arXiv subjects

Azat M. Gainutdinov

Publications and source records attributed to Azat M. Gainutdinov.

At least 19 recordsLinked to original sources

arXiv2026

Crossed-module crossed braided categories

For a crossed module $χ: G \to H$, we introduce the notion of $χ$-crossed braided (resp. ribbon) categories, where the categories are graded by group $G$ and carry an $H$-action. Our definition unifies and generalises several familiar notions: taking $χ= id: G \to G$ with the conjugation action recovers $G$-crossed braided categories; taking $χ: G \to \{*\}$ for abelian $G$ yields $G$-graded braided categories; taking $χ: \{*\} \to G$ leads to braided categories equipped with a $G$-action. The equivalence relation between $χ$-crossed braided categories is typically finer than that between $G$-crossed braided ones. We classify $χ$-crossed braided structures on the category of $G$-graded vector spaces in terms of cohomological data, and give explicit examples for cyclic groups. Given a doubly central algebra with $G$- and $H$-actions in a braided monoidal category, we define a notion of twisted-local modules and show how they give rise to $χ$-crossed braided categories. We furthermore give sufficient conditions so that these categories are additionally $χ$-crossed ribbon or admit an orthogonal $G$-decomposition.

math.CT↗

arXiv2026

Non-semisimple link and manifold invariants for symplectic fermions

We consider the link and three-manifold invariants from arXiv:1912.02063, which are defined in terms of certain non-semisimple finite ribbon categories $\mathcal{C}$ together with a choice of tensor ideal and modified trace. If the ideal is all of $\mathcal{C}$, these invariants agree with those defined by Lyubashenko in the 90's. We show that in that case the invariants depend on the objects labelling the link only through their simple composition factors, so that in order to detect non-trivial extensions one needs to pass to proper ideals. We compute examples of link and three-manifold invariants for $\mathcal{C}$ being the category of $N$ pairs of symplectic fermions. Using a quasi-Hopf algebra realisation of $\mathcal{C}$, we find that the Lyubashenko-invariant of a lens space is equal to the order of its first homology group to the power $N$, a relation we conjecture to hold for all rational homology spheres. For $N \ge 2$, $\mathcal{C}$ allows for tensor ideals $\mathcal{I}$ with a modified trace which are different from all of $\mathcal{C}$ and from the projective ideal. Using the theory of pull-back traces and symmetrised cointegrals, we show that the link invariant obtained from $\mathcal{I}$ can distinguish a continuum of indecomposable but reducible objects which all have the same composition series.

math.QA↗

arXiv2026

Deformations of mixed associators in module categories

We set up a cochain complex $C^\bullet_{\mathrm{mix}}(\mathcal{M})$ whose cohomology controls deformations of the mixed associator of a module category $\mathcal{M}$ over a $\Bbbk$-linear monoidal category $\mathcal{C}$. We show that $C^\bullet_{\mathrm{mix}}(\mathcal{M})$ is isomorphic to the Davydov-Yetter (DY) complex of the representation functor $ρ: \mathcal{C} \to \mathrm{End}(\mathcal{M})$. Using our previous results on DY cohomology (arXiv:2411.19111), we prove that if $\mathcal{C}$ and $\mathcal{M}$ are finite then the cohomology $H^\bullet_{\mathrm{mix}}(\mathcal{M})$ is isomorphic to the relative Ext groups $\mathrm{Ext}^\bullet_{\mathcal{Z}(\mathcal{C}),\mathcal{C}}(\boldsymbol{1},\mathcal{A}_{\mathcal{M}})$ for the usual adjunction between the Drinfeld center $\mathcal{Z}(\mathcal{C})$ and $\mathcal{C}$, where $\mathcal{A}_{\mathcal{M}}$ is the so-called adjoint algebra of $\mathcal{M}$. This allows us to give a dimension formula for $H^n_{\mathrm{mix}}(\mathcal{M})$ in terms of certain Hom spaces in $\mathcal{Z}(\mathcal{C})$, and also to prove that $H^{>0}_{\mathrm{mix}}(\mathcal{C}) = 0$. We also show that the algebra $\mathcal{A}_{\mathcal{M}}$ is the ``full center'' of an algebra in $\mathcal{C}$ realizing $\mathcal{M}$. We furthermore establish a generalized version of Ocneanu rigidity for monoidal functors with coefficients, and provide its application to general (non-exact and non-finite) $\mathcal{C}$-module categories over a fusion category $\mathcal{C}$ such that $\dim(\mathcal{C}) \neq 0$. We spell out these results for module categories defined by finite-dimensional comodule algebras over finite-dimensional Hopf algebras. Examples based on comodule algebras over Sweedler's Hopf algebra are worked out in detail and yield new continuous families of inequivalent non-exact module categories.

math.QA↗

arXiv2026

Fused K-operators and the $q$-Onsager algebra

We study universal solutions to reflection equations with a spectral parameter, so-called K-operators, within a general framework of universal K-matrices - an extended version of the approach introduced by Appel-Vlaar. Here, the input data is a quasi-triangular Hopf algebra $H$, its comodule algebra $B$ and a pair of consistent twists. In our setting, the universal K-matrix is an element of $B\otimes H$ satisfying certain axioms, and we consider the case $H=\mathcal{L} U_q \mathfrak{sl}_2$, the quantum loop algebra for $\mathfrak{sl}_2$, and $B=\mathcal{A}_q$, the alternating central extension of the $q$-Onsager algebra. Considering tensor products of evaluation representations of $\mathcal{L} U_q \mathfrak{sl}_2$ in ''non-semisimple'' cases, the new set of axioms allows us to introduce and study fused K-operators of spin-$j$; in particular, to prove that for all $j\in\frac{1}{2}\mathbb{N}$ they satisfy the spectral-parameter dependent reflection equation. We provide their explicit expression in terms of elements of the algebra ${\mathcal A}_q$ for small values of spin-$j$. The precise relation between the fused K-operators of spin-$j$ and evaluations of a universal K-matrix for ${\mathcal A}_q$ is conjectured based on supporting evidence. We finally discuss implications of our results on the K-operators for quantum integrable systems.

math.QA↗

arXiv2026

Universal TT- and TQ-relations via centrally extended q-Onsager algebra

Let $A_q$ be the alternating central extension of the q-Onsager algebra, a comodule algebra over the quantum loop algebra of $sl_2$. We classify one-dimensional representations of $A_q$, and show that spin-j K-operators constructed in arXiv:2301.00781 act as K-matrices previously obtained in the literature. Using these K-operators and K-matrices, we construct universal spin-j transfer matrices generating commutative subalgebras in $A_q$. Within a technical conjecture, we derive their fusion hierarchy, the so-called universal TT-relations. On spin-chain representations of $A_q$, we show how the universal transfer matrices evaluate to spin-chain transfer matrices, and as a result we get explicit TT-relations for all values of spins for auxiliary and quantum spaces, any inhomogeneities, and general integrable boundary conditions. In particular, we derive previously conjectured TT-relations. Using the TT-relations, we show that n-th local conserved quantities of the spin-j chains of length N are polynomials of total degree 4Njn in two non-local operators of the q-Onsager algebra. As a result, we give an algorithm of explicit calculation of all local conserved quantities in terms of spin operators. Furthermore, using the universal TT-relations we derive exchange relations between spin-j Hamiltonians and the two non-local operators showing non-trivial symmetries for special boundary conditions, that they commute with all Hamiltonian densities. As another application of our universal TT-relations we propose universal T-system, Y-system and universal TQ-relations for $A_q$, and as a result, universal TQ for the q-Onsager algebra. For diagonal boundary conditions, we also obtain universal TT- and TQ-relations for a degenerate version of $A_q$ known as centrally extended augmented q-Onsager algebra. We finally discuss implications of our results for generalized Gibbs ensemble construction.

math.QA↗

arXiv2026

Fully exact and fully dualizable module categories

We define fully exact module categories, a subclass of exact module categories over a finite braided tensor category that is stable under the relative Deligne product. In contrast, we demonstrate with examples in both zero and non-zero characteristic of the base field that the class of exact module categories is not stable under this product. We also observe in examples that fully exact module categories form a dense subset in the class of exact ones. The monoidal 2-category of fully exact module categories strictly contains those of invertible and separable module categories. In fact, we show that each internal algebra of a fully exact module category is projectively separable, a generalization of separable algebras involving projective objects. In the semisimple case, a module category is fully exact if and only if it is separable. In general, fully exact module categories are not dualizable inside their class, but if they are, they are fully dualizable objects in the monoidal 2-category of finite module categories. We call such module categories perfect. We show that perfect module categories form a rigid monoidal 2-subcategory containing all fully dualizable objects. Therefore, we propose perfect module categories as a model for finite tensor 2-categories. If the braiding is symmetric, a module category is fully exact if and only if it is perfect. As a detailed example, we classify fully exact, and hence perfect, module categories over the symmetric tensor category of modules over Sweedler's four-dimensional Hopf algebra and compute their relative Deligne products, and the categories of 1-morphisms. For a general quasi-triangular Hopf algebra, we analyze when the category of finite-dimensional vector spaces is fully exact. We show that this is not the case for both Sweedler's Hopf algebra and Lusztig's factorizable small quantum group of type $A_1$ at an odd root of unity.

math.QA↗

arXiv2024

An adjunction theorem for Davydov-Yetter cohomology and infinitesimal braidings

Davydov-Yetter cohomology $H_{\mathrm{DY}}^{\bullet}(F)$ is associated to a monoidal functor $F: \mathcal{C} \to \mathcal{D}$ between $\Bbbk$-linear monoidal categories where $\Bbbk$ is a field, and its second degree classifies the infinitesimal deformations of the monoidal structure of $F$. Our main result states that if $F$ admits a right adjoint $R$, then there is an object $Γ$ in the Drinfeld center $\mathcal{Z}(\mathcal{C})$ defined in terms of $R$ such that the Davydov-Yetter cohomology of $F$ can be expressed as the Davydov-Yetter cohomology of the identity functor on $\mathcal{C}$ with the coefficient $Γ$. We apply this result in the case when the product functor $\otimes: \mathcal{C} \boxtimes\mathcal{C} \to\mathcal{C}$ has a monoidal structure given by a braiding $c$ on $\mathcal{C}$ and determine explicitly the coefficient $Γ$ as a coend object in $\mathcal{Z}(\mathcal{C}) \boxtimes \mathcal{Z}(\mathcal{C})$. The motivation is that $H^{\bullet}_{\mathrm{DY}}(\otimes)$ contains a ``space of infinitesimal braidings tangent to $c$'' in a way that we describe precisely. For $\mathcal{C} = H\text{-}\mathrm{mod}$, where $H$ is a finite-dimensional Hopf algebra over a field $\Bbbk$, this is the Zariski tangent space to the affine variety of R-matrices for $H$. In the case of perfect $\Bbbk$, we give a dimension formula for this space as an explicit end involving only (low-degree) relative Ext's of the standard adjunction between $\mathcal{Z}(\mathcal{C})$ and $\mathcal{C}$. As a further application of the adjunction theorem, we describe deformations of the restriction functor associated to a Hopf subalgebra and a Drinfeld twist. Both applications are illustrated in the example of bosonization of exterior algebras.

math.QA↗

arXiv2024

Davydov-Yetter cohomology and relative homological algebra

Davydov--Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. In this paper we provide new tools for the computation of the DY cohomology for finite tensor categories and exact functors between them. The key point is to realize DY cohomology as relative Ext groups. In particular, we prove that the infinitesimal deformations of a tensor category $\mathcal{C}$ are classified by the 3-rd self-extension group of the tensor unit of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ relative to $\mathcal{C}$. From classical results on relative homological algebra we get a long exact sequence for DY cohomology and a Yoneda product for which we provide an explicit formula. Using the long exact sequence and duality, we obtain a dimension formula for the cohomology groups based solely on relatively projective covers which reduces a problem in homological algebra to a problem in representation theory, e.g. calculating the space of invariants in a certain object of $\mathcal{Z}(\mathcal{C})$. Thanks to the Yoneda product, we also develop a method for computing DY cocycles explicitly which are needed for applications in the deformation theory. We apply these tools to the category of finite-dimensional modules over a finite-dimensional Hopf algebra. We study in detail the examples of the bosonization of exterior algebras $Λ\mathbb{C}^k \rtimes \mathbb{C}[\mathbb{Z}_2]$, the Taft algebras and the small quantum group of $\mathfrak{sl}_2$ at a root of unity.

math.QA↗

arXiv2023

Integrability of rank-two web models

We continue our work on lattice models of webs, which generalise the well-known loop models to allow for various kinds of bifurcations [arXiv:2101.00282, arXiv:2107.10106]. Here we define new web models corresponding to each of the rank-two spiders considered by Kuperberg [arXiv:q-alg/9712003]. These models are based on the $A_2$, $G_2$ and $B_2$ Lie algebras, and their local vertex configurations are intertwiners of the corresponding $q$-deformed quantum algebras. In all three cases we define a corresponding model on the hexagonal lattice, and in the case of $B_2$ also on the square lattice. For specific root-of-unity choices of $q$, we show the equivalence to a number of three- and four-state spin models on the dual lattice. The main result of this paper is to exhibit integrable manifolds in the parameter spaces of each web model. For $q$ on the unit circle, these models are critical and we characterise the corresponding conformal field theories via numerical diagonalisation of the transfer matrix. In the $A_2$ case we find two integrable regimes. The first one contains a dense and a dilute phase, for which we have analytic control via a Coulomb gas construction, while the second one is more elusive and likely conceals non-compact physics. Three particular points correspond to a three-state spin model with plaquette interactions, of which the one in the second regime appears to present a new universality class. In the $G_2$ case we identify four regimes numerically. The $B_2$ case is too unwieldy to be studied numerically in the general case, but it found analytically to contain a simpler sub-model based on generators of the dilute Birman-Murakami-Wenzl algebra.

hep-th↗

arXiv2023

Algebraic Bethe Ansatz for the Open XXZ Spin Chain with Non-Diagonal Boundary Terms via $U_{\mathfrak{q}}\mathfrak{sl}_2$ Symmetry

We derive by the traditional algebraic Bethe ansatz method the Bethe equations for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [J. Phys. A 37 (2004), 433-440, arXiv:hep-th/0304092]. The technical difficulties due to the breaking of $\mathsf{U}(1)$ symmetry and the absence of a reference state are overcome by an algebraic construction where the two-boundary Temperley-Lieb Hamiltonian is realised in a new $U_{\mathfrak{q}}\mathfrak{sl}_2$-invariant spin chain involving infinite-dimensional Verma modules on the edges [J. High Energy Phys. 2022 (2022), no. 11, 016, 64 pages, arXiv:2207.12772]. The equivalence of the two Hamiltonians is established by proving Schur-Weyl duality between $U_{\mathfrak{q}}\mathfrak{sl}_2$ and the two-boundary Temperley-Lieb algebra. In this framework, the Nepomechie condition turns out to have a simple algebraic interpretation in terms of quantum group fusion rules.

hep-th↗

arXiv2022

Davydov-Yetter cohomology, comonads and Ocneanu rigidity

Davydov-Yetter cohomology classifies infinitesimal deformations of tensor categories and of tensor functors. Our first result is that Davydov-Yetter cohomology for finite tensor categories is equivalent to the cohomology of a comonad arising from the central Hopf monad. This has several applications: First, we obtain a short and conceptual proof of Ocneanu rigidity. Second, it allows to use standard methods from comonad cohomology theory to compute Davydov-Yetter cohomology for a family of non-semisimple finite-dimensional Hopf algebras generalizing Sweedler's four dimensional Hopf algebra.

math.QA↗

arXiv2022

$U_q\mathfrak{sl}_2$-invariant non-compact boundary conditions for the XXZ spin chain

We introduce new $U_q\mathfrak{sl}_2$-invariant boundary conditions for the open XXZ spin chain. For generic values of $q$ we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $q=e^{\frac{iπ}{p}}$ a $2p$-th root of unity $ - $ to its $p$-dimensional analogue. Both cases are parametrised by a continuous "spin" $α\in\mathbb{C}$. To motivate our construction, we first specialise to $q=i$, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the $(η,ξ)$ ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic $q$ we investigate the algebraic properties of the underlying lattice algebras. We show that if $q^α\notin\pm q^{\mathbb{Z}}$, the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to $U_q\mathfrak{sl}_2$. Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of $q$ and $α$. Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.

hep-th↗

arXiv2022

The symplectic fermion ribbon quasi-Hopf algebra and the SL(2,Z)-action on its centre

We introduce a family of factorisable ribbon quasi-Hopf algebras $Q(N)$ for $N$ a positive integer: as an algebra, $Q(N)$ is the semidirect product of $\mathbb{C}\mathbb{Z}_2$ with the direct sum of a Grassmann and a Clifford algebra in $2N$ generators. We show that $Rep Q(N)$ is ribbon equivalent to the symplectic fermion category $SF(N)$ that was computed by the third author from conformal blocks of the corresponding logarithmic conformal field theory. The latter category in turn is conjecturally ribbon equivalent to representations of $V_{ev}$, the even part of the symplectic fermion vertex operator super algebra. Using the formalism developed in our previous paper we compute the projective $SL(2,\mathbb{Z})$-action on the centre of $Q(N)$ as obtained from Lyubashenko's general theory of mapping class group actions for factorisable finite ribbon categories. This allows us to test a conjectural non-semisimple version of the modular Verlinde formula: we verify that the $SL(2,\mathbb{Z})$-action computed from $Q(N)$ agrees projectively with that on pseudo trace functions of $V_{ev}$.

math.QA↗

arXiv2021

3-Dimensional TQFTs From Non-Semisimple Modular Categories

We use modified traces to renormalize Lyubashenko's closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2+1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions of Lyubashenko's mapping class group representations, as discussed in arXiv:2010.14852. This general framework encompasses important examples of non-semisimple modular categories coming from the representation theory of quasi-Hopf algebras, which were left out of previous non-semisimple TQFT constructions.

math.GT↗

arXiv2021

$U_{\mathfrak{q}}(\mathfrak{sl}_3)$ web models: Locality, phase diagram and geometrical defects

We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice H having a U_q(sl_n) quantum group symmetry. We focus here on the n=3 case of cubic webs, based on the Kuperberg A_2 spider, and illustrate its properties by comparisons with the well-known dilute loop model (the n=2 case) throughout. A local vertex-model reformulation is exhibited, analogous to the correspondence between the loop model and a three-state vertex model. The n=3 representation uses seven states per link of H, displays explicitly the geometrical content of the webs and their U_q(sl_3) symmetry, and permits us to study the model on a cylinder via a local transfer matrix. A numerical study of the central charge reveals that for each q $\in \mathbb{C}$ in the critical regime, |q|=1, the web model possesses a dense and a dilute critical point, just like its loop model counterpart. In the dense $q=-e^{i π/4}$ case, the n=3 webs can be identified with spin interfaces of the critical three-state Potts model defined on the triangular lattice dual to H. We also provide another mapping to a $\mathbb{Z}_3$ spin model on H itself, using a high-temperature expansion. We then discuss the sector structure of the transfer matrix, for generic q, and its relation to defect configurations in both the strip and the cylinder geometries. These defects define the finite-size precursors of electromagnetic operators. This discussion paves the road for a Coulomb gas description of the conformal properties of defect webs, which will form the object of a subsequent paper. Finally, we identify the fractal dimension of critical webs in the $q=-e^{i π/3}$ case, which is the n=3 analogue of the polymer limit in the loop model.

cond-mat.stat-mech↗

arXiv2021

Mapping Class Group Representations From Non-Semisimple TQFTs

In [arXiv:1912.02063], we constructed 3-dimensional Topological Quantum Field Theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category $\mathcal{C}$. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of [arXiv:1912.02063] are equivalent to those obtained by Lyubashenko via generators and relations in [arXiv:hep-th/9405167]. Finally, we show that, when $\mathcal{C}$ is the category of finite-dimensional representations of the small quantum group of $\mathfrak{sl}_2$, the action of all Dehn twists for surfaces without marked points has infinite order.

math.GT↗

arXiv2021

Monadic cointegrals and applications to quasi-Hopf algebras

For $\mathcal{C}$ a finite tensor category we consider four versions of the central monad, $A_1, \dots, A_4$ on $\mathcal{C}$. Two of them are Hopf monads, and for $\mathcal{C}$ pivotal, so are the remaining two. In that case all $A_i$ are isomorphic as Hopf monads. We define a monadic cointegral for $A_i$ to be an $A_i$-module morphism $\mathbf{1} \to A_i(D)$, where $D$ is the distinguished invertible object of $\mathcal{C}$. We relate monadic cointegrals to the categorical cointegral introduced by Shimizu (2019), and, in case $\mathcal{C}$ is braided, to an integral for the braided Hopf algebra $\mathcal{L} = \int^X X^\vee \otimes X$ in $\mathcal{C}$ studied by Lyubashenko (1995). Our main motivation stems from the application to finite dimensional quasi-Hopf algebras $H$. For the category of finite-dimensional $H$-modules, we relate the four monadic cointegrals (two of which require $H$ to be pivotal) to four existing notions of cointegrals for quasi-Hopf algebras: the usual left/right cointegrals of Hausser and Nill (1994), as well as so-called $γ$-symmetrised cointegrals in the pivotal case, for $γ$ the modulus of $H$. For (not necessarily semisimple) modular tensor categories $\mathcal{C}$, Lyubashenko gave actions of surface mapping class groups on certain Hom-spaces of $\mathcal{C}$, in particular of $SL(2,\mathbb{Z})$ on $\mathcal{C}(\mathcal{L},\mathbf{1})$. In the case of a factorisable ribbon quasi-Hopf algebra, we give a simple expression for the action of $S$ and $T$ which uses the monadic cointegral.

math.QA↗

arXiv2021

$U_q(\mathfrak{sl}_n)$ web models and $\mathbb{Z}_n$ spin interfaces

This is the first in a series of papers devoted to generalisations of statistical loop models. We define a lattice model of $U_q(\mathfrak{sl}_n)$ webs on the honeycomb lattice, for $n \ge 2$. It is a statistical model of closed, cubic graphs with certain non-local Boltzmann weights that can be computed from spider relations. For $n=2$, the model has no branchings and reduces to the well-known O($N$) loop model introduced by Nienhuis. In the general case, we show that the web model possesses a particular point, at $q=e^{i π/(n+1)}$, where the partition function is proportional to that of a $\mathbb{Z}_n$-symmetric chiral spin model on the dual lattice. Moreover, under this equivalence, the graphs given by the configurations of the web model are in bijection with the domain walls of the spin model. For $n=2$, this equivalence reduces to the well-known relation between the Ising and O($1$) models. We define as well an open $U_q(\mathfrak{sl}_n)$ web model on a simply connected domain with a boundary, and discuss in particular the role of defects on the boundary.

cond-mat.stat-mech↗