A higher-order generalization of group theory (2407.07815v1)

Abstract: The goal of this paper is to show that fundamental concepts in higher-order Fourier analysis can be nauturally extended to the non-commutative setting. We generalize Gowers norms to arbitrary compact non-commutative groups. On the structural side, we show that nilspace theory (the algebraic part of higher-order Fourier analysis) can be naturally extended to include all non-commutative groups. To this end, we introduce generalized nilspaces called "groupspaces" and demonstrate that they possess properties very similar to nilspaces. We study $k$-th order generalizations of groups that are special groupspaces called {\it k-step} groupspaces. One step groupspaces are groups. We show that $k$-step groupspaces admit the structure of an iterated principal bundle with structure groups $G_1,G_2,\dots,G_k$. A similar, but somewhat more technical statement holds for general groupspaces, with possibly infinitely many structure groups. Structure groups of groupspaces are in some sense analogous to higher homotopy groups. In particular we use a version of the Eckmann-Hilton argument from homotopy theory to show that $G_i$ is abelian for $i\geq 2$. Groupspaces also show some similarities with $n$-groups from higher category theory (also used in physics) but the exact relationship between these concepts is a subject of future research.

• The paper introduces groupspaces, a novel framework that generalizes group theory by extending Gowers norms to compact noncommutative structures.
• It establishes groupspaces using combinatorial axioms governing cubes and an iterated principal bundle structure with higher homotopy properties.
• The findings suggest broad applications in algebra, topology, and machine learning by offering a universal approach for analyzing complex noncommutative interactions.

Overview of a Higher-Order Generalization of Group Theory

The paper presents a significant advancement in the paper of non-commutative algebraic structures by introducing a higher-order generalization of group theory. It extends Gowers norms to arbitrary compact non-commutative groups and elaborates on the structural parallelism between these norms and nilspace theory. This generalization is encapsulated through the introduction of "groupspaces," which exploit the frameworks of higher-order structures akin to nilspaces.

The introduction of groupspaces marks a pivotal contribution, as it addresses the properties of cubes in the classical group theory context. Here, cubes serve as a basis for defining the operations and properties within the newly proposed groupspaces. Specifically, a k-dimensional cube in a group G is represented as a map whose form is determined by elements of the group. This notion is central to the understanding of groupspaces, describing them as sequences of cubes governed by combinatorial axioms, including the presheaf, ergodicity, and completion axioms.

Significantly, the paper discusses how k-step groupspaces possess an iterated principal bundle structure, with higher-order homotopy characteristics reminiscent of algebraic topology. This connection is elaborated using a variation of the Eckmann-Hilton argument, which introduces commutative properties in higher homotopy groups.

The theory encompasses practical applications, as groupspaces can generalize Gowers norms beyond Abelian groups to compact groupspaces, thus enabling a broader spectrum of higher-order Hilbert spaces. These structures aid in the interpretation of densities of non-commutative configurations and contribute to the understanding of emergent structures in complex functions over large Abelian groups.

Numerical Results and Bold Claims

One of the bold claims of the work is the extension of the Gowers norms to compact groupspaces, which suggests that groupspaces can serve as universal scaffolds for expressing higher-order analyses. The paper also claims that specific groupspaces that are not nilspaces can be constructed, showcasing the robustness of the groupspace framework.

Implications and Speculation on Future Developments

The development of groupspaces positions itself as a foundational tool in the exploration of higher-order algebraic structures. It is expected to catalyze further research in both theoretical mathematics and applied fields, such as physics. The notion of cubes as fundamental units in this framework opens avenues for studying algebraic transformations and morphisms within complex systems. Future work may explore the relationship between groupspaces and other higher-dimensional algebraic concepts, such as n-groups, and further investigate the connections with category theory and homotopy theory.

Groupspaces could also potentially influence the development of AI, specifically in enhancing the analytical frameworks used in machine learning models that require higher-order interactions and dependencies. Their inherent structure may provide new computational paradigms for processing, categorizing, and predicting data patterns.

Conclusion

The paper provides a comprehensive foundation for higher-order group theory by successfully extending classical group theory concepts to a non-commutative setting. Its introduction of groupspaces and the detailed exploration of their properties in relation to nilspaces puts forth new perspectives and methodologies for analyzing complex algebraic structures, setting a strong precedent for future work in this domain. The mathematical elegance and robustness of groupspaces highlight their potential utility across disciplines, making them worthwhile subjects for ongoing research and application exploration.