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Alexei Davydov

Publications and source records attributed to Alexei Davydov.

At least 19 recordsLinked to original sources
arXiv2026

Adaptive Selection of Codebook Using Assistance Information and Artificial Intelligence for 6G Systems

This paper addresses the problem of adaptive codebook (CB) selection for downlink (DL) precoder quantization in channel state information (CSI) reporting. The accuracy of precoder quantization depends on propagation conditions, requiring independent parameter adaptation for each user equipment (UE). To enable optimal CB selection, this paper proposes UE-assisted CB selection at the base station (BS) using reported by the UE statistical channel properties across time, frequency, and spatial domains. The reported assistance information serves as input to a neural network (NN), which predicts the quantization accuracy of various CB types for each served user. The predicted accuracy is then used to select the optimal CB while considering the associated CSI reporting overhead and precoding performance. System-level simulations demonstrate that the proposed approach reduces total CSI overhead while maintaining the target system throughput performance.

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arXiv2023

Tensor categories

We present here definitions and constructions basic for the theory of monoidal and tensor categories. We provide references to the original sources, whenever possible. Group-theoretical categories are used as examples

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arXiv2021

Braided Picard groups and graded extensions of braided tensor categories

We classify various types of graded extensions of a finite braided tensor category $\cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $\cal B$ by a finite group $A$ correspond to braided monoidal $2$-functors from $A$ to the braided $2$-categorical Picard group of $\cal B$ (consisting of invertible central $\cal B$-module categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane cohomology. We describe in detail braided $2$-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categories.

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arXiv2020

Hopf algebra actions in tensor categories

We prove that commutative algebras in braided tensor categories do not admit faithful Hopf algebra actions unless they come from group actions. We also show that a group action allows us to see the algebra as the regular algebra in the representation category of the acting group.

math.QA↗
arXiv2020

Cosimplicial monoids and deformation theory of tensor categories

We introduce a notion of $n$-commutativity ($0\le n\le \infty$) for cosimplicial monoids in a symmetric monoidal category ${\bf V}$, where $n=0$ corresponds to just cosimplicial monoids in ${\bf V,}$ while $n=\infty$ corresponds to commutative cosimplicial monoids. If ${\bf V}$ has a monoidal model structure we show (under some mild technical conditions) that the total object of an $n$-cosimplicial monoid has a natural $E_{n+1}$-algebra structure. Our main applications are to the deformation theory of tensor categories and tensor functors. We show that the deformation complex of a tensor functor is a total complex of a $1$-commutative cosimplicial monoid and, hence, has an $E_2$-algebra structure similar to the $E_2$-structure on Hochschild complex of an associative algebra provided by Deligne's conjecture. We further demonstrate that the deformation complex of a tensor category is the total complex of a $2$-commutative cosimplicial monoid and, therefore, is naturally an $E_3$-algebra. We make these structures very explicit through a language of Delannoy paths and their noncommutative liftings. We investigate how these structures manifest themselves in concrete examples.

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arXiv2019

Deformation cohomology of Schur-Weyl categories. Free symmetric categories

The deformation cohomology of a tensor category controls deformations of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur-Weyl categories). Using this description we compute the deformation cohomology of free symmetric tensor categories generated by one object with an algebra of endomorphism free of zero-divisors. We compare the answers with the exterior invariants of the general linear Lie algebra.

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arXiv2018

Third Cohomology and Fusion Categories

It was observed recently that for a fixed finite group $G$, the set of all Drinfeld centres of $G$ twisted by 3-cocycles form a group, the so-called group of modular extensions (of the representation category of $G$), which is isomorphic to the third cohomology group of $G$. We show that for an abelian $G$, pointed twisted Drinfeld centres of $G$ form a subgroup of the group of modular extensions. We identify this subgroup with a group of quadratic extensions containing $G$ as a Lagrangian subgroup, the so-called group of Lagrangian extensions of $G$. We compute the group of Lagrangian extensions, thereby providing an interpretation of the internal structure of the third cohomology group of an abelian $G$ in terms of fusion categories.

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arXiv2016

Holomorphic Symplectic Fermions

Let V be the even part of the vertex operator super-algebra of r pairs of symplectic fermions. Up to two conjectures, we show that V admits a unique holomorphic extension if r is a multiple of 8, and no holomorphic extension otherwise. This is implied by two results obtained in this paper: 1) If r is a multiple of 8, one possible holomorphic extension is given by the lattice vertex operator algebra for the even self dual lattice $D_r^+$ with shifted stress tensor. 2) We classify Lagrangian algebras in SF(h), a ribbon category associated to symplectic fermions. The classification of holomorphic extensions of V follows from 1) and 2) if one assumes that SF(h) is ribbon equivalent to Rep(V), and that simple modules of extensions of V are in one-to-one relation with simple local modules of the corresponding commutative algebra in SF(h).

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arXiv2015

Unphysical diagonal modular invariants

A modular invariant for a chiral conformal field theory is physical if there is a full conformal field theory with the given chiral halves realising the modular invariant. The easiest modular invariants are the charge conjugation and the diagonal modular invariants. While the charge conjugation modular invariant is always physical there are examples of chiral CFTs for which the diagonal modular invariant is not physical. Here we give (in group theoretical terms) a necessary and sufficient condition for diagonal modular invariants of $G$-orbifolds of holomorphic conformal field theories to be physical. Mathematically a physical modular invariant is an invariant of a Lagrangian algebra in the product of (chiral) modular categories. The chiral modular category of a $G$-orbifold of a holomorphic conformal field theory is the so-called (twisted) Drinfeld centre ${\cal Z}(G,α)$ of the finite group $G$. We show that the diagonal modular invariant for ${\cal Z}(G)$ is physical if and only if the group $G$ has a {\em double class inverting} automorphism, that is an automorphism $ϕ:G\to G$ with the property that for any commuting $x,y\in G$ there is $g\in G$ such that $ϕ(x) = gx^{-1}g^{-1},\ ϕ(y) = gy^{-1}g^{-1}.$ Groups without double class inverting automorphisms are abundant and provide examples of chiral conformal field theories for which the diagonal modular invariant is unphysical.

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arXiv2014

N=2 minimal conformal field theories and matrix bifactorisations of x^d

We prove a tensor equivalence between full subcategories of a) graded matrix factorisations of the potential x^d-y^d and b) representations of the N=2 minimal super vertex operator algebra at central charge 3-6/d, where d is odd. The subcategories are given by a) permutation-type matrix factorisations with consecutive index sets, and b) Neveu-Schwarz-type representations. The physical motivation for this result is the Landau-Ginzburg / conformal field theory correspondence, where it amounts to the equivalence of a subset of defects on both sides of the correspondence. Our work builds on results by Brunner and Roggenkamp [arXiv:0707.0922], where an isomorphism of fusion rules was established.

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arXiv2013

Z/2Z-extensions of Hopf algebra module categories by their base categories

Starting with a self-dual Hopf algebra H in a braided monoidal category S we construct a Z/2Z-graded monoidal category C = C_0 + C_1. The degree zero component is the category Rep_S(H) of representations of H and the degree one component is the category S. The extra structure on H needed to define the associativity isomorphisms is a choice of self-duality map and cointegral, subject to certain conditions. We also describe rigid, braided and ribbon structures on C in Hopf algebraic terms. Our construction permits a uniform treatment of Tambara-Yamagami categories and categories related to symplectic fermions in conformal field theory.

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arXiv2013

Functoriality of the center of an algebra

The notion of the center of an algebra over a field k has a far reaching generalization to algebras in monoidal categories. The center then lives in the monoidal center of the original category. This generalization plays an important role in the study of bulk-boundary duality of rational conformal field theories. In this paper, we study functorial properties of the center. We show that it gives rise to a 2-functor from the bicategory of semisimple indecomposable module categories over a fusion category to the bicategory of commutative algebras in the monoidal center of this fusion category. Morphism spaces of the latter bicategory are extended from algebra homomorphisms to certain categories of cospans. We conjecture that the above 2-functor arises from a lax 3-functor between tricategories, and that in this setting one can relax the conditions from fusion categories to finite tensor categories. We briefly outline how one is naturally lead to the above 2-functor when studying rational conformal field theory with defects of all codimensions. For example, the cospans of the target bicategory correspond to spaces of defect fields and to the bulk-defect operator product expansions.

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arXiv2013

An alternative description of braided monoidal categories

We give an alternative presentation of braided monoidal categories. Instead of the usual associativity and braiding we have just one constraint (the b-structure). In the unital case, the coherence conditions for a b-structure are shown to be equivalent to the usual associativity, unit and braiding axioms. We also discuss the next dimensional version, that is, b-structures on bicategories. As an application, we show how special b-categories result in the Yang-Baxter equation, and how special b-bicategories produce Zamolodchikov's tetrahedron equation. Finally, we define a cohomology theory (the b-cohomology) which plays a role analogous to the one abelian group cohomology has for braided monoidal categories.

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arXiv2013

A braided monoidal category for symplectic fermions

We describe a class of examples of braided monoidal categories which are built from Hopf algebras in symmetric categories. The construction is motivated by a calculation in two-dimensional conformal field theory and is tailored to contain the braided monoidal categories occurring in the study of the Ising model, their generalisation to Tamabara-Yamagami categories, and categories occurring for symplectic fermions.

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arXiv2012

Twisted derivations of Hopf algebras

In the paper we introduce the notion of twisted derivation of a bialgebra. Twisted derivations appear as infinitesimal symmetries of the category of representations. More precisely they are infinitesimal versions of twisted automorphisms of bialgebras. Twisted derivations naturally form a Lie algebra (the tangent algebra of the group of twisted automorphisms). Moreover this Lie algebra fits into a crossed module (tangent to the crossed module of twisted automorphisms). Here we calculate this crossed module for universal enveloping algebras and for the Sweedler's Hopf algebra.

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arXiv2012

The Picard crossed module of a braided tensor category

For a finite braided tensor category we introduce its Picard crossed module consisting of the group of invertible module categories and the group of braided tensor autoequivalences. We describe the Picard crossed module in terms of braided autoequivalences of the Drinfeld center of the braided tensor category. As an illustration, we compute the Picard crossed module of a braided pointed fusion category.

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