arXiv subjects

Bo Shan Deval

Publications and source records attributed to Bo Shan Deval.

4 recordsLinked to original sources

arXiv2026

Kaluzhnin--Krasner embedding of precrossed modules

The Kaluzhnin--Krasner embedding establishes that, for a group extension, the middle group can be embedded into the wreath product of its kernel and cokernel. Recently, this construction has been generalised in a categorical framework, recovering both the classical group-theoretic result and its analogue for Lie In this article, we apply this categorical framework to two specific cases: precrossed modules and crossed modules (over groups). For precrossed modules, we introduce a Kaluzhnin--Krasner embedding -- previously unformulated -- by rigorously following the steps of the categorical procedure. In contrast, for crossed modules, we encounter substantial obstacles: while we present partial positive results, we also explain why constructing a full-fledged Kaluzhnin--Krasner embedding in this context is far more difficult.

math.CT↗

arXiv2025

Intrinsic tensor products and a Ganea-type extension of the five-term exact sequence

We define an intrinsic symmetric bi-right-exact (and for varieties, bi-cocontinuous) bilinear product on objects of a semi-abelian category, constructed as the cosmash product in the two-nilpotent reflection. When applied to abelian objects, this recovers classical tensor products in many cases. A recognition theorem states that any symmetric bi-cocontinuous bifunctor on an abelian variety of algebras is realised as the bilinear product in the variety of algebras over a suitable 2-nilpotent symmetric operad in the monoidal category of abelian groups. For abelian groups replaced with any commutative ring, the bilinear product of algebras over such an operad is associative as long as the only unary operations are given by multiplication with scalars, but not in general. This relies on a right-exactness theorem for cross-effects of bifunctors, and consequently for cosmash products. We develop basic properties, compare the bilinear product to the Brown-Loday non-abelian tensor product, and prove a categorical version of Ganea's six-term exact homology sequence. We further characterise abelian extensions via internal action cores, obtaining explicit descriptions of bilinear products in categories of representations; in particular, the bilinear product of the associated Beck modules generalises the classical tensor product of representations for groups and Lie algebras.

math.CT↗

arXiv2023

Twisted Commutators and Internal Crossed Modules

We introduce a notion of relative commutator -- an important special case being commutators twisted by an action -- as a straightforward modification of the definition of the Higgins commutator, establish its relation with a new notion of commutativity -- also obtained as a modification of the usual notion -- and show how we can use it to characterise internal crossed modules in the context of a semi-abelian category.

math.CT↗

arXiv2023

A universal Kaluzhnin--Krasner embedding theorem

Given two groups $A$ and $B$, the Kaluzhnin--Krasner universal embedding theorem states that the wreath product $A\wr B$ acts as a universal receptacle for extensions from $A$ to $B$. For a split extension, this embedding is compatible with the canonical splitting of the wreath product, which is further universal in a precise sense. This result was recently extended to Lie algebras and to cocommutative Hopf algebras. The aim of the present article is to explore the feasibility of adapting the theorem to other types of algebraic structures. By explaining the underlying unity of the three known cases, our analysis gives necessary and sufficient conditions for this to happen. From those we may for instance conclude that a version for crossed modules can indeed be attained, while the theorem cannot be adapted to, say, associative algebras, Jordan algebras or Leibniz algebras, when working over an infinite field: we prove that then, amongst non-associative algebras, only Lie algebras admit a universal Kaluzhnin--Krasner embedding theorem.

math.CT↗