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Dmitriy Rumynin

Publications and source records attributed to Dmitriy Rumynin.

35 records · Page 2Linked to original sources
arXiv2019

Covering Groups of Nonconnected Topological Groups and 2-Groups

We investigate the universal cover of a topological group that is not necessarily connected. Its existence as a topological group is governed by a Taylor cocycle, an obstruction in 3-cohomology. Alternatively, it always exists as a topological 2-group. The splitness of this 2-group is also governed by an obstruction in 3-cohomology, a Sinh cocycle. We give explicit formulas for both obstructions and show that they are inverse of each other.

math.QA↗
arXiv2019

Kac-Moody Groups and Their Representations

In this expository paper we review some recent results about representations of Kac-Moody groups. We sketch the construction of these groups. If practical, we present the ideas behind the proofs of theorems. At the end we pose open questions.

math.RT↗
arXiv2019

Integration of Modules I: Stability

We explore the integration of representations from a Lie algebra to its algebraic group in positive characteristic. An integrable module is stable under the twists by group elements. Our aim is to investigate cohomological obstructions for passing from stability to an algebraic group action. As an application, we prove integrability of bricks for a semisimple algebraic group.

math.RT↗
arXiv2018

Kac-Moody Groups and Cosheaves on Davis Building

We investigate smooth representations of complete Kac-Moody groups. We approach representation theory via geometry, in particular, the group action on the Davis realisation of its Bruhat-Tits building. Our results include an estimate on projective dimension, localisation theorem, unimodularity and homological duality.

math.RT↗
arXiv2018

Presentations of affine Kac-Moody groups

How many generators and relations does ${\mathrm SL}_n({\mathbb F}_q[t, t^{-1}])$ need? In this paper we exhibit its explicit presentation with $9$ generators and $44$ relations. We investigate presentations of affine Kac-Moody groups over finite fields. Our goal is to derive finite presentations, independent of the field and with as few generators and relations as we can achieve. It turns out that any simply connected affine Kac-Moody group over a finite field has a presentation with at most 11 generators and 70 relations. We describe these presentations explicitly type by type. As a consequence, we derive explicit presentations of Chevalley groups $G({\mathbb F}_q[t, t^{-1}])$ and explicit profinite presentations of profinite Chevalley groups $G({\mathbb F}_q[[t]])$.

math.GR↗
arXiv2018

2-Groups, 2-Characters, and Burnside Rings

We study 2-representations, i.e., actions of 2-groups on 2-vector spaces. Our main focus is character theory for 2-representations. To this end we employ the technique of extended Burnside rings. Our main theorem is that the Ganter-Kapranov 2-character is a particular mark homomorphism of the Burnside ring. As an application we give a new proof of Osorno formula for the Ganter-Kapranov 2-character of a finite group.

math.CT↗
arXiv2018

D-modules and projective stacks

We study twisted D-modules on the weighted projective stacks. We determine for which values of the twist and the weight the global section functor is an equivalence, thus, proving a version of Beilinson-Bernstein Localisation Theorem.

math.RT↗
arXiv2016

Subgroup Growth in Some Profinite Chevalley Groups

In this article we improve the known uniform bound for subgroup growth of Chevalley groups over $\mathbf{G}(\mathbb{F}_p[[t]])$. We introduce a new parameter, the ridgeline number $v(\mathbf{G})$, and give new bounds for the subgroup growth of $\mathbf{G}(\mathbb{F}_p[[t]])$ expressed through $v(\mathbf{G})$. We achieve this by deriving a new estimate for the codimension of $[U,V]$ where $U$ and $V$ are vector subspaces in the Lie algebra of $\mathbf{G}$.

math.GR↗
arXiv2013

Lie algebras in symmetric monoidal categories

We study algebras defined by identities in symmetric monoidal categories. Our focus is on Lie algebras. Besides usual Lie algebras, there are examples appearing in the study of knot invariants and Rozansky-Witten invariants. Our main result is a proof of Westbury's conjecture for K3-surface: there exists a Lie algebra homomorphism from Vogel's universal simple Lie algebra to the Lie algebra describing the Rozansky-Witten invariants of a K3-surface. Most of the paper involves setting up a proper language to discuss the problem and we formulate nine open questions as we proceed.

math.QA↗
arXiv2012

Cocompact lattices on \tilde{A}_n buildings

Let K be the field of formal Laurent series over the finite field of order q. We construct cocompact lattices Γ'_0 < Γ_0 in the group G = PGL_d(K) which are type-preserving and act transitively on the set of vertices of each type in the building associated to G. The stabiliser of each vertex in Γ'_0 is a Singer cycle and the stabiliser of each vertex in Γ_0 is isomorphic to the normaliser of a Singer cycle in PGL_d(q). We then show that the intersections of Γ'_0 and Γ_0 with PSL_d(K) are lattices in PSL_d(K), and identify the pairs (d,q) such that the entire lattice Γ'_0 or Γ_0 is contained in PSL_d(K). Finally we discuss minimality of covolumes of cocompact lattices in SL_3(K). Our proofs combine a construction of Cartwright and Steger with results about Singer cycles and their normalisers, and geometric arguments.

math.GR↗
arXiv2011

Generalised Burnside Rings, G-categories and Module Categories

This note describes an application of the theory of generalised Burnside rings to algebraic representation theory. Tables of marks are given explicitly for the groups $S_4$ and $S_5$ which are of particular interest in the context of reductive algebraic groups. As an application, the base sets for the nilpotent element $F_4 (a_3)$ are computed.

math.RT↗
arXiv2006

Localization of modules for a semisimple Lie algebra in prime characteristic

We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are the same as the coherent sheaves on (generalized) Springer fibers. The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of $D$-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra. Thus the ``derived'' version of the Beilinson-Bernstein localization Theorem holds in sufficiently large positive characteristic. Next, the algebra of (``crystalline'') differential operators is an Azumaya algebra and its splittings on Springer fibers allow us to pass from D-modules to coherent sheaves. As an application we compute the rank of the Grothendieck group of the category of modules over the Lie algebra with a fixed central character.

math.RT↗
arXiv2001

Subregular representations of $\sl_n$ and simple singularities of type $A_{n-1}$

Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem for type $A$. Using the McKay correspondence, we relate the solution to bases in equivariant $K$-theory introduced by Lusztig.

math.RT↗
arXiv1999

Geometric Representation Theory of Restricted Lie Algebras of Classical Type

We modify the Hochschild $ϕ$-map to construct central extensions of a restricted Lie algebra. Such central extension gives rise to a group scheme which leads to a geometric construction of unrestricted representations. For a classical semisimple Lie algebra, we construct equivariant line bundles whose global sections afford representations with a nilpotent p-character.

math.RT↗
arXiv1997

Algebraic geometry of Hopf-Galois extensions

We continue the study of Hopf-Galois extensions with central invariants for a finite dimensional Hopf algebra. We concentrate on the geometrical side on the subject. We understand how to localize Hopf-Galois extensions and to paste them from local datum.

q-alg↗
arXiv1997

Hopf-Galois extensions with central invariants

We study Hopf-Galois extensions with central invariants for a finite dimensional Hopf algebra. We collect general facts about them and discuss some examples arising in the study of restricted Lie algebras and quantum groups at roots of unity. Our focus is on representation theory and its special feature in this situation, restriction of the central character to the subalgebra of invariants.

q-alg↗