Personal Papers (MS 80-0002)

Letter from Mary Ann Chapman of Interplay magazine enclosing an invoice for the next year's subscription

Semi-abelian categories, cocommutative Hopf algebras and Hopf braces

In this talk I’ll present some recent results on the interactions between Hopf algebra theory and semi-abelian categories. Semi-abelian categories [1] have played a central role in categorical algebra during the last 25 years. After recalling the motivation for studying them, some basic concepts and a few examples, we shall explain why the category of cocommutative Hopf algebras is semi-abelian [2]. We shall then turn our attention to (cocommutative) Hopf braces [3], that extend cocommutative Hopf algebras and can be seen as a Hopf-theoretic generalization of skew braces[4], that are useful to study solutions of the Yang-Baxter equation. In the recent article with Andrea Sciandra [5] we have investigated the exactness properties of the category of Hopf braces, and some natural constructions therein. First, we show that cocommutative Hopf braces form a semi-abelian category, that is also strongly protomodular. When the base field is algebraically closed and has zero characteristic one can find an interesting torsion theory therein, whose torsion-free subcategory is equivalent to the variety of skew braces, which turns out to be also a localization. Finally, we provide some explicit descriptions of the categorical commutator and of the central extensions of Hopf braces, that are likely to be useful for some new applications in non-abelian (co)homology theory. References [1] G. Janelidze, L. Marki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002) 367-386 [2] M. Gran, F. Sterck and J. Vercruysse, A semi-abelian extension of a theorem by Takeuchi, J. Pure Appl. Algebra 223 (2019) 4171-4190 [3] I. Angiono, C. Galindo, L. Vendramin, Hopf braces and Yang-Baxter operators, Proc. Amer. Math. Soc. 145 (2017) 1981-1995 [4] L. Guarnieri, L. Vendramin, Skew braces and the Yang-Baxter equation. Math. Comp. 86 (2017) 2519-2534 [5] M. Gran and A. Sciandra, Hopf braces and semi-abelian categories, preprint, arXiv:2411.19238 (2024

Bankrupts in the Shadows of the ‘Belle Epoque’. Coping with the Failure ; coping with the Shame. Belgium, 1896-1914

In Western Europe, the years 1896-1914 were characterized by a sustained economic growth accompanied by a significant improvement in social conditions. In the shadows of this "Belle Epoque", but often in direct connection with the transformations it bore, bankruptcy still remained a threat to many companies, starting with the smallest ones. This study examines these bankrupts, usually from the petty bourgeoisie, who found themselves bankrupt, i.e. as a person "who ceases his payments and who is no longer creditworthy " . Beyond the judicial procedure and its effects on the business of the bankrupt, bankruptcy is also acutely felt as a personal, family and social failure. The bankrupt is at the center of the scandal of his own bankruptcy, and often threatened by poverty and social downgrading. The bankruptcy petition – or the discovery of the insolvency by the commercial court - constitutes a turning point in the life of this person and his family, characterized by a shift in the emotional register of shame. This research, through a microhistorical approach, proposes to consider how the shame of bankruptcy was experienced and endured by some 21 bankrupts, their relatives and their descendants. The proposed framework is that of a medium-sized provincial town, Namur

Correspondence Analysis for Visualizing

We apply correspondence analysis for visualization of interdependence of pitch class & key and key & composer. A co-occurrence matrix of key & pitch class frequencies is extracted from score (Bach's WTC). Keys are represented as high-dimensional pitch class vectors. Correspondence analysis then projects keys on a planar "keyscape". Vice versa, on "pitchscapes" pitch classes can also be embedded in the key space. In both scenarios a homogenous circle of fifths emerges in the scapes. We employ biplots to embed keys and pitch classes in the keyscape to visualize their interdependence. After a change of co-ordinates the four-dimensional biplots can be interpreted as a configuration on a torus, closely resembling results from music theory and experiments in listener models