Mathematics - Quantum Algebra Publications (50)

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Mathematics - Quantum Algebra Publications

We define a family of quantum invariants of closed oriented $3$-manifolds using spherical multi-fusion categories. The state sum nature of this invariant leads directly to $(2+1)$-dimensional topological quantum field theories ($\text{TQFT}$s), which generalize the Turaev-Viro-Barrett-Westbury ($\text{TVBW}$) $\text{TQFT}$s from spherical fusion categories. The invariant is given as a state sum over labeled triangulations, which is mostly parallel to, but richer than the $\text{TVBW}$ approach in that here the labels live not only on $1$-simplices but also on $0$-simplices. Read More


We give explicit descriptions of the adjoint group of the Coxeter quandle associated with a Coxeter group $W$. The adjoint group turns out to be an intermediate group between $W$ and the corresponding Artin group, and fits into a central extension of $W$ by a finitely generated free abelian group. We construct $2$-cocycles corresponding to the central extension. Read More


The goal of this note is to describe a class of formal deformations of a symplectic manifold $M$ in the case when the base ring of the deformation problem involves parameters of non-positive degrees. The interesting feature of such deformations is that these are deformations "in $A_{\infty}$-direction" and, in general, their description involves all cohomology classes of $M$ of degrees $\ge 2$. Read More


We describe a normal surface algorithm that decides whether a knot satisfies the Strong Slope Conjecture. We also establish a relation between the Jones period of a knot and the number of sheets of the surfaces that satisfy the Strong Slope Conjecture (Jones surfaces). Read More


We prove a long-standing conjecture by B. Feigin et al. that certain screening operators on a conformal field theory obey the algebra relations of the Borel part of a quantum group (and more generally a diagonal Nichols algebra). Read More


We construct explicit families of right coideal subalgebras of quantum groups, where all irreducible representations are one-dimensional and which are maximal with this property. We have previously called such a right coideal subalgebra a Borel subalgebra. Conversely we can prove that any tringular Borel subalgebra fulfilling a certain non-degeneracy property is of the form we construct; this classification requires a key assertion about Weyl groups which we could only prove in type $A_n$. Read More


We complete the classification of conformal embeddings of a maximally reductive subalgebra $\mathfrak k$ into a simple Lie algebra $\mathfrak g$ at non-integrable non-critical levels $k$ by dealing with the case when $\mathfrak k$ has rank less than that of $\mathfrak g$. We describe some remarkable instances of decomposition of the vertex algebra $V_{k}(\mathfrak g)$ as a module for the vertex subalgebra generated by $\mathfrak k$. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. Read More


We introduce intertwining operators among twisted modules or twisted intertwining operators associated to not-necessarily-commutative automorphisms of a vertex operator algebra. Let $V$ be a vertex operator algebra and let $g_{1}$, $g_{2}$ and $g_{3}$ be automorphisms of $V$. We prove that for $g_{1}$-, $g_{2}$- and $g_{3}$-twisted $V$-modules $W_{1}$, $W_{2}$ and $W_{3}$, respectively, such that the vertex operator map for $W_{3}$ is injective, if there exists a twisted intertwining operator of type ${W_{3}\choose W_{1}W_{2}}$ such that the images of its component operators span $W_{3}$, then $g_{3}=g_{1}g_{2}$. Read More


In this letter we prove that the unrolled small quantum group, appearing in quantum topology, is a Hopf subalgebra of Lusztig's quantum group of divided powers. We do so by writing down non-obvious primitive elements with the right adjoint action. We also construct a new larger Hopf algebra that contains the full unrolled quantum group. Read More


In this article we construct three explicit natural subgroups of the Brauer-Picard group of the category of representations of a finite-dimensional Hopf algebra. In examples the Brauer Picard group decomposes into an ordered product of these subgroups, somewhat similar to a Bruhat decomposition. Our construction returns for any Hopf algebra three types of braided autoequivalences and correspondingly three families of invertible bimodule categories. Read More


The famous pentagon identity for quantum dilogarithms has a generalization for every Dynkin quiver, due to Reineke. A more advanced generalization is associated with a pair of alternating Dynkin quivers, due to Keller. The description and proof of Keller's identities involves cluster algebras and cluster categories, and the statement of the identity is implicit. Read More


In this note we compute several invariants (e.g. algebraic K-theory, cyclic homology and topological Hochschild homology) of the noncommutative projective schemes associated to Koszul algebras of finite global dimension. Read More


Gasper & Rahman's multivariate $q$-Racah polynomials are shown to arise as connection coefficients between families of multivariable $q$-Hahn or $q$-Jacobi polynomials. The families of $q$-Hahn polynomials form bases for irreducible components of multifold tensor product representations of $su_{q}(1,1)$ and can be considered as generalized Clebsch-Gordan coefficients. This gives an interpretation of the multivariate $q$-Racah polynomials in terms of $3nj$ symbols. Read More


We study pre-Lie pairs, by which we mean a pair of a homotopy Lie algebra and a pre-Lie algebra with a compatible pre-Lie action. Such pairs provide a wealth of algebraic structure, which in particular can be used to analyze the homotopy Lie part of the pair. Our main application and the main motivation for this development are the dg Lie algebras of hairy graphs computing the rational homotopy groups of the mapping spaces of the $E_n$ operads. Read More


We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the $\mathfrak{sl}_N$ link homology categorifying the $\mathfrak{sl}_N$ link polynomial. We also provide connections to the equivariant cohomology rings of partial flag manifolds. Read More


In this paper, we prove that for any odd prime larger than 3, the modular group representation associated to the SO$(p)_2$-TQFT can be defined over the ring of integers of a cyclotomic field. We will provide explicit integral bases. In the last section, we will relate these representations to the Weil representations over finite fields. Read More


We consider various $A_{\infty}$-algebras of differential (super)forms, which are related to gauge theories and demonstrate explicitly how certain reformulations of gauge theories lead to the transfer of the corresponding $A_{\infty}$-structures. In $N=2$ 3D space we construct the homotopic counterpart of the de Rham complex, which is related to the superfield formulation of the $N=2$ Chern-Simons theory. Read More


By a generalized Yangian we mean a Yangian-like algebra of one of two classes. One of these classes consists of the so-called braided Yangians, introduced in our previous paper. The braided Yangians are in a sense similar to the reflection equation algebra. Read More


We give an introduction to vertex algebras using elementary forward difference methods originally due to Isaac Newton. Read More


We study actions of discrete groups on 2-categories. The motivating examples are actions on the 2-category of representations of finite tensor categories and their relation with the extension theory of tensor categories by groups. Associated to a group action on a 2-category, we construct the 2-category of equivariant objects. Read More


We propose that Borcherds' Fake Monster Lie algebra is a broken symmetry of heterotic string theory compactified on $T^7 \times T^2$. As evidence, we study the fully flavored counting function for BPS instantons contributing to a certain loop amplitude. The result is controlled by $\Phi_{12}$, an automorphic form for $O(2, 26, \mathbb{Z})$. Read More


The goal of the present paper is to introduce a smaller, but equivalent version of the Deligne-Hinich-Getzler $\infty$-groupoid associated to a homotopy Lie algebra. In the case of differential graded Lie algebras, we represent it by a universal cosimplicial object. Read More


We compute all Nichols algebras of rigid vector spaces of dimension 2 that admit a non-trivial quadratic relation. Read More


In this note we recall the construction of two chain level lifts of the gravity operad, one due to Getzler-Kapranov and one due to Westerland. We prove that these two operads are formal and that they indeed have isomorphic homology. Read More


We introduce a Dirac operator $D$ for the quantum group $U_q(\mathfrak{sl}_2)$, as an element of the tensor product of $U_q(\mathfrak{sl}_2)$ with the Clifford algebra on two generators. We study the properties of $D$, including an analogue of Vogan's conjecture. We compute the cohomology of $D$ acting on various $U_q(\mathfrak{sl}_2)$-modules. Read More


We introduce the notions of categorical integrals and categorical cointegrals of a finite tensor category $\mathcal{C}$ by using a certain adjunction between $\mathcal{C}$ and its Drinfeld center $\mathcal{Z}(\mathcal{C})$. These notions can be identified with integrals and cointegrals of a finite-dimensional Hopf algebra $H$ if $\mathcal{C}$ is the representation category of $H$. We generalize basic results on integrals and cointegrals of a finite-dimensional Hopf algebra (such as the existence, the uniqueness, and the Maschke theorem) to finite tensor categories. Read More


We give a Pieri-Chevalley type formula for the equivariant (with respect to an Iwahori subgroup) $K$-theory of semi-infinite flag manifolds, which can be regarded as a semi-infinite analog of a result for the torus equivariant $K$-theory of ordinary finite-dimensional flag manifolds, due to Pittie-Ram. Our formula describes the product in the $K$-theory of the structure sheaf of a semi-infinite Schubert variety with a line bundle (associated to a dominant integral weight) over the semi-infinite flag manifold, in terms of semi-infinite Lakshmibai-Seshadri (LS for short) paths; the main ingredient in our proof is the combinatorial version of standard monomial theory for semi-infinite LS paths, which is also proved in this paper. Read More


Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$, and set $\lambda: = \Lambda_1 - \Lambda_2$, where $\Lambda_1, \Lambda_2$ are the fundamental weights for $\mathfrak{g}$; note that $\lambda$ is neither dominant nor antidominant. Let $\mathbb{B}(\lambda)$ be the crystal of all Lakshmibai-Seshadri paths of shape $\lambda$. We prove that (the crystal graph of) $\mathbb{B}(\lambda)$ is connected. Read More


We review different constructions of the supersymmetry subalgebras of the chiral de Rham complex on special holonomy manifolds. We describe the difference between the holomorphic-anti-holomorphic sectors based on a local free ghost system vs the decomposition in left-right sectors from a local Boson-Fermion system. We describe the topological twist in the case of $G_2$ and $Spin_7$ manifolds. Read More


In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types and on mapping spaces between coalgebras and algebras. When considering tensor products, such algebraic structures extend the Lie algebra or associative algebra structures that can be obtained by means of the Manin products of operads. These new homotopy algebra structures are proven by to compatible with the concepts of homotopy theory: $\infty$-morphisms and the Homotopy Transfer Theorem. Read More


We argue that the Hamiltonians for A_{2n}^(2) open quantum spin chains corresponding to two choices of integrable boundary conditions have the symmetries U_q(B_n) and U_q(C_n), respectively. The deformation of C_n is novel, with a nonstandard coproduct. We find a formula for the Dynkin labels of the Bethe states (which determine the degeneracies of the corresponding eigenvalues) in terms of the numbers of Bethe roots of each type. Read More


Let $(W,S)$ be an arbitrary Coxeter system, and let $J$ be the asymptotic Hecke algebra associated to $(W,S)$ via Kazhdan-Lusztig polynomials by Lusztig. We study a subalgebra $J_C$ of $J$ corresponding to the subregular cell $C$ of $W$ and prove a factorization theorem that allows us to compute products in $J_C$ without inputs from Kazhdan-Lusztig theory. We discuss two applications of this result. Read More


We study the universal Hopf algebra L of Majid and Lyubashenko in the case that the underlying ribbon category is the category of representations of a finite dimensional ribbon quasi-Hopf algebra A. We show that L=A* with coadjoint action and compute the Hopf algebra structure morphisms of L in terms of the defining data of A. We give explicitly the condition on A which makes Rep(A) factorisable and compute Lyubashenko's projective SL(2,Z)-action on the centre of A in this case. Read More


Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a braided monoidal functor $Z_q^{\varphi}:\mathcal{B}_q \to\widehat{\mathbf{A}}_q^{\varphi}$ for each Drinfeld associator $\varphi$. Here $\mathcal{B}_q $ is the category of bottom $q$-tangles in handlebodies, and $\widehat{\mathbf{A}}_q^{\varphi}$ is the degree-completion $\widehat{\mathbf{A}}$ of the $\mathbb{K}$-linear category $\mathbf{A}$ of Jacobi diagrams in handlebodies (with $\mathbb{K}$ a field of characteristic $0$), equipped with a braided monoidal structure associated to $\varphi$. This functor refines the LMO functor on the category of Lagrangian cobordisms of surfaces, introduced in a prior joint work with Cheptea. Read More


The Kontsevich deformation quantization combines Poisson dynamics, noncommutative geometry, number theory, and calculus of oriented graphs. To manage the algebra and differential calculus of series of weighted graphs, we present software modules: these allow generating the Kontsevich graphs, expanding the noncommutative $\star$-product by using a priori undetermined coefficients, and deriving linear relations between the weights of graphs. Throughout this text we illustrate the assembly of the Kontsevich $\star$-product up to order 4 in the deformation parameter $\hbar$. Read More


In this paper, we study the relation between an anomaly-free $n+$1D topological order, which are often called $n+$1D topological order in physics literature, and its $n$D gapped boundary phases. We argue that the $n+$1D bulk anomaly-free topological order for a given $n$D gapped boundary phase is unique. This uniqueness defines the notion of the "bulk" for a given gapped boundary phase. Read More


In this paper, we prove quantum analogues of the Chamber Ansatz formulae for unipotent cells. These formulae imply that the quantum twist automorphisms, constructed by Kimura and the author, are generalizations of Berenstein-Rupel's quantum twist automorphisms for unipotent cells associated with the squares of acyclic Coxeter elements. This conclusion implies that the known compatibility between quantum twist automorphisms and dual canonical bases corresponds to the property conjectured by Berenstein and Rupel. Read More


The 4-dimensional Sklyanin algebras are a well-studied 2-parameter family of non-commutative graded algebras, often denoted A(E,tau), that depend on a quartic elliptic curve E in P^3 and a translation automorphism tau of E. They are graded algebras generated by four degree-one elements subject to six quadratic relations and in many important ways they behave like the polynomial ring on four indeterminates apart from the minor difference that they are not commutative. They are elliptic analogues of the enveloping algebra of sl(2,C) and the quantized enveloping algebras U_q(gl_2). Read More


Let $\mathcal{A}$ and $\mathcal{B}$ be monoidal categories and let $\left( L:\mathcal{B}\rightarrow \mathcal{A},R:\mathcal{A}\rightarrow \mathcal{B}\right) $ be a pair of adjoint functors. Supposing that $R$ is moreover a lax monoidal functor (or, equivalently, that $L$ is colax monoidal), $R$ induces a functor $\overline{R}:{\sf Alg}({\mathcal{A}})\rightarrow {\sf Alg}({\mathcal{B}})$ and $L$ colifts to a functor $\underline{L}: {\sf Coalg}({\mathcal{B}})\rightarrow {\sf Coalg}({\mathcal{A}})$, as is well-known. An adjoint pair of such functors $(L,R)$ is called "liftable" if the functor $\overline{R}$ has a left adjoint and if the functor $\underline{L}$ has a right adjoint. Read More


The vector space of all polygons with configurations of diagonals is endowed with an operad structure. This is the consequence of a functorial construction $\mathsf{C}$ introduced here, which takes unitary magmas $\mathcal{M}$ as input and produces operads. The obtained operads involve regular polygons with configurations of arcs labeled on $\mathcal{M}$, called $\mathcal{M}$-decorated cliques and generalizing usual polygons with configurations of diagonals. Read More


We give results and observations which allow the application of the logarithmic tensor category theory of Lepowsky, Zhang and the author ([HLZ1]--[HLZ9]) to more general vertex (operator) algebras and their module categories than those studied in a paper by the author ([H3]). Read More


We propose a new approach to the topological recursion of Eynard-Orantin based on the notion of Airy structure, which we introduce in the paper. We explain why Airy structure is a more fundamental object than the one of the spectral curve. We explain how the concept of quantization of Airy structure leads naturally to the formulas of topological recursion as well as their generalizations. Read More


We prove that a finite non-degenerate involutive set-theoretic solution (X,r) of the Yang-Baxter equation is a multipermutation solution if and only if its structure group G(X,r) admits a left ordering or equivalently it is poly-(infinite cyclic). Read More


We use Coulomb branch indices of Argyres-Douglas theories on $S^1 \times L(k,1)$ to quantize moduli spaces ${\cal M}_H$ of wild/irregular Hitchin systems. In particular, we obtain formulae for the "wild Hitchin characters" -- the graded dimensions of the Hilbert spaces from quantization -- for four infinite families of ${\cal M}_H$, giving access to many interesting geometric and topological data of these moduli spaces. We observe that the wild Hitchin characters can always be written as a sum over fixed points in ${\cal M}_H$ under the $U(1)$ Hitchin action, and a limit of them can be identified with matrix elements of the modular transform $ST^kS$ in certain two-dimensional chiral algebras. Read More


We introduce a simple diagrammatic 2-category $\mathscr{A}$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of $\mathfrak{sl}_\infty$. We show that $\mathscr{A}$ is equivalent to a truncation of the Khovanov--Lauda categorified quantum group $\mathscr{U}$ of type $A_\infty$, and also to a truncation of Khovanov's Heisenberg 2-category $\mathscr{H}$. This equivalence is a categorification of the principal realization of the basic representation of $\mathfrak{sl}_\infty$. Read More


We study a relationship between the graded characters of generalized Weyl modules $W_{w \lambda}$, $w \in W$, over the positive part of the affine Lie algebra and those of specific quotients $V_{w}^- (\lambda) / X_{w}^- (\lambda)$, $w \in W$, of the Demazure submodules $V_{w}^- (\lambda)$ of the extremal weight modules $V(\lambda)$ over the quantum affine algebra, where $W$ is the finite Weyl group and $\lambda$ is a dominant weight. More precisely, we prove that a specific quotient of the Demazure submodule is a quantum analog of a generalized Weyl module. Read More


In 2007, Alekseev-Meinrenken proved that there exists a Ginzburg-Weinstein diffeomorphism from the dual Lie algebra ${\rm u}(n)^*$ to the dual Poisson Lie group $U(n)^*$ compatible with the Gelfand-Zeitlin integrable systems. In this paper, we explicitly construct such diffeomorphisms via Stokes phenomenon and Boalch's dual exponential maps. Then we introduce a relative version of the Ginzburg-Weinstein linearization motivated by irregular Riemann-Hilbert correspondence, and generalize the results of Enriquez-Etingof-Marshall to this relative setting. Read More


We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomial of the link. As an application we give the first examples for which the volume conjecture of Chen and the third named author\,\cite{Chen-Yang} is verified. Namely, we show that the asymptotics of the Turaev-Viro invariants of the Figure-eight knot and the Borromean rings complement determine the corresponding hyperbolic volumes. Read More


We show that the Khovanov complex of a rational tangle has a very simple representative whose backbone of non-zero morphisms forms a zig-zag. We find that the bigradings of the subobjects of such a representative can be described by matrix actions, which after a change of basis is the reduced Burau representation of $B_3$. Read More