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Yakov Veryovkin

Publications and source records attributed to Yakov Veryovkin.

7 recordsLinked to original sources
arXiv2025

The commutator subalgebra of the Lie algebra associated with a right-angled Coxeter group

We study the graded Lie algebra $L(RC_K)$ associated with the lower central series of a right-angled Coxeter group. We construct a surjective homomorphism from the polynomial ring over an explicit Lie algebra $N_K$ to the commutator subalgebra of $L(RC_K)$, and conjecture that it is an isomorphism. The homomorphism is defined in terms of a new operation in Lie algebras associated with groups generated by involutions, which corresponds to the squaring and has an analogue in homotopy theory. We show that the universal enveloping algebra $U(N_K)$ is isomorphic to the mod 2 loop homology algebra of the corresponding moment-angle complex $ZK$. This allows us to give a presentation of the Lie algebra $N_K$ by generators and relations.

math.GR↗
arXiv2022

Graded components of the Lie algebra associated with the lower central series of a right-angled Coxeter group

The lower central series of the rgiht-angled Coxeter group $RC_\mathcal K$ and the corresponding graded Lie algebra $L(RC_\mathcal K)$ associated with the lower central series of a right-angled Coxeter group are studied. Relations are obtained in the graded components of the Lie algebra $L(RC_\mathcal K)$. A basis of the fourth graded component $L(RC_\mathcal K)$ for groups with at most $4$ generators was described.

math.GR↗
arXiv2018

On the commutator subgroup of a right-angled Artin group

We use polyhedral product models to analyse the structure of the commutator subgroup of a right-angled Artin group. In particular, we provide a minimal set of generators for the commutator subgroup, consisting of special iterated commutators of canonical generators.

math.GR↗
arXiv2016

Polyhedral products and commutator subgroups of right-angled Artin and Coxeter groups

We construct and study polyhedral product models for classifying spaces of right-angled Artin and Coxeter groups, general graph product groups and their commutator subgroups. By way of application, we give a criterion of freeness for the commutator subgroup of a graph product group, and provide an explicit minimal set of generators for the commutator subgroup of a right-angled Coxeter group.

math.GR↗
arXiv2016

Pontryagin algebras of some moment-angle-complexes

We consider the problem of describing the Pontryagin algebra (loop homology) of moment-angle complexes and manifolds. The moment-angle complex Z_K is a cell complex built of products of polydiscs and tori parametrised by simplices in a finite simplicial complex K. It has a natural torus action and plays an important role in toric topology. In the case when K is a triangulation of a sphere, Z_K is a topological manifold, which has interesting geometric structures. Generators of the Pontryagin algebra H_*(ΩZ_K) when K is a flag complex have been described in the work of Grbic, Panov, Theriault and Wu. Describing relations is often a difficult problem, even when K has a few vertices. Here we describe these relations in the case when K is the boundary of a pentagon or a hexagon. In this case, it is known that Z_K is a connected sum of products of spheres with two spheres in each product. Therefore H_*(ΩZ_K) is a one-relator algebra and we describe this one relation explicitly, therefore giving a new homotopy-theoretical proof of McGavran's result. An interesting feature of our relation is that it includes iterated Whitehead products which vanish under the Hurewicz homomorphism. Therefore, the form of this relation cannot be deduced solely from the result of McGavran.

math.AT↗