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Representations of the orbifold VOAS $L_{\hat{\frak{sl}_2}}(k,0)^{K}$ and the commutant VOAS $C_{{L_{\hat{\mathfrak{so}_m}}(1,0)}^{\otimes 3}}({L_{\hat{\mathfrak{so}_m}}(3,0)})$Sep 18 2019For the Klein group $K$, $k\in\mathbb{Z}_{\geqslant 1}$ and $m\in\mathbb{Z}_{\geqslant 4}$, we study the representations of the orbifold vertex operator algebra $L_{\hat{\mathfrak{sl}_2}}(k,0)^{K}$ and the commutant vertex operator algebra of $L_{\hat{\mathfrak{so}_m}}(3,0)$ ... More
Interplay between symmetries of quantum 6-j symbols and the eigenvalue hypothesisSep 17 2019The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of $U_q(sl_N)$ is uniquely determined by eigenvalues of the corresponding quantum $\cal{R}$-matrices. If this hypothesis turns out to be true, then it ... More
Group-theoretical property of non-degenerate fusion categories of FP-dimension $p^2q^3$ and $p^3q^3$Sep 16 2019In this paper, we show that non-degenerate fusion categories of FP-dimensions $p^2q^3d$ and $p^3q^3d$ are group-theoretical, where $p, q$ are odd primes, $d$ is a square-free integer such that $(pq,d) = 1$.
A Variation of the Goldman-Millson Theorem for Filtered $L_\infty$ AlgebrasSep 14 2019In this paper, we extend the Goldman-Millson Theorem for $L_\infty$ algebras. We consider $L$ and $\tilde{L}$, two $L_\infty$ algebras endowed with descending, bounded above and complete filtrations compatible with the $L_\infty$ structures and $U:L \rightarrow ... More
Equivariant quantum differential equation and $qKZ$ equations for a projective space: Stokes bases as exceptional collections, Stokes matrices as Gram matrices, and B-TheoremSep 14 2019In arXiv:1901.02990v1 the equivariant quantum differential equation ($qDE$) for a projective space was considered and a compatible system of difference $qKZ$ equations was introduced; the space of solutions to the joint system of the $qDE$ and $qKZ$ equations ... More
The Conformal Packing ProblemSep 12 2019We formulate the conformal packing problem and dual packing problem in analogy to similar problems for binary codes and lattices. We obtain explicit numerical upper bounds for the minimal dual conformal weight of a unitary strongly-rational vertex operator ... More
Non-commutative first order differential calculus over ginitely generated associative algebrasSep 11 2019In this review article the construction of first order coordinate differential calculi on finitely generated and finitely related associative algebras are considered and explicit construction of the bimodule of one form over such algebras is presented. ... More
Non-commutative first order differential calculus over finitely generated associative algebrasSep 11 2019Sep 12 2019In this review article the construction of first order coordinate differential calculi on finitely generated and finitely related associative algebras are considered and explicit construction of the bimodule of one form over such algebras is presented. ... More
Podleś Spheres for the Braided Quantum $\operatorname{SU}(2)$Sep 11 2019Starting with the braided quantum group $\operatorname{SU}_q(2)$ for a complex deformation parameter $q$ we perform the construction of the quotient $\operatorname{SU}_q(2)/\mathbb{T}$ which serves as a model of a quantum sphere. Then we follow the reasoning ... More
On factorization and vector bundles of conformal blocks from vertex algebrasSep 10 2019Modules over conformal vertex algebras give rise to sheaves of coinvariants and conformal blocks on moduli of stable pointed curves. We show that under certain natural hypotheses, these sheaves satisfy the factorization property, a reflection of their ... More
Heisenberg double and Drinfeld double of the quantum superplaneSep 10 2019We study infinite dimensional generalisations of the Heisenberg doubles of the Borel half of $U_q(sl(2))$ and of $U_q(osp(1|2))$ and find associated canonical elements which satisfy pentagon equation. The former reproduces the canonical element, expressed ... More
Noncommutative tensor triangular geometrySep 10 2019We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$\Delta$Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, ... More
On symmetries of peculiar modules; or, $δ$-graded link Floer homology is mutation invariantSep 10 2019We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant of 4-ended tangles which the author introduced in [arXiv:1712.05050]. In particular, we give an almost complete answer to the geography problem for components of peculiar ... More
The Chiral Anomaly of the Free Fermion in Functorial Field TheorySep 10 2019When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, ... More
Yang-Baxter endomorphismsSep 09 2019Every unitary solution of the Yang-Baxter equation (R-matrix) in dimension $d$ can be viewed as a unitary element of the Cuntz algebra ${\mathcal O}_d$ and as such defines an endomorphism of ${\mathcal O}_d$. These Yang-Baxter endomorphisms restrict and ... More
Boundary N=2 Theory, Floer Homologies, Affine Algebras, and the Verlinde FormulaSep 09 2019Generalizing our ideas in [arXiv:1006.3313], we explain how topologically-twisted N=2 gauge theory on a four-manifold with boundary, will allow us to furnish purely physical proofs of (i) the Atiyah-Floer conjecture, (ii) Munoz's theorem relating quantum ... More
Not even Khovanov homologySep 09 2019We construct a supercategory that can be seen as a skew version of (thickened) KLR algebras for the type $A$ quiver. We use our supercategory to construct homological invariants of tangles and show that for every link our invariant gives a link homology ... More
The Roger-Yang skein algebra and the decorated Teichmuller spaceSep 06 2019Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and ... More
Generalized topological state-sum constructions and their universalitySep 06 2019We formalize and generalize the concept of a topological state-sum construction using the language of tensor networks. We give examples for constructions that are possibly more general than all state-sum constructions in the literature that we are aware ... More
Fusion rules for permutation extensions of modular tensor categoriesSep 06 2019We give a construction and algorithmic description of the fusion ring of permutation extensions of an arbitrary modular tensor category using a combinatorial approach inspired by the physics of anyons and symmetry defects in bosonic topological phases ... More
Computing fusion rules for spherical G-extensions of fusion categoriesSep 06 2019A $G$-graded extension of a fusion category $\mathcal{C}$ yields a categorical action $\rho\colon \underline{G}\rightarrow \underline{\operatorname{Aut}}^{\mathrm{br}}_{\otimes}( Z(\mathcal C))$. If the extension admits a spherical structure, we provide ... More
Representations for three-point Lie algebras of genus zeroSep 06 2019In this paper, we study representations for three-point Lie algebras of genus zero based on the Cox-Jurisich's presentations. We construct two functors which transform simple restricted modules with nonzero levels over the standard affine algebras into ... More
On representations of semidirect products of a compact quantum group with a finite groupSep 05 2019We study unitary representations of semidirect products of a compact quantum group with a finite group. We give a classification of all irreducible unitary representations, a description of the conjugate representation of irreducible unitary representations ... More
State sums for some super quantum link invariantsSep 05 2019We present state sums for quantum link invariants arising from the representation theory of $U_q(\mathfrak{gl}_{N|M})$. We investigate the case of the $N$-th exterior power of the standard representation of $U_q(\mathfrak{gl}_{N|1})$ and explicit the ... More
Connectedness and irreducibility of compact quantum groupsSep 04 2019We show that a natural notion of irreducibility implies connectedness in the Compact Quantum Group setting. We also investigate the converse implication and show it is related to Kaplansky's conjectures on group algebras.
Thick morphisms of supermanifolds, quantum mechanics, and spinor representationAug 31 2019Sep 04 2019"Thick" or "microformal" morphisms of supermanifolds generalize ordinary maps. They were discovered as a tool for homotopy algebras. Namely, the corresponding pullbacks provide $L_{\infty}$-morphisms for $S_{\infty}$ or Batalin--Vilkovisky algebras. It ... More
Biquandle Brackets and KnotoidsAug 31 2019Biquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this paper we use biquandle ... More
On the Yang-Baxter equation and associated algebraic structuresAug 30 2019To study set-theoretic solutions of the Yang-Baxter equation several authors introduced algebraic structures. Rump and Ced\'o, Jespers and Okni\'nski introduced braces, Guarnieri and Vendramin introduced skew braces and Catino, Colazzo and Stefanelli ... More
Higher rank relations for the Askey-Wilson and $q$-Bannai-Ito algebraAug 30 2019The higher rank Askey-Wilson algebra was recently constructed in the $n$-fold tensor product of $U_q(\mathfrak{sl}_2)$. In this paper we prove a class of identities inside this algebra, which generalize the defining relations of the rank one Askey-Wilson ... More
Orthogeometries and AW*-algebrasAug 29 2019Based on results of Harding, Heunen, Lindenhovius and Navara, (2019), we give a connection between the category of AW*-algebras and their normal Jordan homomorphisms and a category COG of orthogemetries, which are structures that are somewhat similar ... More
Flagged Littlewood-Richardson tableaux and branching rule for classical groupsAug 29 2019We give a new formula for the branching rule from ${\rm GL}_n$ to ${\rm O}_n$ generalizing the Littlewood's restriction formula. The formula is given in terms of Littlewood-Richardson tableaux with certain flag conditions which vanish in a stable range. ... More
Supersymmetry and the Suzuki chainAug 29 2019We classify $N{=}1$ SVOAs with no free fermions and with bosonic subalgebra a simply connected WZW algebra which is not of type $\mathrm{E}$. The latter restriction makes the classification tractable; the former restriction implies that the $N{=}1$ automorphism ... More
The Batalin-Vilkovisky Formalism and the Determinant Line BundleAug 28 2019Given a smooth family of massless free fermions parametrized by a base manifold $B$, we show that the (mathematically rigorous) Batalin-Vilkovisky quantization of the observables of this family gives rise to the determinant line bundle for the corresponding ... More
Generalised Taft algebras and pairs in involutionAug 28 2019A class of finite-dimensional Hopf algebras which generalise the notion of Taft algebras is studied. We give necessary and sufficient conditions for these Hopf algebras to omit a pair in involution, that is, to not have a group-like and a character implementing ... More
On Indecomposable Vertex Algebras associated with Vertex AlgebroidsAug 27 2019Let $A$ be a finite dimensional unital commutative associative algebra and let $B$ be a finite dimensional vertex $A$-algebroid such that its Levi factor is isomorphic to $sl_2$. Under suitable conditions, we construct an indecomposable non-simple $\mathbb{N}$-graded ... More
Comparison of two notions of weak crossed productAug 27 2019We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in \cite{BB} with the notion of crossed product with a weak Hopf algebra introduced in~\cite{AG}
Legendrian skein algebras and Hall algebrasAug 27 2019We compare two associative algebras which encode the "quantum topology" of Legendrian curves in contact threefolds of product type $S\times\mathbb R$. The first is the skein algebra of graded Legendrian links and the second is the Hall algebra of the ... More
Deformation cohomology of Schur-Weyl categories. Free symmetric categoriesAug 24 2019The 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 ... More
Legendrian DGA representations and the colored Kauffman polynomialAug 23 2019For any Legendrian knot $K$ in standard contact $\R^3$ we relate counts of ungraded ($1$-graded) representations of the Legendrian contact homology DG-algebra $(\mathcal{A}(K),\partial)$ with the $n$-colored Kauffman polynomial. To do this, we introduce ... More
Non-extremal weight modules for quantized universal enveloping algebrasAug 23 2019For quantized universal enveloping algebras we construct weight modules by inducing representations of the centralizer of the Cartan subalgebra in the quantized universal enveloping algebra. The induced modules arising from finite-dimensional weight modules ... More
Tensor hierarchy algebras and extended geometry I: Construction of the algebraAug 23 2019Tensor hierarchy algebras constitute a class of non-contragredient Lie superalgebras, whose finite-dimensional members are the "Cartan-type" Lie superalgebras in Kac's classification. They have applications in mathematical physics, especially in extended ... More
A tale of two shuffle algebrasAug 22 2019As a quantum affinization, the quantum toroidal algebra is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations. In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations ... More
M. Kontsevich's graph complexes and universal structures on graded symplectic manifolds IAug 22 2019In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds. This universal deformation complex takes the form of a ... More
Groups of extended affine Lie typeAug 21 2019We construct certain Steinberg groups associated to extended affine Lie algebras and their root systems. Then by the integration methods of Kac and Peterson for integrable Lie algebras, we associate a group to every tame extended affine Lie algebra. Afterwards, ... More
Construction of Hopf algebroidsAug 21 2019For arbitrary algebras $L$, we construct Hopf algebroids $A_\sigma$ with base rings $L$ by means of $\sigma^{ab}_{cd}\in L$ satisfying suitable properties.
On Cores in Yetter-Drinfel'd Hopf AlgebrasAug 20 2019By constructing explicit examples, we show that the core of a group-like element in a cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra over the group ring of a finite abelian group is not always completely trivial.
Minimal modular extensions for super-Tannakian categoriesAug 20 2019In this paper, we continue with the ideas presented in [GVR17]. In this opportunity, we apply the fermionic action concept to classify in cohomology terms the minimal modular extensions of a super-Tannakian category. For this goal, we study some properties ... More
Level one Weyl modules for toroidal Lie algebrasAug 20 2019We identify level one global Weyl modules for toroidal Lie algebras with certain twists of modules constructed by Moody-Eswara Rao-Yokonuma via vertex operators for type ADE and by Iohara-Saito-Wakimoto and Eswara Rao for general type. The twist is given ... More
Classification of Rank 6 Modular Categories with Galois Group $\langle (012)(345)\rangle$Aug 20 2019Modular Tensor Categories (MTC's) arise in the study of certain condensed matter systems. There is an ongoing program to classify MTC's of low rank, up to modular data. We present an overview of the methods to classify modular tensor categories of low ... More
HOMFLYPT homology for links in handlebodies via type A Soergel bimodulesAug 19 2019We define a triply-graded invariant of links in a genus g handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-ball. Our main tools are the description of these links in terms of a subgroup of the classical braid group, and a family ... More
Finite spectral triples for the fuzzy torusAug 19 2019Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different ... More
Derived invariants of the fixed ring of enveloping algebras of semi-simple Lie algebrasAug 19 2019Let $\mathfrak{g}$ be semi-simple complex Lie algebra, and $W\subset Aut(U(\mathfrak{g}))$ be a finite subgroups of $\mathbb{C}$-algebra automorphisms. We show that the derived category of $U(\mathfrak{g})^W$-modules determines isomorphism classes of ... More
Derived invariants of the fixed ring of enveloping algebras of semi-simple Lie algebrasAug 19 2019Sep 08 2019Let $\mathfrak{g}$ be semi-simple complex Lie algebra, and $W\subset Aut(U(\mathfrak{g}))$ be a finite subgroups of $\mathbb{C}$-algebra automorphisms. We show that the derived category of $U(\mathfrak{g})^W$-modules determines isomorphism classes of ... More
Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate ringsAug 18 2019The elliptic algebras in the title are connected graded $\mathbb{C}$-algebras, denoted $Q_{n,k}(E,\tau)$, depending on a pair of relatively prime integers $n>k\ge 1$, an elliptic curve $E$, and a point $\tau\in E$. For fixed $n$ and $k$, they form a flat ... More
On the quantum affine vertex algebra associated with trigonometric $R$-matrixAug 18 2019We apply the theory of $\phi$-coordinated modules, developed by H.-S. Li, to the Etingof--Kazhdan quantum affine vertex algebra associated with the trigonometric $R$-matrix of type $A$. We prove, for a certain associate $\phi$ of the one-dimensional additive ... More
Demazure slices of type $A_{2l}^{(2)}$Aug 18 2019We consider a Demazure slice of type $A_{2l}^{(2)}$, that is an associated graded piece of an infinite-dimensional version of a Demazure module. We show that a global Weyl module of a hyperspecial current algebra of type $A_{2l}^{(2)}$ is filtered by ... More
Poisson vertex algebras in supersymmetric field theoriesAug 15 2019A large class of supersymmetric quantum field theories, including all theories with $\mathcal{N} = 2$ supersymmetry in three dimensions and theories with $\mathcal{N} = 2$ supersymmetry in four dimensions, possess topological-holomorphic sectors. We formulate ... More
Actions of quantum linear spaces on quantum algebrasAug 14 2019We study actions of bosonizations of quantum linear spaces on quantum algebras. Under mild conditions, we classify actions on quantum affine spaces and quantum matrix algebras. In the former case, it is shown that all actions of generalized Taft algebras ... More
The finiteness conjecture for skein modulesAug 14 2019Sep 04 2019We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of ... More
The finiteness conjecture for skein modulesAug 14 2019We give a new, algebraically computable, formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of ... More
Revisiting the Askey--Wilson algebra with the universal R-matrix of $U_q(sl(2))$Aug 13 2019A description of the embedding of the universal Askey--Wilson algebra, AW(3), in $U_q(sl_2)^{\otimes 3}$ is given in terms of the universal R-matrix of $U_q(sl_2)$. The generators of the centralizer of $U_q(sl_2)$ in its three-fold product are naturally ... More
Cyclotomic Expansion of Generalized Jones PolynomialsAug 12 2019In previous work of the first and third authors, we proposed a conjecture that the Kauffman bracket skein module of any knot in $S^3$ carries a natural action of the rank 1 double affine Hecke algebra $SH_{q,t_1, t_2}$ depending on 3 parameters $q, t_1, ... More
On $q$-deformed real numbersAug 12 2019We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a "$q$-analogue of a real." The construction is based on the notion of $q$-deformed rational number introduced in arXiv:1812.00170. Extending ... More
On $q$-deformed real numbersAug 12 2019Aug 20 2019We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a "$q$-analogue of a real." The construction is based on the notion of $q$-deformed rational number introduced in arXiv:1812.00170. Extending ... More
On the Universal ellipsitomic KZB connectionAug 11 2019We construct a twisted version of the genus one universal Knizhnik--Zamolodchikov--Bernard (KZB) connection introduced by Calaque--Enriquez--Etingof, that we call the ellipsitomic KZB connection. This is a flat connection on a principal bundle over the ... More
Q-systems and extensions of completely unitary vertex operator algebrasAug 09 2019We relate extensions of completely unitary VOAs and (commutative) Q-systems. As an application, we show that any unitary extension of a completely unitary VOA is completely unitary. We also study the $C^*$-tensor categories of the unitary bimodules of ... More
2-positive almost order zero maps and decomposition rankAug 09 2019We consider 2-positive almost order zero (disjointness preserving) maps on C*-algebras. Generalizing the argument of M. Choi for multiplicative domains, we give an internal characterization of almost order zero for 2-positive maps. It is also shown that ... More
Skew braces of size $pq$Aug 08 2019We construct all skew braces of size $pq$ ($p$, $q$ being primes) by using Byott's classification of Hopf-Galois extensions of the same order. For $p\neq 1 \pmod{q}$ there exists only $2$ skew braces which are the trivial ones. When $p= 1 \pmod{q}$, we ... More
Skew braces of size $pq$Aug 08 2019Sep 07 2019We construct all skew braces of size $pq$ ($p$, $q$ being primes) by using Byott's classification of Hopf-Galois extensions of the same order. For $p\neq 1 \pmod{q}$ there exists only one skew brace which is the trivial one. When $p= 1 \pmod{q}$, we have ... More
Branching rules for cell modules on a tower of the partition algebrasAug 08 2019Partition algebras with non-zero parameters are cellularly stratified and thus have the features of both cellular algebras and stratified algebras. Also, partition algebras form a tower of algebras. In this paper, we provide a diagrammatic approach to ... More
Kitaev's quantum double model as an error correcting codeAug 07 2019Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case ... More
A Note on the "Third Life of Quantum Logic"Aug 07 2019The purpose of this note is to discuss some of the questions raised by Dunn, J. Michael; Moss, Lawrence S.; Wang, Zhenghan in Editors' introduction: the third life of quantum logic: quantum logic inspired by quantum computing.
A Note on the "Third Life of Quantum Logic"Aug 07 2019Aug 26 2019The purpose of this note is to discuss some of the questions raised by Dunn, J. Michael; Moss, Lawrence S.; Wang, Zhenghan in Editors' introduction: the third life of quantum logic: quantum logic inspired by quantum computing.
The Elliptic Kashiwara-Vergne Lie algebra in low weightsAug 07 2019In this paper, we study the elliptic Kashiwara-Vergne Lie Algebra $\mathfrak{krv}$, which is a certain Lie subalgebra of the Lie algebra of derivations of the free Lie algebra in two generators. It has a natural bigrading, such that the Lie bracket is ... More
Generalized Shen-Larsson bifunctors and cohomologies of crossed homomorphismsAug 07 2019Aug 14 2019Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In ... More
Generalized Rudakov-Shen-Larsson bifunctors and cohomologies of crossed homomorphismsAug 07 2019Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In ... More
Integrals along bimonoid homomorphismsAug 05 2019In this paper, we introduce a notion of integral along bimonoid homomorphism. It simultaneously generalizes the notions of integrals and cointegrals of bimonoids. Moreover, we introduce a notion of normalized integral and generator integral. We give a ... More
Algebraic realization of noncommutative near-group fusion categoriesAug 05 2019Noncommutative near-group fusion categories were completely classified in the previous work of the first named author by using an operator algebraic method (and hence under the assumption of unitarity), and they were shown to be group theoretical though ... More
An algebra of distributions related to a star product with separation of variablesAug 04 2019Aug 07 2019Given a star product with separation of variables $\star$ on a pseudo-K\"ahler manifold $M$ and a point $x_0 \in M$, we construct an associative algebra of formal distributions supported at $x_0$. We use this algebra to express the formal oscillatory ... More
An algebra of distributions related to a star product with separation of variablesAug 04 2019Given a star product with separation of variables $\star$ on a pseudo-K\"ahler manifold $M$ and a point $x_0 \in M$, we construct an associative algebra of formal distributions supported at $x_0$. We use this algebra to express the formal oscillatory ... More
An algebra of distributions related to a star product with separation of variablesAug 04 2019Aug 09 2019Given a star product with separation of variables $\star$ on a pseudo-K\"ahler manifold $M$ and a point $x_0 \in M$, we construct an associative algebra of formal distributions supported at $x_0$. We use this algebra to express the formal oscillatory ... More
A note on cohomology for multiplier Hopf algebrasAug 02 2019In this note we discuss the possibility of constructing the cosimplicial complex for the multiplier Hopf algebras and extending the cyclicity operator to obtain the Hopf-cyclic cohomology for them. We show that the definition of modular pairs in involution ... More
The Betti side of the double shuffle theory. III. Double shuffle relations for associatorsAug 01 2019We give interpretations of the "module" counterpart of the harmonic coproduct and of its Betti version in terms of the geometry of the moduli spaces $\mathfrak M_{0,4}$ and $\mathfrak M_{0,5}$. Based on these interpretations, we show that any associator ... More
Invariants of long knotsJul 31 2019By using the notion of a rigid R-matrix in a monoidal category and the Reshetikhin--Turaev functor on the category of tangles, we review the definition of the associated invariant of long knots. In the framework of the monoidal categories of relations ... More
Mirror symmetry and line operatorsJul 31 2019We study half-BPS line operators in 3d N=4 gauge theories, focusing in particular on the algebras of local operators at their junctions. It is known that there are two basic types of such line operators, distinguished by the SUSY subalgebras that they ... More
Mirror symmetry and line operatorsJul 31 2019Sep 05 2019We study half-BPS line operators in 3d N=4 gauge theories, focusing in particular on the algebras of local operators at their junctions. It is known that there are two basic types of such line operators, distinguished by the SUSY subalgebras that they ... More
Braided Cartan Calculi and Submanifold AlgebrasJul 31 2019It is a necessity of derivation based Cartan calculi on noncommutative algebras to employ central bimodules. In analogy to differential geometry we construct a noncommutative Cartan calculus for any braided commutative algebra in the symmetric braided ... More
Quantum affine algebras and GrassmanniansJul 31 2019Aug 05 2019We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory $\mathcal{C}_{\ell}$ of $U_q(\widehat{\mathfrak{sl}_n})$-modules ... More
Quantum affine algebras and GrassmanniansJul 31 2019We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory $\mathcal{C}_{\ell}$ of $U_q(\widehat{\mathfrak{sl}_n})$-modules ... More
Homotopy Rota-Baxter operators, homotopy $\mathcal{O}$-operators and homotopy post-Lie algebrasJul 31 2019Rota-Baxter operators, $\mathcal{O}$-operators on Lie algebras and their interconnected pre-Lie and post-Lie algebras are important algebraic structures with applications in mathematical physics. This paper introduces the notions of a homotopy Rota-Baxter ... More
Quantum principal bundles on projective basesJul 30 2019The purpose of this paper is to propose a sheaf theoretic approach to the theory of quantum principal bundles over non affine bases. We study the quantization of principal bundles G -> G/P, where G is a complex simple group and P a parabolic subgroup. ... More
String-net models for non-spherical pivotal fusion categoriesJul 29 2019A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this ... More
Realizing IR theories by projections in the UVJul 29 2019We study bulk RG flows in the context of TQFTs and show how IR theories can be entirely represented within the respective UV theories by means of codimension-one projection defects. What is more, RG flows of the bulk theory can be described in terms of ... More
Incarnations of XXX $\widehat{\frak{sl}_N}$ Bethe ansatz equations and integrable hierarchiesJul 29 2019We consider the space of solutions of the Bethe ansatz equations of the $\widehat{\frak{sl}_N}$ XXX quantum integrable model, associated with the trivial representation of $\widehat{\frak{sl}_N}$. We construct a family of commuting flows on this space ... More
Invariants of 4-manifolds from Khovanov-Rozansky link homologyJul 29 2019We use Khovanov-Rozansky gl(N) link homology to define pivotal 4-categories, which give rise to invariants of oriented smooth 4-manifolds. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies ... More
Invariants of 4-manifolds from Khovanov-Rozansky link homologyJul 29 2019Aug 14 2019We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, ... More
A remark on matrix product operator algebras, anyons and subfactorsJul 29 2019We show that the mathematical structures in a recent work of Bultinck-Mari\"ena-Williamson-\c Sahino\u glu-Haegemana-Verstraete are the same as those of flat symmetric bi-unitary connections and the tube algebra in subfactor theory. More specifically, ... More
The matched product of set-theoretical solutions associated with shelvesJul 27 2019We investigate the matched product of solutions associated with right and left shelves. First, we prove that the requirements to provide the matched product of solutions that come from shelves can be simplified. Then we give conditions for left non-degeneracy ... More
The Elliptic Tail KernelJul 27 2019We introduce and study a new family of $q$-translation-invariant determinantal point processes on the two-sided $q$-lattice. We prove that these processes are limits of the $q$-$zw$ measures, which arise in the $q$-deformation of harmonic analysis on ... More
On Indecomposable Non-Simple $\mathbb{N}$-graded Vertex AlgebrasJul 26 2019In this paper, we study an impact of Leibniz algebras on the algebraic structure of $\mathbb{N}$-graded vertex algebras. We provide easy ways to characterize indecomposable non-simple $\mathbb{N}$-graded vertex algebras $\oplus_{n=0}^{\infty}V_{(n)}$ ... More