Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2109.00981

On the Gottlieb group, Drinfeld centre and the centre of a crossed module

Mariam Pirashvili
View original source ↗Read source document ↗Export metadata ↓

Abstract

This new version includes a connection of the main construction to the Gottlieb group, which was absent in the previous versions. However, the first version included material about Lie algebras which will become available soon as a separate paper. The aim of this paper is to introduce the concept of the centre of a crossed module $\G_* = (\G_1\to \G_0)$. This centre is closely related to the Gottlieb group of the classifying space of a crossed module and also to the Drinfeld centre of a monoidal category introduced independently by Drinfeld and Joyal and Street. Our definition of the centre is based on certain crossed homomorphisms $\G_0\to \G_1$, which makes it easy to relate it to group cohomology. This connection is used to relate the Gottlieb group of a 2-type to its Whitehead centre.

Explore related subjects

math.KTmath.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mariam Pirashvili. 2025-01-17. On the Gottlieb group, Drinfeld centre and the centre of a crossed module. https://arxiv.org/abs/2109.00981

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Gerstenhaber algebra structure on the Hochschild cohomology ring of the Xu--Snashall algebra

Let $A$ be a finite dimensional algebra and let $\rmHH^*(A)$ be its Hochschild cohomology ring, which is a Gerstenhaber algebra. Denote by $\calN$ (resp. $G$, $\calG$) the ideal (resp. weak Gerstenhaber ideal, Gerstenhaber ideal) generated by all homogeneous nilpotent elements. Motivated by their work on support varieties via Hochschild cohomology, Snashall and Solberg conjectured that $\rmHH^*(A)/\calN$ is a finitely generated algebra. Xu constructed a counterexample to the Snashall-Solberg conjecture over a base field of characteristic two, and Snashall generalized this example to arbitrary characteristic. Hermann further asked whether $\rmHH^*(A)/G$ is a finitely generated algebra and suggested considering first the Xu--Snashall algebra. In this paper, we answer this question for the Xu--Snashall algebra. In fact, by explicitly computing the Gerstenhaber algebra structure on the Hochschild cohomology ring, we show that $G=\calN$; hence $\rmHH^*(A)/G=\rmHH^*(A)/\calN$ is not a finitely generated algebra. Furthermore, we show that $\rmHH^*(A)/\calG\cong K$. Therefore, one may ask whether, for a finite dimensional algebra $A$, $\rmHH^*(A)/\calG$ is always a finitely generated algebra. Our main tools are two-sided Anick resolutions and weak self-homotopies.

math.KT↗
arXiv2026

K-theory of Matroids and Monoid Schemes

This paper continues the study of the $K$-theory of monoid schemes, using it to give a useful definition of the higher $K$-theory of a matroid via its Bergman fan.

math.KT↗
arXiv2026

Improved injective stability for relative $\mathrm{K_1Sp}$-groups

We prove a relative version of Vorst's theorem concerning the equality of the group of all invertible matrices and the group of all elementary matrices over $R[X]$ with respect to an ideal $I\subset R$ such that $R/I$ is regular, where $R$ is a regular $k$-spot. We then introduce a relative version of the symplectic elementary Witt group and show that it fits into a relative version of the Karoubi periodicity sequence. Combining these results, we improve the existing injective stability bounds for relative linear and symplectic $\mathrm{K_1}$-groups of smooth affine algebras over various base fields. As an application, we give a necessary and sufficient condition for the freeness of stably free modules over smooth real $4$-folds with empty real locus.

math.KT↗