Graded Symmetry Groups: Plane and Simple
This work provides a foundational insight into symmetry groups for researchers in mathematics and physics, generalizing a conjecture by M. Riesz and offering closed-form formulas for exponential and logarithmic functions in Spin groups.
The paper tackled the problem of understanding symmetries described by Pin groups by analyzing them using geometric algebra, which revealed a graded structure on Lie groups not evident in matrix representations, and proved the invariant decomposition theorem that any composition of k linearly independent reflections decomposes into ⌈k/2⌉ commuting factors, each a product of at most two reflections.
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, which is not evident in their matrix representation. By embracing this graded structure, the invariant decomposition theorem was proven: any composition of $k$ linearly independent reflections can be decomposed into $\lceil k/2 \rceil$ commuting factors, each of which is the product of at most two reflections. This generalizes a conjecture by M. Riesz, and has e.g. the Mozzi-Chasles' theorem as its 3D Euclidean special case. To demonstrate its utility, we briefly discuss various examples such as Lorentz transformations, Wigner rotations, and screw transformations. The invariant decomposition also directly leads to closed form formulas for the exponential and logarithmic function for all Spin groups, and identifies element of geometry such as planes, lines, points, as the invariants of $k$-reflections. We conclude by presenting novel matrix/vector representations for geometric algebras $\mathbb{R}_{pqr}$, and use this in E(3) to illustrate the relationship with the classic covariant, contravariant and adjoint representations for the transformation of points, planes and lines.