arXiv subjects

Pragya Belwal

Publications and source records attributed to Pragya Belwal.

4 recordsLinked to original sources

arXiv2026

Skew braces and Rota-Baxter operators on semi-direct products

This paper examines the connections between (relative) Rota--Baxter groups, skew left braces, and enlargements of these structures on naturally associated semi-direct products. Given a skew left brace, we define a new skew left brace, referred to as its square, on the natural semi-direct product of its additive and multiplicative groups. Further, the square construction is distinct from the previously known double construction arising as a special case of matched pairs of skew braces. This provides a method to construct a new bijective, non-degenerate solution to the Yang--Baxter equation from an existing solution arising from a skew left brace. We show that the square construction is functorial and integrates naturally into both the cohomological and extension-theoretic frameworks for (relative) Rota--Baxter groups and skew left braces. Furthermore, we provide a sufficient condition under which two isoclinic skew left braces yield isoclinic squares.

math.QA↗

arXiv2024

A Wells-like exact sequence for abelian extensions of relative Rota--Baxter groups

Relative Rota--Baxter groups, a generalization of Rota--Baxter groups, are closely connected to skew left braces, which play a fundamental role in understanding non-degenerate set-theoretical solutions to the Yang-Baxter equation. In this paper, we explore the inducibility problem for automorphisms of abelian extensions of relative Rota--Baxter groups. This problem is intricately linked to the recently introduced second cohomology of relative Rota--Baxter groups. Specifically, we prove a Wells-like exact sequence for abelian extensions of relative Rota--Baxter groups. The sequence establishes a connection among the group of derivations, certain automorphism group, and the second cohomology of relative Rota--Baxter groups, thereby giving precise structural relationships between these groups.

math.GR↗

arXiv2023

Schur multiplier and Schur covers of relative Rota-Baxter groups

Relative Rota-Baxter groups are generalizations of Rota-Baxter groups and share a close connection with skew left braces. These structures are well-known for offering bijective non-degenerate set-theoretical solutions to the Yang-Baxter equation. This paper builds upon the recently introduced extension theory and low-dimensional cohomology of relative Rota-Baxter groups. We prove an analogue of the Hochschild-Serre exact sequence for central extensions of relative Rota-Baxter groups. We introduce the Schur multiplier $M_{RRB}(\mathcal{A})$ of a relative Rota-Baxter group $\mathcal{A} =(A,B,β,T)$, and prove that the exponent of $M_{RRB}(\mathcal{A})$ divides $|A||B|$ when $\mathcal{A}$ is finite. We define weak isoclinism of relative Rota-Baxter groups, introduce their Schur covers, and prove that any two Schur covers of a finite bijective relative Rota-Baxter group are weakly isoclinic. The results align with recent results of Letourmy and Vendramin for skew left braces.

math.QA↗

arXiv2023

Cohomology and Extensions of Relative Rota-Baxter groups

Relative Rota-Baxter groups are generalisations of Rota-Baxter groups and recently shown to be intimately related to skew left braces, which are well-known to yield bijective non-degenerate solutions to the Yang-Baxter equation. In this paper, we develop an extension theory of relative Rota-Baxter groups and introduce their low dimensional cohomology groups, which are distinct from the ones known in the context of Rota-Baxter operators on Lie groups. We establish an explicit bijection between the set of equivalence classes of extensions of relative Rota-Baxter groups and their second cohomology. Further, we delve into the connections between this cohomology and the cohomology of associated skew left braces. We prove that for bijective relative Rota-Baxter groups, the two cohomologies are isomorphic in dimension two.

math.QA↗