing as planes, lines, points, as the invariants of :-reflections. We conclude by exponentials and logarithms, much easier. presenting novel matrix/vector representations for geometric algebras R?@A , and use this in E¹3º to illustrate the relationship with the classic covariant, Paul Dirac famously remarked that his research work was done contravariant and adjoint representations for the transformation of points, in pictures, and that he often thought projective geometry the most planes and lines. useful, but Additional Key Words and Phrases: Lie groups, Lie algebras, Invariant de- “When I came to publish the results I suppressed the composition, Pseudo-Euclidean group, Conformal group, closed form expo- projective geometry as the results could be expressed nential and logarithmic formulas, Wigner rotation, Mozzi-Chasles’ theorem, more concisely in analytic form.” – P. A. M. Dirac [5] Baker-Campbell-Hausdorff formula, Lorentz group, Geometric gauge But the pictures provide additional geometric insights which are not easily gained purely from algebra. For example, while the Clifford- 1 INTRODUCTION Lipschitz group and the twisted Clifford-Lipschitz group might be Central to this paper is the generalisation of a conjecture by M. algebraically isomorphic [24], the geometrical interpretation makes Riesz [20], stating that a bivector of an n-dimensional geometric it clear that the conjugation law of the twisted Clifford-Lipschitz = group has to be used to apply reflections, see section 3.3. The empha- algebra R?@ can always be decomposed into at most b 2 c simple commuting orthogonal bivectors. We extend this conjecture to the sis on geometry is motivated further by recent advances in computer graphics [11, 12], which demonstrate that vectors can be identified wider class of algebras R?@A , which includes A null basis vectors, and consider the group of all reflections therein, Pin¹?, @, Aº. The with (hyper)planes instead of points, an idea dating back to Michel resulting theorem then states: Chasles [4], but not often considered. A pictorial approach will help to underscore the importance of this insight. Theorem 1 (invariant decomposition). A product of : reflec- In section3 we introduce Clifford algebras, whose graded struc- : tions * = D1D2 ···D: can be decomposed into exactly d 2 e commuting ture and concise expression for reflections makes them the ideal : factors. These are b 2 c products of two reflections, and, for odd :, one algebraic framework for formalizing graded symmetry groups. In extra reflection. These factors are called simple. section4 we establish blades as the natural choice to represent the primitive elements of geometry such as points, lines, spheres, etc. We 4 In the three dimensional Euclidean group this says that every - then turn our attention to Spin Lie algebras, and their identification arXiv:2107.03771v1 [math-ph] 8 Jul 2021 reflection can be decomposed into two commuting 2-reflections, as bivector algebras in section5. A proof and novel algorithm for better known as the Mozzi-Chasles’ theorem: the invariant decomposition is presented in section6. In sections7,9 “Every three dimensional rigid body motion can be and 9.3 we use this invariant decomposition to present closed form decomposed as a translation along a line followed or solutions for exponentials, factorization of group elements, and loga- preceded by a rotation around the same line.” rithms, respectively. We conclude in section 10 with a novel method Our generalisation shows that this is - in contrast to popular belief - to construct efficient matrix-matrix and matrix-vector representa- not the simplest example, and a similar statement can be made for tions of graded symmetry groups, and show how these contain the the 3-reflections in the two dimensional Euclidean group. classic covariant, contravariant and adjoint matrix representations. All the analytical results of this work were also condensed into a ∗Both authors contributed equally to the paper Authors’ addresses: Martin Roelfs, Department of Physics, KU Leuven, Kortrijk, Bel- cheat sheet, which can be found on the last page of this manuscript. gium, [email protected]; Steven De Keninck, Informatics Institute, University of Amsterdam, Amsterdam, The Netherlands, [email protected]. 1 Roelfs and De Keninck Fig. 3. When viewed in 3D, translations (right) can be seen as rotations Fig. 1. Reflections in 2D. From left to right, a single line reflection, two inter- (left) around infinite points. secting line reflections forming a rotation, and two parallel line reflections forming a translation. plane, the composition of three reflections still produces a new isom- etry, the glide reflection. But when four reflections are combined, we 2 GEOMETRIC INTUITION 2.1 The Euclidean Group E¹2º To build an intuition that will carry over from the Orthogonal group all the way to the Conformal group, we first study the distance preserving transformations of the plane. We build on Hamilton’s observation that rotations and translations can be constructed by composing reflections, and use that idea as our guiding principle. Figure1 illustrates how the reflection of a shape in a line (left) is the basic building block from which both the rotations (middle) and Fig. 4. (left) Three line reflections produce a glide reflection. Example 9.3 translations (right) can be constructed through composition. The shows that in E¹2º the seemingly distinct rotoreflections are also just glide resulting bireflection will transform the shape with twice the angle reflections. (right) Four line reflections produce no new isometry, but instead or distance between the lines respectively. again a rotation/translation. Figure2 illustrates how, when creating a bireflection, only the in- can factorize these such that two adjacent reflections are identical. tersection point and the relative separation between the reflections Two identical reflections leave the entire plane unchanged, andasa matters. Indeed, the same rotation can be created using any of the consequence every composition of four reflections in a plane can configurations in fig.2. This is important, as this gauge degree of be simplified down to two reflections, as demonstrated infig.5. freedom allows us to select a favorable factorization of any bireflec- tion. Because of the associativity of reflections, this means that : reflections have : − 1 gauge degrees of freedom. We also note at Fig. 5. Given four line reflections 0123, the gauge symmetry allows the pairs 01 and 23 to be rotated into 0010 and 2030, until 10 and 20 are incident and vanish from the expression. The result is a bireflection. Fig. 2. Three different sets of line reflections that intersect in the same 2.3 Graded Symmetry Groups point and at the same angle create the same rotation around that point. Before we continue our geometric view, it makes sense at this point this point how this approach unifies the treatment of rotations and to introduce the usual group theoretic notation. A group G is a translations, showing clearly not only how small changes in one set of elements and a single binary operation, written here with of the elementary reflections creates the continuous rotational and juxtaposition, that satisfies the following requirements: translational symmetries, but that indeed rotations and translations (1) For any ordered pair 0,1 2 G : 01 2 G (closure) are part of the same continuous manifold. This relation is illustrated (2) The associative law holds. 80,1, 2 2 G : ¹01º2 = 012 = 0¹12º further in fig.3, which shows how translations can be understood (3) There is an identity element 1 2 G, such that as rotations around infinite or vanishing points. 80 2 G : 10 = 0 = 01 (4) Each non-degenerate 0 2 G has an inverse 2.2 Cartan–Dieudonné 0−1 2 G : 00−1 = 1 = 0−10 Having constructed our translations and rotations as bireflections, Because compositions of reflections satisfy all these conditions, they we ask which isometries can be created by composing a larger num- form a group.