Bisch and Jones proposed the classification of planar algebras by simple generators and relations. They investigated with the second author the classification of planar algebras generated by 2-boxes. In this paper, we classify singly-generated Thurston-relation planar algebras, defined as subfactor planar algebras generated by a 3-box satisfying a relation proposed by Dylan Thurston. Our main result shows that such subfactor planar algebras are either the E6 subfactor planar algebras or belong to a two-parameter family of planar algebras arising from the representations of type A quantum groups. We introduce a new method for determining positivity of the Markov trace of planar algebras in this family.

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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We study the tensor product decomposition of the split real quantum group Uqq~(sl(2,R)) from the perspective of finite dimensional representation theory of compact quantum groups. It is known that the class of positive representations of Uqq~(sl(2,R)) is closed under taking tensor product. In this paper, we show that one can derive the corresponding Hilbert space decomposition, given explicitly by quantum dilogarithm transformations, from the Clebsch-Gordan coefficients of the tensor product decomposition of finite dimensional representations of the compact quantum group Uq(sl2) by solving certain functional equations and using normalization arising from tensor products of canonical basis. We propose a general strategy to deal with the tensor product decomposition for the higher rank split real quantum group Uqq~(gR)

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 $U_{q}(\mathfrak {gl}(1\mid1))$ .

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.

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Some filtrations of the tensor product of a highest weight module and a lowest weight module over quantum group Uq(g)are constructed in[1]and one can use them to define a two-sided ideal of the modified quantized enveloping algebra. It is shown that the quotient algebra inherits a canonical basis from the modified quantized enveloping algebra and is dual to the quantum coordinate algebra defined by Kashiwara for a symmetrizable Kac–Moody algebra g.

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

Let V⊆A be a conformal inclusion of vertex operator algebras and let C be a category of grading-restricted generalized V-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We give conditions under which C inherits semisimplicity from the category of grading-restricted generalized A-modules in C, and vice versa. The most important condition is that A be a rigid V-module in C with non-zero categorical dimension, that is, we assume the index of V as a subalgebra of A is finite and non-zero. As a consequence, we show that if A is strongly rational, then V is also strongly rational under the following conditions: A contains V as a V-module direct summand, V is C_2-cofinite with a rigid tensor category of modules, and A has non-zero categorical dimension as a V-module. These results are vertex operator algebra interpretations of theorems proved for general commutative algebras in braided tensor categories. We also generalize these results to the case that A is a vertex operator superalgebra.

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for C a braided tensor category, we give a detailed construction of the canonical algebra in C⊠C^rev: if C is semisimple but not necessarily finite or rigid, then ⨁X∈Irr(C) X′⊠X is a commutative algebra, with X′ a representing object for Hom_C(∙⊗_C X,1_C). Conversely, let A=⨁i∈I U_i ⊠ V_i be a simple commutative algebra in U⊠V with U semisimple and rigid but not necessarily finite, and V rigid but not necessarily semisimple. If the unit objects of U and V form a commuting pair in A, we show there is a braid-reversed equivalence between subcategories of U and V sending U_i to (V_i)*. When U and V are module categories for simple vertex operator algebras U and V, we glue U and V along U⊠V via a map τ: Irr(U)→Obj(V) such that τ(U)=V to create A=⨁X∈Irr(U) X′⊗τ(X). Thus under certain conditions, τ extends to a braid-reversed equivalence between U and V if and only if A is a simple conformal vertex algebra extending U⊗V. As examples, we glue Kazhdan-Lusztig categories at generic levels to obtain new vertex algebras extending the tensor product of two affine vertex algebras, and we prove braid-reversed equivalences between certain module categories for affine vertex algebras and W-algebras at admissible levels.

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.

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For each simple Lie algebra g, we construct an algebra embedding of the quantum group Uq(g) into certain quantum torus algebra Dg via the positive representations of split real quantum group. The quivers corresponding to Dg is obtained from an amalgamation of two basic quivers, each of which is mutation equivalent to one describing the cluster structure of the moduli space of framed G-local system on a disk with 3 marked points on its boundary when G is of classical type. We derive a factorization of the universal R-matrix into quantum dilogarithms of cluster monomials, and show that conjugation by the R-matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in a twice punctured disk.

We found an explicit construction of a representation of the positive quantum group and its modular double by positive essentially self-adjoint operators. Generalizing Lusztig's parametrization, we found a Gauss type decomposition for the totally positive quantum group parametrized by the standard decomposition of the longest element . Under this parametrization, we found explicitly the relations between the standard quantum variables, the relations between the quantum cluster variables, and realizing them using non-compact generators of the q-tori by positive essentially self-adjoint operators. The modular double arises naturally from the transcendental relations, and an space in the von Neumann setting can also be defined.

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.

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 $L$.

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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 $R^{w}\hat{R}^{w}R^{w}=R^{w}$.

We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra g^, from g-module homomorphisms. When g=sl_2, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to g^, and we identify obstructions to the construction arising from the possible non-existence of series solutions having a prescribed form.

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.

Let $\mathbf{U}$ be the quantum group and $\mathbf{f}$ be the Lusztig's algebra associated with a symmetrizable generalized Cartan matrix. The algebra $\mathbf{f}$ can be viewed as the positive part of $\mathbf{U}$. Lusztig introduced some symmetries $T_i$ on $\mathbf{U}$ for all $i\in I$. Since $T_i(\mathbf{f})$ is not contained in $\mathbf{f}$, Lusztig considered two subalgebras ${_i\mathbf{f}}$ and ${^i\mathbf{f}}$ of $\mathbf{f}$ for any $i\in I$, where ${_i\mathbf{f}}=\{x\in\mathbf{f}\,\,|\,\,T_i(x)\in\mathbf{f}\}$ and ${^i\mathbf{f}}=\{x\in\mathbf{f}\,\,|\,\,T^{-1}_i(x)\in\mathbf{f}\}$. The restriction of $T_i$ on ${_i\mathbf{f}}$ is also denoted by $T_i:{_i\mathbf{f}}\rightarrow{^i\mathbf{f}}$. The geometric realization of $\mathbf{f}$ and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig's symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of ${_i\mathbf{f}}$, ${^i\mathbf{f}}$ and $T_i:{_i\mathbf{f}}\rightarrow{^i\mathbf{f}}$ by using the language 'quiver with automorphism' introduced by Lusztig.

Let e be an arbitrary even nilpotent element in the general linear Lie super- algebra glM|N and let We be the associated finite W-superalgebra. Let Ym|n be the super Yangian associated to the Lie superalgebra glm|n. A subalgebra of Ym|n, called the shifted super Yangian and denoted by Ym|n(σ), is defined and studied. Moreover, an explicit iso- morphism between We and a quotient of Ym|n(σ) is established.

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.