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We introduce the notions of $(G,q)$-opers and Miura $(G,q)$-opers, where $G$ is a simply-connected complex simple Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of $(G,q)$-opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. This may be viewed as a $q$DE/IM correspondence between the spectra of a quantum integrable model (IM) and classical geometric objects ($q$-differential equations). If $\mathfrak{g}$ is simply-laced, the Bethe Ansatz equations we obtain coincide with the equations that appear in the quantum integrable model of XXZ-type associated to the quantum affine algebra $U_q \widehat{\mathfrak{g}}$. However, if $\mathfrak{g}$ is non-simply laced, then these equations correspond to a different integrable model, associated to $U_q {}^L\widehat{\mathfrak{g}}$ where $^L\widehat{\mathfrak{g}}$ is the Langlands dual (twisted) affine algebra. A key element in this $q$DE/IM correspondence is the $QQ$-system that has appeared previously in the study of the ODE/IM correspondence and the Grothendieck ring of the category ${\mathcal O}$ of the relevant quantum affine algebra., v3: 44 pages, minor revisions, to appear in the Journal of the European Mathematical Society

We construct a generalization of Poisson-Chern-Simons theory, defined on any supermanifold equipped with an appropriate filtration of the tangent bundle. Our construction recovers interacting eleven-dimensional supergravity in Cederwall's formulation, as well as all possible twists of the theory, and does so in a uniform and geometric fashion. Among other things, this proves that Costello's description of the maximal twist is the twist of eleven-dimensional supergravity in its pure spinor description. It also provides a pure spinor lift of the interactions in the minimally twisted theory. Our techniques enhance the BV formulation of the interactions of each theory to a homotopy Poisson structure by defining a compatible graded-commutative product; this suggests interpretations in terms of deformations of geometric structures on superspace, and provides some concrete evidence for a first-quantized origin of the theories., 35 pages. One table. Comments welcome!

This paper is a fundamental study of comodules and contramodules over a comonoid in a symmetric closed monoidal category. We study both algebraic and homotopical aspects of them. Algebraically, we enrich the comodule and contramodule categories over the original category, construct enriched functors between them and enriched adjunctions between the functors. Homotopically, for simplicial sets and topological spaces, we investigate the categories of comodules and contramodules and the relations between them., Comment: Version 2: major revision to make the paper easier to read. Version 3: another major revision: we refer to various old results on the functorial semantics that makes paper shorter and more accessible. Version 4: yet another major revision: we replace a biclosed monoidal category with a symmetric closed monoidal category, simplifying many proofs. Version 5: minor edits, final journal version

The inclusion of the unit in a braided tensor category $\mathcal{V}$ induces a 1-morphism in the Morita 4-category of braided tensor categories $BrTens$. We give criteria for the dualizability of this morphism.\\ When $\mathcal{V}$ is a semisimple (resp. non-semisimple) modular category, we show that the unit inclusion induces under the Cobordism Hypothesis a (resp. non-compact) relative 3-dimensional topological quantum field theory. Following Jordan--Safronov, we conjecture that these relative field theories together with their bulk theories recover Witten--Reshetikhin--Turaev (resp. De Renzi--Gainutdinov--Geer--Patureau-Mirand--Runkel) theories, in a fully extended setting. In particular, we argue that these theories can be obtained by the Cobordism Hypothesis., 28 pages. Comments welcome!

We study the braided tensor structure of line operators in the topological A and B twists of abelian 3d $\mathcal{N}=4$ gauge theories, as accessed via boundary vertex operator algebras (VOA's). We focus exclusively on abelian theories. We first find a non-perturbative completion of boundary VOA's in the B twist, which start out as certain affine Lie superalebras; and we construct free-field realizations of both A and B-twist VOA's, finding an interesting interplay with the symmetry fractionalization group of bulk theories. We use the free-field realizations to establish an isomorphism between A and B VOA's related by 3d mirror symmetry. Turning to line operators, we extend previous physical classifications of line operators to include new monodromy defects and bound states. We also outline a mechanism by which continuous global symmetries in a physical theory are promoted to higher symmetries in a topological twist -- in our case, these are infinite one-form symmetries, related to boundary spectral flow, which structure the categories of lines and control abelian gauging. Finally, we establish the existence of braided tensor structure on categories of line operators, viewed as non-semisimple categories of modules for boundary VOA's. In the A twist, we obtain the categories by extending modules of symplectic boson VOA's, corresponding to gauging free hypermultiplets; in the B twist, we instead extend Kazhdan-Lusztig categories for affine Lie superalgebras. We prove braided tensor equivalences among the categories of 3d-mirror theories. All results on VOA's and their module categories are mathematically rigorous; they rely strongly on recently developed techniques to access non-semisimple extensions., 158 pages, comments welcome!

In 2019, Jung-Weber gave an example of a concrete magic unitary $M$, which defines a $C^*$-algebraic model of the quantum permutation group $S_4^+$. We show with the help of a computer that there exist no polynomials up to degree $50$ separating the entries of $M$ from the generators of $C(S_4^+)$. This indicates that the magic unitary $M$ might already define a faithful model of $S_4^+$.

Let $\mathbb{K}$ be a field. We study the free bialgebra $\mathcal{T}$ generated by the coalgebra $C=\mathbb{K} g \oplus \mathbb{K} h$ and its quotient bialgebras (or Hopf algebras) over $\mathbb{K}$. We show that the free noncommutative Fa\`a di Bruno bialgebra is a sub-bialgebra of $\mathcal{T}$, and the quotient bialgebra $\overline{\mathcal{T}}:=\mathcal{T}/(E_{\alpha}|~\alpha(g)\ge 2)$ is an Ore extension of the well-known Fa\`a di Bruno bialgebra. The image of the free noncommutative Fa\`a di Bruno bialgebra in the quotient $\overline{\mathcal{T}}$ gives a more reasonable non-commutative version of the commutative Fa\`a di Bruno bialgebra from the PBW basis point view. If char $\mathbb{K}=p>0$, we obtain a chain of quotient Hopf algebras of $\overline{\mathcal{T}}$: $\overline{\mathcal{T}} \twoheadrightarrow \overline{\mathcal{T}}_{n}\twoheadrightarrow \overline{\mathcal{T}}_{n}'(p)\twoheadrightarrow \overline{\mathcal{T}}_{n}(p)\twoheadrightarrow \overline{\mathcal{T}}_{n}(p;d_{1}) \twoheadrightarrow \ldots \twoheadrightarrow \overline{\mathcal{T}}_{n}(p;d_{j},d_{j-1},\ldots,d_{1}) \twoheadrightarrow \ldots \twoheadrightarrow \overline{\mathcal{T}}_{n}(p;d_{p-2},d_{p-3},\ldots,d_{1})$ with finite GK-dimensions. Furthermore, we study the homological properties and the coradical filtrations of those quotient Hopf algebras., Comment: 27 pages, typos corrected

For all Frobenius groups and a large class of finite multiply transitive permutation groups, we show that the corresponding group-subgroup subfactors are completely characterized by their principal graphs. The class includes all the sharply $k$-transitive permutation groups for $k=2,3,4$, and in particular the Mathieu group $M_{11}$ of degree 11.

In this note we give an introduction to Drinfel'd's associator coming from the Knizhnik-Zamolodchikov connections and a self-contained proof of the hexagon and pentagon equations by means of minimal amounts of analysis or differential geometry: we rather use limits of concrete parallel transports.

We study the $Q$-Kostka polynomials $L_{\lambda\mu}(t)$ by the vertex operator realization of the $Q$-Hall-Littlewood functions $G_{\lambda}(x;t)$ and derive new formulae for $L_{\lambda\mu}(t)$. In particular, we have established stability property for the Q-Kostka polynomials. We also introduce spin Green polynomials $Y^{\lambda}_{\mu}(t)$ as both an analogue of the Green polynomials and deformation of the spin irreducible characters of $\mathfrak S_n$. Iterative formulas of the spin Green polynomials are given and some favorable properties parallel to the Green polynomials are obtained. Tables of $Y^{\lambda}_{\mu}(t)$ are included for $n\leq7.$, Comment: 5 tables

The quantum Grothendieck ring of a certain category of finite-dimensional modules over a quantum loop algebra associated with a complex finite-dimensional simple Lie algebra $\mathfrak{g}$ has a quantum cluster algebra structure of skew-symmetric type. Partly motivated by a search of a ring corresponding to a quantum cluster algebra of {\em skew-symmetrizable} type, the quantum {\em virtual} Grothendieck ring, denoted by $\mathfrak{K}_q(\mathfrak{g})$, is recently introduced by Kashiwara--Oh \cite{KO23} as a subring of the quantum torus based on the $(q,t)$-Cartan matrix specialized at $q=1$. In this paper, we prove that $\mathfrak{K}_q(\mathfrak{g})$ indeed has a quantum cluster algebra structure of skew-symmetrizable type. This task essentially involves constructing distinguished bases of $\mathfrak{K}_q(\mathfrak{g})$ that will be used to make cluster variables and generalizing the quantum $T$-system associated with Kirillov--Reshetikhin modules to establish a quantum exchange relation of cluster variables. Furthermore, these distinguished bases naturally fit into the paradigm of Kazhdan--Lusztig theory and our study of these bases leads to some conjectures on quantum positivity and $q$-commutativity., We corrected typos, added a conjecture and revised

We use semi-derived Ringel-Hall algebras of quivers with loops to realize the whole quantum Borcherds-Bozec algebras and quantum generalized Kac-Moody algebras., Comment: v2, 12 pages, minor edits, accepted by Proc. AMS

We provide a mathematical formulation of the idea of a defect for a field theory, in terms of the factorization algebra of observables and using the BV formalism. Our approach follows a well-known ansatz identifying a defect as a boundary condition along the boundary of a blow-up, but it uses recent work of Butson-Yoo and Rabinovich on boundary conditions and their associated factorization algebras to implement the ansatz. We describe how a range of natural examples of defects fits into our framework., Comment: 19 pages. Comments welcome!

In this article, we discuss some recent developments of Zariski Cancellation Problem in the setting of noncommutative algebras and Poisson algebras.

In this article, we investigate the representations of the Drinfeld doubles $D(R_{mn}(q))$ of the Radford Hopf algebras $R_{mn}(q)$ over an algebraically closed field $\Bbbk$, where $m>1$ and $n>1$ are integers and $q\in\Bbbk$ is a root of unity of order $n$. Under the assumption ${\rm char}(\Bbbk)\nmid mn$, all the finite dimensional indecomposable modules over $D(R_{mn}(q))$ are displayed and classified up to isomorphism. The Auslander-Reiten sequences in the category of finite dimensional $D(R_{mn}(q))$-modules are also all displayed. It is shown that $D(R_{mn}(q))$ is of tame representation type., This is a research paper, 30 pages

There are known two different constructions of contractible dg 2-operads, providing a weak 2-category structure on the following dg 2-quiver of small dg 2-categories. Its vertices are small dg 2-categories over a given field, arrows are dg functors, and the 2-arrows $F\Rightarrow G$ are defined as the Hochschild cochains of $C$ with coefficients in $C$-bimodule $D(F(-),G(=))$, where $F,G\colon C\to D$ are dg functors, $C,D$ small dg categories. It is known that such definition is correct homotopically, but, on the other hand, the corresponding dg 2-quiver fails to be a strict 2-category. The question ``What do dg categories form'' is the question of finding a weak 2-category structure on it, in an appropriate sense. One way of phrasing it out is to make it an algebra over a contractible 2-operad, in the sense of M.Batanin [Ba1,2] (in turn, there are many compositions of 2-arrows for a given diagram, but their totality forms a contractible complex) . In [T], D.Tamarkin proposed a contractible $\Delta$-colored 2-operad in Sets, whose dg condensation solves the problem. In our recent paper arXiv:1807.04305 we constructed contractible dg 2-operad, called the twisted tensor product operad, acting on the same 2-quiver (the construction uses the twisted tensor product of small dg categories introduced in arXiv:1803.01191). In this paper, we compare the two constructions., Comment: 25 pages It is an improved and corrected version of Appendix B of the previous version of arXiv:1807.04305, which has been removed from the most recent version

In present paper, some properties of Hopf action on vertex algebras will be given. We prove that $V#H$ is an $\mathcal{S}$-local vertex algebra but it is not a vertex algebra when $V$ is an $H$-module vertex algebra. In addition, we extend the $\mathcal{S}$-locality to the quantum vertex algebras., 20 pages

We provide an intrinsic formulation of the noncommutative differential geometry developed earlier by Chaichian, Tureanu, R. B. Zhang and the second author. This yields geometric definitions of covariant derivatives of noncommutative metrics and curvatures, as well as the noncommutative version of the first and the second Bianchi identities. Moreover, if a noncommutative metric and chiral coefficients satisfy certain conditions which hold automatically for quantum fluctuations given by isometric embedding, we prove that the two noncommutative Ricci curvatures are essentially equivalent. Finally, we show that the quantum fluctuations and their curvatures have close forms if (pseudo-) Riemannian metrics are given by certain type of spherically symmetric isometric embeddings. Hence the quantization of gravity is renormalizable in this case., 28 pages

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra K_{n} obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, the well-known Fomin-Kirillov algebras appear as Nichols algebras over K_{3}. As a byproduct we obtain new Hopf algebras of dimension 216., 24 pages. Comments are welcome!

We establish a cluster theoretical interpretation of the isomorphisms of [F.-H.-O.-O., J. Reine Angew. Math., 2022] among quantum Grothendieck rings of representations of quantum loop algebras. Consequently, we obtain a quantization of the monoidal categorification theorem of [Kashiwara-Kim-Oh-Park, arXiv:2103.10067]. We establish applications of these new ingredients. First we solve long-standing problems for any non-simply-laced quantum loop algebras: the positivity of $(q,t)$-characters of all simple modules, and the analog of Kazhdan-Lusztig conjecture for all reachable modules (in the cluster monoidal categorification). We also establish the conjectural quantum $T$-systems for the $(q,t)$-characters of Kirillov-Reshetikhin modules. Eventually, we show that our isomorphisms arise from explicit birational transformations of variables, which we call substitution formulas. This reveals new non-trivial relations among $(q, t)$-characters of simple modules., v1 60 pages; v2 58 pages, the proofs in Section 7 are simplified

We characterize cyclic algebras over the associative and the framed little 2-disks operad in any symmetric monoidal bicategory. The cyclicity is appropriately treated in a coherent way, i.e. up to coherent isomorphism. When the symmetric monoidal bicategory is specified to be a certain symmetric monoidal bicategory of linear categories subject to finiteness conditions, we prove that cyclic associative and cyclic framed little 2-disks algebras, respectively, are equivalent to pivotal Grothendieck-Verdier categories and ribbon Grothendieck-Verdier categories, a type of category that was introduced by Boyarchenko-Drinfeld based on Barr's notion of a $\star$-autonomous category. We use these results and Costello's modular envelope construction to obtain two applications to quantum topology: I) We extract a consistent system of handlebody group representations from any ribbon Grothendieck-Verdier category inside a certain symmetric monoidal bicategory of linear categories and show that this generalizes the handlebody part of Lyubashenko's mapping class group representations. II) We establish a Grothendieck-Verdier duality for the category extracted from a modular functor by evaluation on the circle (without any assumption on semisimplicity), thereby generalizing results of Tillmann and Bakalov-Kirillov., 66 pages, lots of figures and diagrams; v3: minor changes in terminology to be consistent with follow-up papers (ribbon GV instead of balanced braided GV, auxiliary notion of self-dual algebra), comment on invertibility of maps of modular algebras included, minor changes in the presentation in several places; v4: some changes following comments by the referee, to appear in Quart. J. Math

We propose an algebraic expression for $U_q(\mathfrak{sl}_3)$ quantum $3j$ symbols (quantum Clebsch-Gordan coefficients) appearing in the decomposition of tensor product of symmetric representations. Our compact form will be useful to write the spectral parameter dependent $R$-matrix elements for any bi-partite vertex model whose edges carry states of the symmetric representations.

This paper extends an algorithm and canonical embedding by Cauchon to a large class of quantum algebras. It applies to iterated Ore extensions over a field satisfying some suitable assumptions which cover those of Cauchon's original setting but also allows for roots of unity. The extended algorithm constructs a quantum affine space $A'$ from the original quantum algebra $A$ via a series of change of variables within the division ring of fractions $\mathrm{Frac}(A)$. The canonical embedding takes a completely prime ideal $P\lhd A$ to a completely prime ideal $Q\lhd A'$ such that when $A$ is a PI algebra, ${\rm PI}\text{-}{\rm deg}(A/P) = {\rm PI}\text{-}{\rm deg}(A'/Q)$. When the quantum parameter is a root of unity we can state an explicit formula for the PI degree of completely prime quotient algebras. This paper ends with a method to construct a maximum dimensional irreducible representation of $A/P$ given a suitable irreducible representation of $A'/Q$ when $A$ is PI., 27 pages

This document is a thesis presented for the ``Habilitation \`a diriger des recherches''. The first chapter provides some background and sketch the story of the classical Schur-Weyl duality and its quantum analogue involving the Hecke algebra. Quantum groups and their centralisers are discussed along with their connections with the braid group, the Yang-Baxter equation and the construction of link invariants. A quick review of several related notions of Hecke algebras concludes the first chapter. The second chapter presents the fused Hecke algebras and an application to the construction of solutions for the Yang-Baxter equation. The third chapter summarises recent works on diagonal centralisers of Lie algebras, and gives an application to the study of the missing label for sl(3). The last chapter discusses algebraic results on the Yokonuma-Hecke algebras and their applications to the study of link invariants., Comment: 81 pages

We apply Majid's transmutation procedure to Hopf algebra maps $H \to \mathbb C[T]$, where $T$ is a compact abelian group, and explain how this construction gives rise to braided Hopf algebras over quotients of $T$ by subgroups that are cocentral in $H$. This allows us to unify and generalize a number of recent constructions of braided compact quantum groups, starting from the braided $SU_q(2)$ quantum group, and describe their bosonizations., 19 pages; v2: minor corrections and improvements, a discussion of Mrozinski's quantum groups added

The Lie superalgebra $\mathfrak{psl}(2|2)$ is recognized as a pretty special one in both mathematics and theoretical physics. In this paper, we present the Drinfeld realization of the Yangian algebra associated with the centrally extended Lie superalgebra $\mathfrak{psl}(2|2)$. Furthermore, we show that it possesses the Hopf algebra structures, particularly the coproducts. The idea to prove the existence of the manifest coproducts is the following. Firstly, we shall introduce them to Levendorskii's realization, a system of a finite truncation of the Drinfeld generators. Secondly, we show that Levendorskii's realization is isomorphic to the Drinfeld realization by induction., 44 pages, v2: typos corrected, references added

An averaging operator on an associative algebra $A$ is an algebraic abstraction of the time average operator on the space of real-valued functions defined in time-space. In this paper, we consider relative averaging operators on a bimodule $M$ over an associative algebra $A$. A relative averaging operator induces a diassociative algebra structure on the space $M$. The full data consisting of an associative algebra, a bimodule and a relative averaging operator is called a relative averaging algebra. We define bimodules over a relative averaging algebra that fits with the representations of diassociative algebras. We construct a graded Lie algebra and a $L_\infty$-algebra that are respectively controlling algebraic structures for a given relative averaging operator and relative averaging algebra. We also define cohomologies of relative averaging operators and relative averaging algebras and find a long exact sequence connecting various cohomology groups. As applications, we study deformations and abelian extensions of relative averaging algebras. Finally, we define homotopy relative averaging algebras and show that they induce homotopy diassociative algebras., 36pages

We develop a string-net construction for the (2,1)-dimensional part of a $G$-equivariant three-dimensional topological field theory based on a $G$-graded spherical fusion category. In this construction, a $G$-equivariant generalization of the Ptolemy groupoid enters. We compute the associated cylinder categories and show that, as expected, the model is closely related to the $G$-equivariant Turaev-Viro theory., 44 pages, lots of figures

We propose a definition by generators and relations of the rank $n-2$ Askey--Wilson algebra $aw(n)$ for any integer $n$, generalising the known presentation for the usual case $n=3$. The generators are indexed by connected subsets of $\{1,\dots,n\}$ and the simple and rather small set of defining relations is directly inspired from the known case of $n=3$. Our first main result is to prove the existence of automorphisms of $aw(n)$ satisfying the relations of the braid group on $n+1$ strands. We also show the existence of coproduct maps relating the algebras for different values of $n$. An immediate consequence of our approach is that the Askey--Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the $n$-fold tensor product of the quantum group $U_q(sl_2)$ or, equivalently, onto the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. We also obtain a family of central elements of the Askey--Wilson algebras which are shown, as a direct by-product of our construction, to be sent to $0$ in the realisation in the $n$-fold tensor product of $U_q(sl_2)$, thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements., 35 pages

We present the quantum teleportation and superdense coding protocols in the context of topological qudits, as realised by anyons. The simplicity of our proposed realisation hinges on the monoidal structure of Tambara-Yamagami categories, which readily allows for the generation of maximally entangled qudits. In particular, we show that both protocols can be performed without any braiding of anyons. Our exposition makes use of the graphical calculus for braided fusion categories, a medium in which the protocols find a natural interpretation. We also find a braid-free realisation of the Pauli gates using Ising anyons.

The Deligne-Getzler-Hinich--$\infty$-groupoid or Maurer-Cartan simplicial set of an $L_\infty$-algebra plays an important role in deformation theory and many other areas of mathematics. Unfortunately, this construction only works over a field of characteristic $0$. The goal of this paper is to show that the notions of Maurer-Cartan equation and Maurer-Cartan simplicial set can be defined for a much larger number of operads than just the $L_\infty$-operad. More precisely, we show that the Koszul dual of every unital Hopf cooperad (a cooperad in the category of unital associative algebras) with an arity $0$ operation admits a twisting procedure, a natural notion of Maurer-Cartan equation and under some mild additional assumptions can also be integrated to a Maurer-Cartan simplicial set. In particular, we show that the Koszul dual of the Barratt-Eccles operad and its $E_n$-suboperads admit Maurer-Cartan simplicial sets. In this paper, we will work over arbitrary rings., 23 pages

We classify all two-dimensional, unitary, rational conformal field theories with two primaries, central charge $c, 62 pages, 7 tables, 1 figure. Several typos corrected. References updated

Extended affine root systems appear as the root systems of extended affine Lie algebras. A subclass of extended affine root systems, whose elements are called ``minimal" turns out to be of special interest mostly because of the geometric properties of their Weyl groups; they posses the so called ``the presentation by conjugation". In this work we give a characterization of minimal extended affine root systems in terms of ``minimal reflectable bases" which resembles the concept of the ``base" for finite and affine root systems. As an application, we construct elliptic Lie algebras by means of a Serre's type generators and relations., 30 pages

In the TQFT formalism of Moore-Tachikawa for describing Higgs branches of theories of class $\mathcal{S}$, the space associated to the unpunctured sphere in type $\mathfrak{g}$ is the universal centraliser $\mathfrak{Z}_G$, where $\mathfrak{g}=Lie(G)$. In more physical terms, this space arises as the Coulomb branch of pure $\mathcal{N}=4$ gauge theory in three dimensions with gauge group $\check{G}$, the Langlands dual. In the analogous formalism for describing chiral algebras of class $\mathcal{S}$, the vertex algebra associated to the sphere has been dubbed the \emph{chiral universal centraliser}. In this paper, we construct an open, symplectic embedding from a cover of the Kostant-Toda lattice of type $\mathfrak{g}$ to the universal centraliser of $G$, extending a classic result of Kostant. Using this embedding and some observations on the Poisson algebraic structure of $\mathfrak{Z}_G$, we propose a free field realisation of the chiral universal centraliser for any simple group $G$. We exploit this realisation to develop free field realisations of chiral algebras of class $\mathcal{S}$ of type $\mathfrak{a}_1$ for theories of genus zero with up to six punctures. These realisations make generalised $S$-duality completely manifest, and the generalisation to more than six punctures is conceptually clear, though technically burdensome., 50 pages, one appendix

Conformal Quantum Field Theories (CFT) in 1 or 1+1 spacetime dimensions (respectively called chiral and full CFTs) admit several "axiomatic" (mathematically rigorous and model-independent) formulations. In this note, we deal with the von Neumann algebraic formulation due to Haag and Kastler, mainly restricted to the chiral CFT setting. Irrespectively of the chosen formulation, one can ask the questions: given a theory $\mathcal{A}$, how many and which are the possible extensions $\mathcal{B} \supset \mathcal{A}$ or subtheories $\mathcal{B} \subset \mathcal{A}$? How to construct and classify them, and study their properties? Extensions are typically described in the language of algebra objects in the braided tensor category of representations of $\mathcal{A}$, while subtheories require different ideas. In this paper, we review recent structural results on the study of subtheories in the von Neumann algebraic formulation (conformal subnets) of a given chiral CFT (conformal net). Furthermore, building on [BDG23], we provide a "quantum Galois theory" for conformal nets analogous to the one for Vertex Operator Algebras (VOA). We also outline the case of 3+1 dimensional Algebraic Quantum Field Theories (AQFT). The aforementioned results make use of families of (extreme) vacuum state preserving unital completely positive maps acting on the net of von Neumann algebras, hereafter called quantum operations. These are natural generalizations of the ordinary vacuum preserving gauge automorphisms, hence they play the role of "generalized global gauge symmetries". Quantum operations suffice to describe all possible conformal subnets of a given conformal net with the same central charge., Invited contribution to the conference proceedings of Functional Analysis, Approximation Theory and Numerical Analysis, FAATNA20>22, Matera, Italy, 2022

Let $V$ be a vertex algebra of countable dimension, $G$ a subgroup of ${\rm Aut} V$ of finite order, $V^{G}$ the fixed point subalgebra of $V$ under the action of $G$, and ${\mathscr S}$ a finite $G$-stable set of inequivalent irreducible twisted weak $V$-modules associated with possibly different automorphisms in $G$. We show a Schur--Weyl type duality for the actions of ${\mathscr A}_{\alpha}(G,{\mathscr S})$ and $V^G$ on the direct sum of twisted weak $V$-modules in ${\mathscr S}$ where ${\mathscr A}_{\alpha}(G,{\mathscr S})$ is a finite dimensional semisimple associative algebra associated with $G,{\mathscr S}$, and a $2$-cocycle $\alpha$ naturally determined by the $G$-action on ${\mathscr S}$. It follows as a natural consequence of the result that for any $g\in G$ every irreducible $g$-twisted weak $V$-module is a completely reducible weak $V^G$-module., Comment: 13 pages

We classify discrete quantum subgroups in the quantum double of the $q$-deformation of a compact semisimple Lie group, regarded as the complexification. We also record their classifications in some variants of quantum groups. Along the way, we show that quantum doubles of non-Kac type compact quantum groups do not admit the quantum analog of lattices considered by Brannan-Chirvasitu-Viselter., 15 pages, minor modifications

Given a tensor functor between tensor categories $\mathcal{C}$ and $\mathcal{D}$, we give criteria that, under certain assumptions, the Frobeniusness of $\mathcal{C}$ or $\mathcal{D}$ implies the Frobeniusness of the other one. We also give an affirmative answer to Natale's question asking if the class of Frobenius tensor categories is closed under exact sequences., 22 pages

We introduce a noncommutative Poisson random measure on a von Neumann algebra. This is a noncommutative generalization of the classical Poisson random measure. We call this construction Poissonization. Poissonization is a functor from the category of von Neumann algebras with normal semifinite faithful weights to the category of von Neumann algebras with normal faithful states. Poissonization is a natural adaptation of the second quantization to the context of von Neumann algebras. The construction is compatible with normal (weight-preserving) homomorphisms and unital normal completely positive (weight-preserving) maps. We present two main applications of Poissonization. First Poissonization provides a new framework to construct algebraic quantum field theories that are not generalized free field theories. Second Poissonization permits straight-forward calculations of quantum relative entropies (and other quantum information quantities) in the case of type III von Neumann algebras.

We notice that the famous pentagon identity for quantum dilogarithm functions and the five-term relation for certain operators related to Macdonald polynomials discovered by Garsia and Mellit can both be understood as specific cases of a general "master pentagon identity" for group-like elements in the Ding-Iohara-Miki (or quantum toroidal, or elliptic Hall) algebra. We perform some checks of this remarkable identity and discuss its implications., 10 pages

We present a rich source of Hopf algebras starting from a cofinite central extension of a Noetherian Hopf algebra and a subgroup of the algebraic group of characters of the central Hopf subalgebra. The construction is transparent from a Tannakian perspective. We determine when the new Hopf algebras are co-Frobenius, or cosemisimple, or Noetherian, or regular, or have finite Gelfand-Kirillov dimension., 25 pages, comments are welcome

To every Hopf heap or quantum cotorsor of Grunspan a Hopf algebra of translations is associated. This translation Hopf algebra acts on the Hopf heap making it a Hopf-Galois co-object. Conversely, any Hopf-Galois co-object has the natural structure of a Hopf heap with the translation Hopf algebra isomorphic to the acting Hopf algebra. It is then shown that this assignment establishes an equivalence between categories of Hopf heaps and Hopf-Galois co-objects., 14 pages

In this paper, we study quasi-homomorphisms of quantum cluster algebras, which are quantum analogy of quasi-homomorphisms of cluster algebras introduced by Fraser. Quasi-homomorphisms of quantum cluster algebras appeared in a work by Kimura, Qin, and Wei in 2022 which they called ``variation maps''. For a quantum Grassmannian cluster algebra $\CC_q[\Gr(k,n)]$, we show that there is an associated braid group and each generator $\sigma_i$ of the braid group preserves the quasi-commutative relations of quantum Pl\"{u}cker coordinates and exchange relations of the quantum Grassmannian cluster algebra. We conjecture that $\sigma_i$ also preserves $r$-term ($r \ge 4$) quantum Pl\"{u}cker relations of $\CC_q[\Gr(k,n)]$. Up to this conjecture, we show that $\sigma_i$ is a quasi-automorphism of $\CC_q[\Gr(k,n)]$ and the braid group acts on $\CC_q[\Gr(k,n)]$.

Action Lie groupoids are used to model spaces of orbits of actions of Lie groups on manifolds. For each such action groupoid $M\rtimes H$ we construct a locally convex bialgebroid $\mathord{\mathrm{Dirac}}(M\rtimes H)$ with an antipode over $\mathord{\mathcal{C}^{\infty}_{c}}(M)$, from which the groupoid $M\rtimes H$ can be reconstructed as its spectral action Lie groupoid $\mathord{\mathcal{AG}_{\mathit{sp}}}(\mathord{\mathrm{Dirac}}(M\rtimes H))$.

An algebraic iterative formula for the spin Kostka-Foulkes polynomial $K^-_{\xi\mu}(t)$ is given using vertex operator realizations of Hall-Littlewood symmetric functions and Schur's Q-functions. Based on the operational formula, more favorable properties are obtained parallel to the Kostka polynomial. In particular, we obtain some formulae for the number of (unshifted) marked tableaux. As an application, we confirmed a conjecture of Aokage on the expansion of the Schur $P$-function in terms of Schur functions. Tables of $K^-_{\xi\mu}(t)$ for $|\xi|\leq6$ are listed., Comment: 19 pages, 5 tables (correction of authors' names)

In this note we are dealing with a particular class of quadratic algebras -- the so-called quantum matrix algebras. The well-known examples are the algebras of quantized functions on classical Lie groups (the RTT algebras). We consider the problem of constructing some projectors on homogenous components of such algebras, which are analogs of the usual symmetrizers. The main objective of this note is to present a method, which hopefully enables one to construct symmetrizers on all homogenous components of the RTT algebras. We illustrate the construction by two low-dimensional examples. A way of extending this method onto other quantum matrix algebras is also discussed.

Given a Hopf algebra $H$, Brzezi\'nski and Militaru have shown that each braided commutative Yetter-Drinfeld $H$-module algebra $A$ gives rise to an associative $A$-bialgebroid structure on the smash product algebra $A \sharp H$. They also exhibited an antipode map making $A\sharp H$ the total algebra of a Lu's Hopf algebroid over $A$. However, the published proof that the antipode is an antihomomorphism covers only a special case. In this paper, a complete proof of the antihomomorphism property is exhibited. Moreover, a new generalized version of the construction is provided. Its input is a compatible pair $A$ and $A^{\mathrm{op}}$ of braided commutative Yetter-Drinfeld $H$-module algebras, and output is a symmetric Hopf algebroid $A\sharp H \cong H\sharp A^{\mathrm{op}}$ over $A$. This construction does not require that the antipode of $H$ is invertible., Comment: 34 pages

For the Klein group $K$ and a positive integr $k$, irreducible modules of the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{K}$ have been classified and constructed in \cite{JWa}. In this paper, we determine completely the fusion rules of $L_{\widehat{\frak{sl}_2}}(k,0)^{K}$.

Let $(X,r_X)$ and $(Y,r_Y)$ be finite nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation, and let $A_X = A(\textbf{k}, X, r_X)$ and $A_Y= A(\textbf{k}, Y, r_Y)$ be their quadratic Yang-Baxter algebras over a field $\textbf{k}.$ We find an explicit presentation of the Segre product $A_X\circ A_Y$ in terms of one-generators and quadratic relations. We introduce analogues of Segre maps in the class of Yang-Baxter algebras and find their images and their kernels. The results agree with their classical analogues in the commutative case., 21 pages. arXiv admin note: substantial text overlap with arXiv:2204.08850

We classify the finite dimensional semi-weight representations of the reduced stated skein algebras at odd roots of unity of connected marked surfaces which either have a boundary component with at least two boundary arcs or which do not have any unmarked boundary component. We deduce computations of the PI-degrees and Azumaya loci of unreduced stated skein algebras of essential surfaces having at most one boundary arc per boundary component and of the unrestricted quantum moduli algebras of lattice gauge field theory.