Varagnolo and Vasserot conjectured an equivalence between the category O for CRDAHA’s and a subcategory of an affine parabolic category O of type A. We prove this conjecture. As applications, we prove a conjecture of Rouquier on the dimension of simple modules of CRDAHA’s and a conjecture of Chuang–Miyachi on the Koszul duality for the category O of CRDAHA’s.

We introduce fusion bialgebras and their duals and systematically study their Fourier analysis. As an application, we discover new efficient analytic obstructions on the unitary categorification of fusion rings. We prove the Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and their duals. We show that the Schur product property, Young's inequality and the sum-set estimate hold for fusion bialgebras, but not always on their duals. If the fusion ring is the Grothendieck ring of a unitary fusion category, then these inequalities hold on the duals. Therefore, these inequalities are analytic obstructions of categorification. We classify simple integral fusion rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less than 4080. We find 34 ones, 4 of which are group-like and 28 of which can be eliminated by applying the Schur product property on the dual. In general, these inequalities are obstructions to subfactorize fusion bialgebras.

We give a new type of Schur-Weyl duality for the representations of a family of quantum subgroups and their centralizer algebra. We define and classify singly-generated, Yang-Baxter relation planar algebras. We present the skein theoretic construction of a new parameterized planar algebra. We construct infinitely many new subfactors and unitary fusion categories, and compute their trace formula as a closed-form expression, in terms of Young diagrams.

In this paper, we generalize the categorifical construction of a quantum group and its canonical basis by Lusztig (\cite{Lusztig,Lusztig2}) to the generic form of whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula in \cite{Green} and the restriction functor in \cite{Lusztig2}. By a geometric way to prove Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework.

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Let $\U_q(\fg)$ be the quantum supergroup of $\gl_{m|n}$ or the modified quantum supergroup of $\osp_{m|2n}$ over the field of rational functions in $q$, and let $V_q$ be the natural module for $\U_q(\fg)$. There exists a unique tensor functor, associated with $V_q$, from the category of ribbon graphs to the category of finite dimensional representations of $\U_q(\fg)$, which preserves ribbon category structures. We show that this functor is full in the cases $\fg=\gl_{m|n}$ or $\osp_{2\ell+1|2n}$. For $\fg=\osp_{2\ell|2n}$, we show that the space $\Hom_{\U_q(\fg)}(V_q^{\otimes r}, V_q^{\otimes s})$ is spanned by images of ribbon graphs if $r+s< 2\ell(2n+1)$. The proofs involve an equivalence of module categories for two versions of the quantisation of $\U(\fg)$.

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The first author constructed a q-parameterized spherical category $\sC$ over C(q) in [Liu15], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of $\sC$, using Littlewood-Richardson coefficients, as well as the characters (including a generating function), using symmetric functions with infinite variables.

We present the third in the series of papers describing Poisson properties of planar directed networks in the disk or in the annulus. In this paper we concentrate on special networks$N$_{$u,v$}in the disk that correspond to the choice of a pair ($u, v$) of Coxeter elements in the symmetric group$S$_{$n$}and the corresponding networks $N_{u,v}^\circ$ in the annulus. Boundary measurements for$N$_{$u,v$}represent elements of the Coxeter double Bruhat cell$G$^{$u,v$}⊂GL_{$n$}. The Cartan subgroup$H$acts on$G$^{$u,v$}by conjugation. The standard Poisson structure on the space of weights of$N$_{$u,v$}induces a Poisson structure on$G$^{$u,v$}, and hence on the quotient$G$^{$u,v$}/$H$, which makes the latter into the phase space for an appropriate Coxeter–Toda lattice. The boundary measurement for $N_{u,v}^\circ$ is a rational function that coincides up to a non-zero factor with the Weyl function for the boundary measurement for$N$_{$u,v$}. The corresponding Poisson bracket on the space of weights of $N_{u,v}^\circ$ induces a Poisson bracket on the certain space $ {\mathcal{R}_n} $ of rational functions, which appeared previously in the context of Toda flows.

This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six. Among 72 possible fusion rings, 25 ones are filtered out by using categorification criteria. Each of the remaining 47 fusion rings admits a unitary complex categorification. We found 6 new Grothendieck rings, categorified by applying a localization approach of the Pentagon Equation.

In this note, we discuss the notion of symmetric self-duality of shaded planar algebras, which allows us to lift shadings on subfactor planar algebras to obtain Z/2Z-graded unitary fusion categories. This finishes the proof that there are unitary fusion categories with fusion graphs 4442 and 3333.

Inspired by the quantum McKay correspondence, we consider the classical ADE Lie theory as a quantum theory over sl2. We introduce anti-symmetric characters for representations of quantum groups and investigate the Fourier duality to study the spectral theory. In the ADE Lie theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix. We formalize related notions and prove such a correspondence for representations of Verlinde algebras of quantum groups: this includes generalized Dynkin diagrams over any simple Lie algebra g at any level k. This answers a recent comment of Terry Gannon on an old question posed by Victor Kac in 1994.

In this paper, we introduce a motivic version of To¨en’s derived Hall algebra. Then we point out that the two kinds of Hall algebras in the sense of To¨en and Kontsevich–Soibelman, respectively, are Drinfeld dual pairs, not only in the classical case (by counting over finite fields) but also in the motivic version. Consequently they are canonically isomorphic. All proofs, including that for the most important associative property, are deduced in a self-contained way by analyzing the symmetry properties around the octahedral axiom, a method we used previously

We investigate a generalization of Hopf algebra $\mathfrak{sl}_{q}\left( 2\right) $ by weakening the invertibility of the generator $K$, i.e. exchanging its invertibility $KK^{-1}=1$ to the regularity $K\overline{K}K=K$. This leads to a weak Hopf algebra $w\mathfrak{sl}_{q}\left( 2\right) $ and a $J$-weak Hopf algebra $v\mathfrak{sl}_{q}\left( 2\right) $ which are studied in detail. It is shown that the monoids of group-like elements of $w\mathfrak{sl}_{q}\left( 2\right) $ and $v\mathfrak{sl}_{q}\left( 2\right) $ are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, from $w\mathfrak{sl}_{q}\left( 2\right) $ a quasi-braided weak Hopf algebra $\overline{U}_{q}^{w}$ is constructed and it is shown that the corresponding quasi-$R$-matrix is regular $R^{w}\hat{R}^{w}R^{w}=R^{w}$.

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We develop a general theory of canonical bases for quantum symmetric pairs (U, U^i) with parameters of arbitrary finite type. We construct new canonical bases for the simple integrable U-modules and their tensor products regarded as U^i-modules. We also construct a canonical basis for the modified form of the i-quantum group U^i. To that end, we establish several new structural results on quantum symmetric pairs, such as bilinear forms, braid group actions, integral forms, Levi subalgebras (of real rank one), and integrality of the intertwiners.

Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framework. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of newly constructed exactly solvable models with translation-invariant local Hamiltonians. Moreover, the Hamiltonian is gapped in the large limit when the underlying group is infinite.

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The subject for investigation in this note is concerned with strongly homotopy (SH) Lie algebras, also known as $L_\infty$-algebras, or $L_\infty[1]$-algebras. We extract an intrinsic character, the Atiyah class, when a smaller $L_\infty[1]$-algebra $A$ is extended to a bigger one $L$. In fact, we establish that given such an SH Lie pair $(L,A)$, and any $A$-module $E$, there associates a canonical cohomology class, the Atiyah class $[\alpha^E]$, living in $H^2_{\mathrm{CE}}(A, A^\perp\otimes End(E))$, which generalizes earlier known Atiyah classes out of Lie algebra pairs. In particular, the quotient space $L/A$ is a canonical $A$-module and there associates the Atiyah class $[\alpha^{L/A}]\in H^2_{\mathrm{CE}}(A, Hom(L/A\otimes L/A,L/A))$. The nature of such classes is that they induce on $H^\bullet_{\mathrm{CE}}(A,L/A[-2])$ a Lie algebra structure, and on $H^\bullet_{\mathrm{CE}}(A,E[-2])$ a Lie algebra module structure. An alternative point of view is that the process of taking Atiyah classes defines a functor from the category of $A$-modules to the category of $H^\bullet_{\mathrm{CE}}(A,L/A[-2])$-modules. Moreover, Atiyah classes are invariant under gauge equivalent $A$-compatible infinitesimal deformations of $L$.

We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov--Witten (GW) invariants of the Fano orbifold projective curve P^1_{a1,a2,a3}. The vertex operators in our construction are given in terms of the K-theory of P^1_{a1,a2,a3} via Iritani's Γ-class modification of the Chern character map. We also identify our HQEs with an appropriate Kac--Wakimoto hierarchy of ADE type. In particular, we obtain a generalization of the famous Toda conjecture about the GW invariants of P^1 .

In this paper, we associate quantum vertex algebras to a certain family of associative algebras $\widetilde{\A}(g)$ which are essentially Ding-Iohara algebras. To do this, we introduce another closely related family of associative algebras $\A(h)$. The associated quantum vertex algebras are based on the vacuum modules for $\A(h)$, whereas $\phi$-coordinated modules for these quantum vertex algebras are associated to $\widetilde{A}(g)$-modules. Furthermore, we classify their irreducible $\phi$-coordinated modules.

For any unital separable simple infinite-dimensional nuclear$C$^{∗}-algebra with finitely many extremal traces, we prove that $$ \mathcal{Z} $$ -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear$C$^{∗}-algebra with tracial rank zero is approximately divisible, and hence is $$ \mathcal{Z} $$ -absorbing.

In these notes we collect some results about finite-dimensional representations of $U_{q}(\mathfrak {gl}(1\mid1))$ and related invariants of framed tangles, which are well-known to experts but difficult to find in the literature. In particular, we give an explicit description of the ribbon structure on the category of finite-dimensional $U_{q}(\mathfrak {gl}(1\mid1))$ -representations and we use it to construct the corresponding quantum invariant of framed tangles. We explain in detail why this invariant vanishes on closed links and how one can modify the construction to get a non-zero invariant of framed closed links. Finally we show how to obtain the Alexander polynomial by considering the vector representation of $U_{q}(\mathfrak {gl}(1\mid1))$ .

We present a 3D topological picture-language for quantum information. Our approach combines charged excitations carried by strings, with topological properties that arise from embedding the strings in the interior of a 3D manifold with boundary. A quon is a composite that acts as a particle. Specifically, a quon is a hemisphere containing a neutral pair of open strings with opposite charge. We interpret multiquons and their transformations in a natural way. We obtain a type of relation, a string–genus “joint relation,” involving both a string and the 3D manifold. We use the joint relation to obtain a topological interpretation of the C* Hopf algebra relations, which are widely used in tensor networks. We obtain a 3D representation of the controlled NOT (CNOT) gate that is considerably simpler than earlier work, and a 3D topological protocol for teleportation.

In this paper, we give geometric realizations of Lusztig's symmetries. We also give projective resolutions of a kind of standard modules. By using the geometric realizations and the projective resolutions, we obtain the categorification of the formulas of Lusztig's symmetries.