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A noncommutative algebra of the complex $q$-twistors and their differentials is considered on the basis of the quantum $GL_q (4)\times SL_q (2)$ group. Real and pseudoreal $q$-twistors are discussed too. |
In this paper, we investigate the differential smoothness of skew PBW extensions over commutative polynomial rings on one and two indeterminates. |
Let $\mathcal{F}_{25}$ be the family of irreducible highest weight modules for the Virasoro algebra of central charge $25$ which are not isomorphic to Verma modules. Let $L(25,0)$ be the Virasoro vertex operator algebra of central charge 25. |
We consider two new families of quantum vertex algebras which are associated with the type $A$ trigonometric $R$-matrix and elliptic $R$-matrix of the eight-vertex model. We show that their $\phi$-coordinated representation theory is governed by the so-called FRT-operator, $h$-adically restricted operator satisfying t... |
In Mulevičius-Runkel, <a href="https://arxiv.org/abs/2002.00663" data-arxiv-id="2002.00663" class="link-https">arXiv:2002.00663</a>, it was shown how a so-called orbifold datum $\mathbb{A}$ in a given modular fusion category (MFC) $\mathcal{C}$ produces a new MFC $\mathcal{C}_{\mathbb{A}}$. Examples of these associate... |
Using the second Drinfeld formulation of the quantized universal enveloping algebra $U_q(\widehat{sl_2})$ we introduce a family of its Heisenberg-type elements which are endowed with a deformed commutator and satisfy properties similar to generators of a Heisenberg subalgebra. Explicit expressions for new family of ge... |
In this paper, we completely determine the group of algebra automorphisms for the two-parameter Hopf algebra ${\check U}_{r,s}^{\geq 0}({\mathfrak sl_{3}})$. As a result, the group of Hopf algebra automorphisms is determined for $\V$ as well. |
We prove that braided fusion categories of Frobenius-Perron $p^mq^nd$ or $p^2q^2r^2$ are weakly group-theoretical, where $p,q,r$ are distinct prime numbers, $d$ is a square-free natural number such that $(pq,d)=1$. As an application, we obtain that weakly integral braided fusion categories of Frobenius-Perron dimensio... |
We discuss two-parameter deformations of an universal enveloping algebra $U(g[u])$ of a polynomial loop algebra $g[u]$, where $g$ is a finite-dimensional complex simple Lie algebra (or superalgebra). These deformations are Hopf algebras. |
This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello-Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. |
We study modules over the algebroid stack $\W[\stx]$ of deformation quantization on a complex symplectic manifold $\stx$ and recall some results: construction of an algebra for $\star$-products, existence of (twisted) simple modules along smooth Lagrangian submanifolds, perversity of the complex of solutions for regula... |
Theory of motivic superpolynomials is developed, including its extension to algebraic links colored by rows, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis, and recurrences for iterated torus links. The key theme is the conjectural coincide... |
Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G = GL(4,R) is e... |
Applying the Fedosov connections constructed in our previous work, we find a (dense) subsheaf of smooth functions on a Kähler manifold $X$ which admits a non-formal deformation quantization. When $X$ is prequantizable and the Fedosov connection satisfies an integrality condition, we prove that this subsheaf of functio... |
We show that the combinatorial matter of graph coloring is, in fact, quantum in the sense of satisfying the sum over all the possible intermediate state properties of a path integral. In our case, the topological field theory (TFT) with defects gives meaning to it. |
We describe a Gauss decomposition for the Yangian Y(gl_{m|n}) of the general linear Lie superalgebra. This gives a connection between this Yangian and the Yangian of the classical Lie superalgebra Y(A(m-1,n-1)) (with m and n not equal) defined and studied in papers by Stukopin, and suggests natural definitions for the... |
Motivated by topological quantum field theory, we investigate the geometric aspects of unitary 2-representations of finite groups on 2-Hilbert spaces, and their 2-characters. We show how the basic ideas of geometric quantization are `categorified' in this context: just as representations of groups correspond to eq... |
We consider actions of the current Lie algebras $\mathfrak{gl}_{n}[t]$ and $\mathfrak{gl}_{k}[t]$ on the space of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1}\dots z_{k})$ and $\bar{\alpha}=(\alpha_{1}\dots \alpha_{n})$, respectively. |
We give a thermodynamical interpretation of the moment map for toric varieties. |
In this paper we study conjugacy classes for pivotal fusion categories. In particular we prove a Burnside type formula for the structure constants concerning the product of two conjugacy class sums of a such fusion category. |
To some braiding R of Hecke type (a Hecke symmetry) we put into correspondence an associative algebra called the modified Reflection Equation Algebra (mREA). We construct a series of matrices L_(m), m=1,2,... with entries belonging to mREA such that each of them satisfies a version of the Cayley-Hamilton identity with... |
We define and compute the continuous orbifold partition function and a generating function for all $n$-point correlation functions for the rank two free fermion vertex operator superalgebra on a genus two Riemann surface formed by self-sewing a torus. The partition function is proportional to an infinite dimensional d... |
The dually conjugate Hopf algebras $Fun_{p,q}(R)$ and $U_{p,q}(R)$ associated with the two-parametric $(p,q)$-Alexander-Conway solution $(R)$ of the Yang-Baxter equation are studied. Using the Hopf duality construction, the full Hopf structure of the quasitriangular enveloping algebra $U_{p,q}(R)$ is extracted. |
We study the deformation theory aspects of Matricial Factorizations, possibly with an orthogonal or symplectic structure. We discuss and extend the Knörrer and Hori-Walcher periodicity theorems |
In this work, we focus on the set-theoretical solutions of the Yang-Baxter equation which are of finite order and not necessarily bijective. We use the matched product of solutions as a unifying tool for treating these solutions of finite order, that also include involutive and idempotent solutions. |
We classify Drinfeld twists for the quantum Borel subalgebra u_q(b) in the Frobenius-Lusztig kernel u_q(g), where g is a simple Lie algebra over C and q an odd root of unity. More specifically, we show that alternating forms on the character group of the group of grouplikes for u_q(b) generate all twists for u_q(b), u... |
Representations of the quantum superalgebra U_q[osp(1/2)] and their relations to the basic hypergeometric functions are investigated. We first establish Clebsch-Gordan decomposition for the superalgebra U_q[osp(1/2)] in which the representations having no classical counterparts are incorporated. |
We demonstrate our recent general results on the Casimir construction and moduli space of all bicovariant calculi by means of some detailed examples, including finite-difference and 2-jet cacluli on $\R^n$ and full details of the Casimir construction of the 4D calculus of $SU_q(2)$. We likewise demonstrate our previou... |
The objects of study in this paper are Hopf algebras $H$ which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra by a finite dimensional Hopf algebra. Basic structural and homological properties are recalled and classes of examples are listed. |
We explain how existing results (such as categorical sl(n) actions, associated braid group actions and infinite twists) can be used to define a triply graded link invariant which categorifies the HOMFLY polynomial of links coloured by arbitrary partitions. The construction uses a categorified HOMFLY clasp defined via ... |
In this dissertation the notion of deformation quantization of principal fibre bundles is established and investigated in order to find a geometric formulation of classical gauge theories on noncommutative space-times. As a generalization, the notion of deformation quantization of surjective submersions is also discus... |
Given a finite set of roots of unity, we show that all power sums are non-negative integers iff the set forms a group under multiplication. The main argument is purely combinatorial and states that for an arbitrary finite set system the non-negativity of certain alternating sums is equivalent to the set system being a... |
The goal of the present paper is to classify an interesting class of elementary quasi-Hopf algebras, or equivalently, finite-dimensional pointed Majid algebras. By a Tannaka-Krein type duality, this determines a big class of pointed finite tensor categories. |
Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object Int H is invertible. |
Graph cocycles for star-products are investigated from the combinatorial point of view, using Connes-Kreimer renormalization techniques. <br>The Hochschild complex, controlling the deformation theory of associative algebras, is the ``Kontsevich representation'' of a DGLA of graphs coming from a pre-Lie algebra... |
We develop a theory of descent and forms of tensor categories over arbitrary fields. We describe the general scheme of classification of such forms using algebraic and homotopical language, and give examples of explicit classification of forms. |
We prove that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. This result may be used to test the local geometric Langland... |
Looking for the universal covering of the smooth non-commutative torus leads to a curve of associative multiplications on the space $\Cal O_M'(\Bbb R^{2n})\cong \Cal O_C(\Bbb R^{2n})$ of Laurent Schwartz which is smooth in the deformation parameter $\hbar$. The Taylor expansion in $\hbar$ leads to the formal Moyal... |
We use an explicit isomorphism from the representation ring of the quantum group U_q(sl(N)) to the Homfly skein of the annulus, to determine an element of the skein which is the image of the mth Adams operator, \psi_m, on the fundamental representation, c_1. This element is a linear combination of m very simple m-stri... |
Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum Yang-Baxter equation, or more properly the braid equation, is investigated in this setting. |
The simple finite-dimensional modules over the Drinfeld double of the bosonization of the Nichols algebra $\mathfrak{ufo}(7)$ are classified. |
We introduce algebraic operads $\mathcal{P}^{\text{ch}N_W=N}$ and $\mathcal{P}^{\text{ch}N_K=N}$ encoding the structures of $N_W=N$ and $N_K=N$ SUSY vertex algebras, and study the corresponding cohomology theory. Our operad is a SUSY analogue of the operad $P^{\text{ch}}$ introduced by Bakalov, De Sole, Heluani and Ka... |
We introduce a new method to construct 4-dimensional Artin-Schelter regular algebras as normal extensions of (not necessarily noetherian) 3-dimensional ones. The method produces large classes of new 4-dimensional Artin-Schelter regular algebras. |
In this paper, irreducible modules of the diagonal coset vertex operator algebra $C(L_{\mathfrak{g}}(k+l,0),L_{\mathfrak{g}}(k,0)\otimes L_{\mathfrak{g}}(l,0))$ are classified under the assumption that $C(L_{\mathfrak{g}}(k+l,0),L_{\mathfrak{g}}(k,0)\otimes L_{\mathfrak{g}}(l,0))$ is rational, $C_2$-cofinite and certai... |
We introduce theta-functions of VOA-modules and show that the space spanned by them has a modular invariance property. |
In this paper we give a topological interpretation and diagrammatic calculus for the rank $(n-2)$ Askey-Wilson algebra by proving there is an explicit isomorphism with the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. To do this we consider the Askey-Wilson algebra in the braided tensor product of $n... |
The purpose of this paper is to provide new constructions of Hom-associative algebras using Hom-analogues of certain operators called twistors and pseudotwistors, by deforming a given Hom-associative multiplication into a new Hom-associative multiplication. As examples, we introduce Hom-analogues of the twisted tensor... |
Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The obtained combinatorial structure fits perfectly into an existing framework of generalized root systems associated to a family of Cartan matrices,... |
Definition of Dirac operators on the quantum group $SU_{q}(2)$ and the quantum sphere $S^{2}_{q \mu}$ are discussed. In both cases similar $SU_{q}(2)$-invariant form is obtained. |
The paper studies the Karoubi envelope of a one-dimensional topological theory with defects and inner endpoints, defined over a field. It turns out that the Karoubi envelope is determined by a symmetric Frobenius algebra K associated to the theory. |
We analyse the possibility of defining complex valued Knot invariants associated with infinite dimensional unitary representations of $SL(2,R)$ and the Lorentz Group taking as starting point the Kontsevich Integral and the notion of infinitesimal character. This yields a family of knot invariants whose target space is... |
In this paper, we study a certain deformation $D$ of the Virasoro algebra that was introduced and called $q$-Virasoro algebra by Nigro,in the context of vertex algebras. Among the main results, we prove that for any complex number $\ell$, the category of restricted $D$-modules of level $\ell$ is canonically isomorphic... |
We present an explicit form of braided symmetries of the quantum spheres, by introducing a braided quantum Hopf algebra $\cU_{q, \phi}$ and demonstrating that they are braided Hopf modules over this braided Hopf algebra. To obtain this result, we systematically develop braided versions of structures like pairing betwe... |
We study the universal integrable modules W_q(m) of level zero for quantum affine sl_2 and a family of maximal finite--dimensional quotients of these modules. We show that these all have dimension 2^m. |
Given a premodular category $\mathcal{C}$, we show that its $R$-symbol can be recovered from its $T$-matrice, fusion coefficients and some 2nd generalized Frobenius-Schur indicators. In particular, if $\mathcal{C}$ is modular, its $R$-symbols for a certain gauge choice are completely determined by its modular data. |
I propose a definition of left/right connection along a strong homotopy Lie-Rinehart algebra. This allows me to generalize simultaneously representations up to homotopy of Lie algebroids and actions of strong homotopy Lie algebras on graded manifolds. |
We study classical twists of Lie bialgebra structures on the polynomial current algebra $\mathfrak{g}[u]$, where $\mathfrak{g}$ is a simple complex finite-dimensional Lie algebra. We focus on the structures induced by the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. |
We study Cayley graphs of abelian groups from the perspective of quantum symmetries. We develop a general strategy for determining the quantum automorphism groups of such graphs. |
We classify semisimple rigid monoidal categories with two isomorphism classes of simple objects over the field of complex numbers. In the appendix written by <a href="http://P.Etingof" rel="external noopener nofollow" class="link-external link-http">this http URL</a> it is proved that the number of semisimple Hopf alg... |
We prove that the action of the Grothendieck-Teichmüller group on the genus completed properad of (homotopy) Lie bialgebras commutes with the reversing directions involution of the latter. We also prove that every universal quantization of Lie bialgebras is homotopy equivalent to the one which commutes with the dualit... |
We show that the bases of irreducible integrable highest weight module of a non-symmetric Kac-Moody algebra, which is associated to a quiver with a nontrivial admissible automorphism, can be naturally identified with a set of certain invariant Langrangian irreducible subvarieties of certain varieties associated with th... |
We review briefly the existing vertex-operator-algebraic constructions of various tensor category structures on module categories for affine Lie algebras. We discuss the results first conjectured in the work of Moore and Seiberg that led us to the construction of the modular tensor category structure in the positive i... |
The Hamiltonian of the trigonometric Calogero-Sutherland model coincides with some limit of the Hamiltonian of the elliptic Calogero-Moser model. In other words the elliptic Hamiltonian is a perturbed operator of the trigonometric one. |
We show that the quantum algebra $U_q(sl_2)$ has a presentation with generators $x,x^{-1},y,z$ and relations <br>$x x^{-1}=1$, $x^{-1} x=1$, <br>$\frac{qxy-q^{-1}yx}{q-q^{-1}}=1$, <br>$\frac{qyz-q^{-1}zy}{q-q^{-1}}=1$, <br>$\frac{qzx-q^{-1}xz}{q-q^{-1}}=1$. <br>We call this the equitable presentation. |
The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of the Yangian or the quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. |
The most common geometric interpretation of the Yang-Baxter equation is by braids, knots and relevant Reidemeister moves. So far, cubes were used for connections with the third Reidemeister move only. |
In this paper, we develop the PBW theory for the bosonic extension $\qbA{\g}$ of a quantum group $\mathcal{U}_q(\g)$ of \emph{any} finite type. When $\g$ belongs to the class of \emph{simply-laced type}, the algebra $\qbA{\g}$ arises from the quantum Grothendieck ring of the Hernandez-Leclerc category over quantum aff... |
Belov-Kanel and Kontsevich conjectured that the group of automorphisms of the n'th Weyl algebra and the group of polynomial symplectomorphisms of C^2 are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity. |
Grothendieck-Verdier duality is a powerful and ubiquitous structure on monoidal categories, which generalises the notion of rigidity. Hopf algebroids are a generalisation of Hopf algebras, to a non-commutative base ring. |
The modified q-Bessel functions and the q-Bessel-Macdonald functions of the first and second kind are introduced. Their definition is based on representations as power series. |
We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. |
The q-monopole bundle introduced previously is extended to a general construction for quantum group bundles with non-universal differential calculi. We show that the theory applies to several other classes of bundles as well, including bicrossproduct quantum groups, the quantum double and combinatorial bundles associa... |
We give a direct elementary proof of the existence of traces for arbitrary star products on a symplectic manifold. We follow the approach we used in \cite{refs:GuttRaw}, solving first the local problem. |
Within the framework of complex supergeometry and motivated by two-dimensional genus-zero holomorphic N=1 superconformal field theory, we define the moduli space of N=1 genus-zero super-Riemann surfaces with oriented and ordered half-infinite tubes, modulo superconformal equivalence. We define a sewing operation on th... |
We prove that the trace of the $n$th power of the antipode of a Hopf algebra with the Chevalley property is a gauge invariant, for each integer $n$. As a consequence, the order of the antipode, and its square, are invariant under Drinfeld twists. |
A simple proof of the higher Capelli identities is given. |
We study in this article the concepts of algebra up to homotopy for a structure defined by two operations $ \pt $ and $[, ]$. Having determined the structure of $ G_\infty $ algebras and $ P_\infty $ algebras, we generalize this construction and we define a structure of $ (a, b)$-algebra up to homotopy. |
We describe points on Nakajima varieties and Weyl group actions on them via complexes of semisimple and projective modules over certain finite-dimensional algebras. |
We prove that the family of non-linear $W$-algebras $SW(3/2,2)$ which are extensions of the $N=1$ superconformal algebra by a primary supercurrent of conformal weight $2$ can be realized as a quantum Hamiltonian reduction of the Lie superalgebra $osp(3|2)$. In consequence we obtain an explicit free field realization o... |
By applying Miyamoto's $\mathbb{Z}_{3}$-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices and their automorphisms of order 3, we construct holomorphic vertex operator algebras of central charge 24 whose Lie algebras of the weight one spaces are of types $A_{2,3}^6$, $E_{6... |
We study finite-dimensional representations of quantum affine algebras of type $B_N$. We show that a module is tame if and only if it is thin. |
We construct Drinfeld realisations for the quantum affine superalgebras associated with the osp(1|2n)^{(1)}, Sl(1|2n)^{(2)} and osp(2|2n)^{(2)} series of affine Lie superalgebras. |
We study the conformal vertex algebras which naturally arise in relation to the Nakajima-Yoshioka blow-up equations. |
We consider the q-hypergeometric equation with q^{N}=1 and $\alpha, \beta, \gamma \in {\Bbb Z}$. We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. |
The quantum Yang-Baxter equation is a braiding condition on vector spaces which is of high relevance in several fields of mathematics, such as knot theory and quantum group theory. Their combinatorial counterpart are set-theoretic solutions to the Yang--Baxter equation, whose investigation is strongly driven by the st... |
We find a group-theoretical condition under which a twist of a group algebra, in Movshev's way, admits an integral Hopf order. Let $K$ be a (large enough) number field with ring of integers $R$. |
Solutions of the qKZ equation associated with the quantum affine algebra $U_q(\hat{sl}_2)$ and its two dimensional evaluation representation are studied. The integral formulae derived from the free field realization of intertwining operators of $q$-Wakimoto modules are shown to coincide with those of Tarasov and Varch... |
The grouplike elements of a coalgebra over a field are known to be linearly independent over said field. Here we prove three variants of this result. |
The present paper is a continuation of [5], where Lie bialgebra structures on g[u] were studied. These structures fall into different classes labelled by the vertices of the extended Dynkin diagram of g. |
We use categories of representations of finite dimensional quantum groupoids (weak Hopf algebras) to construct ribbon and modular categories that give rise to invariants of knots and 3-manifolds. |
In this paper, we first classify all irreducible modules of the vertex algebra $V_L^+$ when $L$ is a negative definite even lattice of arbitrary rank. In particular, we show that any irreducible $V_L^+$-module is isomorphic to a submodule of an irreducible twisted $V_L$-module. |
The minimal model $\mathfrak{osp}(1|2)$ vertex operator superalgebras are the simple quotients of affine vertex operator superalgebras constructed from the affine Lie super algebra $\widehat{\mathfrak{osp}}(1|2)$ at certain rational values of the level $k$. We classify all isomorphism classes of $\mathbb{Z}_2$-graded ... |
We establish an explicit formula for the n-point correlation functions in the sense of Bloch-Okounkov for the irreducible representations of $\hat{gl}_\infty$ and $W_{1+\infty}$ of arbitrary positive integral levels. |
Hopf algebra structure on the differential algebra of the extended $q$-plane is defined. An algebra of forms which is obtained from the generators of the extended $q$-plane is introduced and its Hopf algebra structure is given. |
We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy alg... |
Let $H$ be a weak Hopf algebra with a bijective antipode and $A$ an $H$-comodule Poisson algebra. In this paper, we mainly generalize the fundamental theorem of Poisson Hopf modules to the case of weak Hopf algebras. |
This article serves a two-fold purpose. On the one hand, it is a survey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of graded Hopf algebras. |
Let ${\mathfrak g}$ be a complex semisimple Lie algebra, and $Y_h({\mathfrak g})$, $U_q(L{\mathfrak g})$ the corresponding Yangian and quantum loop algebra, with deformation parameters related by $q=\exp(\pi i h)$. When $h$ is not a rational number, we constructed in <a href="https://arxiv.org/abs/1310.7318" data-arxi... |
Scaling limits when q tends to 1 of the elliptic vertex algebras A_qp(sl(N)) are defined for any N, extending the previously known case of N=2. They realise deformed, centrally extended double Yangian structures DY_r(sl(N)). |
The classical algebra $\Lambda$ of symmetric functions has a remarkable deformation $\Lambda^*$, which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by shifted Schur functions $s^*_\mu$, where $\mu$ ranges over the set of all partitions. |
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