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Nous définissons et entamons l’étude d’analogues infinitésimaux des quotients principaux (algèbres de Temperley-Lieb, Hecke, Birman-Wenzl-Murakami) de l’algèbre de groupe du groupe d’Artin $B_{n}$ . Ce sont des algèbres de Hopf qui correspondent à des groupes réductifs, et permettent de donner un cadre général aux représentations dérivées des représentations classiques de $B_{n}$ . Nous décomposons complètement l’algèbre de Temperley-Lieb infinitésimale, et en déduisons plusieurs résultats d’irréductibilité.

Self-similar Lie algebras

Laurent Bartholdi (2015)

Journal of the European Mathematical Society

We give a general definition of branched, self-similar Lie algebras, and show that important examples of Lie algebras fall into that class. We give sufficient conditions for a self-similar Lie algebra to be nil, and prove in this manner that the self-similar algebras associated with Grigorchuk’s and Gupta–Sidki’s torsion groups are nil as well as self-similar.We derive the same results for a class of examples constructed by Petrogradsky, Shestakov and Zelmanov.

The abelianization of the Johnson kernel

Alexandru Dimca, Richard Hain, Stefan Papadima (2014)

Journal of the European Mathematical Society

We prove that the first complex homology of the Johnson subgroup of the Torelli group $T_{g}$ is a non-trivial, unipotent $T_{g}$ -module for all $g \geq 4$ and give an explicit presentation of it as a $S y m_{.} H_{1} (T_{g}, C)$ -module when $g \geq 6$ . We do this by proving that, for a finitely generated group $G$ satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that the first homology of its Johnson kernel is a nilpotent module over the corresponding Laurent polynomial ring, isomorphic to the...

The graded Lie algebra of a Kähler group.

Rüdiger Plantiko (1996)

Forum mathematicum

The Hughes subgroup

Robert Bryce (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let $G$ be a group and $p$ a prime. The subgroup generated by the elements of order different from $p$ is called the Hughes subgroup for exponent $p$ . Hughes [3] made the following conjecture: if $H_{p} (G)$ is non-trivial, its index in $G$ is at most $p$ . There are many articles that treat this problem. In the present Note we examine those of Strauss and Szekeres [9], which treats the case $p = 3$ and $G$ arbitrary, and that of Hogan and Kappe [2] concerning the case when $G$ is metabelian, and $p$ arbitrary. A common proof is...