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We apply Takesaki’s and Connes’s ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory. [ABSTRACT FROM AUTHOR]

We give a recursive method for computing all values of a basis of Whittaker functions for unramified principal series invariant under an Iwahori or parahoric subgroup of a split reductive group G over a nonarchimedean local field F. Structures in the proof have surprising analogies to features of certain solvable lattice models. In the case G = GL r we show that there exist solvable lattice models whose partition functions give precisely all of these values. Here ‘solvable’ means that the models have a family of Yang–Baxter equations which imply, among other things, that their partition functions satisfy the same recursions as those for Iwahori or parahoric Whittaker functions. The R-matrices for these Yang–Baxter equations come from a Drinfeld twist of the quantum group U q (gl ^ (r | 1)) , which we then connect to the standard intertwining operators on the unramified principal series. We use our results to connect Iwahori and parahoric Whittaker functions to variations of Macdonald polynomials. [ABSTRACT FROM AUTHOR]

We introduce the doubled Hecke algebra, which is an infinite-dimensional algebra generated by two Hecke algebras. This concept originates from the degenerate doubled Hecke algebra in the theory of Schur-Weyl duality related to enhanced reductive algebraic groups. We study the finite-dimensional natural representation of the doubled Hecke algebra on tensor space and prove that the doubled Hecke algebra forms a duality with the quantum group of Levi type. [ABSTRACT FROM AUTHOR]

We define quantum automorphisms and isomorphisms of Hadamard matrices. We show that every Hadamard matrix of size N\ge 4 has quantum symmetries and that all Hadamard matrices of a fixed size are mutually quantum isomorphic. These results pass also to the corresponding Hadamard graphs. We also define quantum Hadamard matrices acting on quantum spaces and bring an example thereof over matrix algebras. [ABSTRACT FROM AUTHOR]

This is part of an ongoing project of formulating notion(s) of quantum group of outer automorphisms of a C ∗ or von Neumann algebra. Motivated by the fact that the group of outer automorphism of a I I 1 factor can be viewed as a subgroup of the group of group-like or invertible objects in the category of Hilbert bimodules of finite ranks, we explore a natural class of objects in the bimodule category of a finite dimensional (i.e. direct sum of matrix algebras) von Neumann algebra A which may come from the (co-action) of a discrete quantum group. In particular, we prove that any discrete quantum group giving an outer quantum symmetry on A in a sense defined by us must be a finite dimensional quantum group. We relate the analysis of such quantum groups or the corresponding fusion rings with certain combinatorial objects involving matrices with nonnegative integer entries and do some explicit computations in a few simple examples. [ABSTRACT FROM AUTHOR]

AbstractWe explore two modules F and K over a subalgebra U of the quantum group Uq(gl(∞)). The first U-module is the Fock space related to the natural module over the quantum group. If we let O+ to be the category of finite-dimensional osp(2|2)-modules of integral weights, the second U-module K is the vector space obtained by tensoring the Grothendieck group K(O+) by the field Q. Using thick tensor ideals and a U-isomorphism, we compute the socles of F and K. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a basis for the center of the infinitesimal q-Schur algebra s q (2 , r) 1 in terms of its certain basis. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a basis for the center of the infinitesimal q-Schur algebra s q (2 , r) 1 in terms of its certain basis. [ABSTRACT FROM AUTHOR]

Part I of this paper established the basic properties of quantum predicate logic as the internal logic of discrete quantum structures. We now show that a majority of the established quantum generalizations of discrete structures are naturally axiomatizable within this internal logic. In particular, we axiomatize the quantum graphs of Duan, Severini, and Winter, the quantum metric spaces of Kuperberg and Weaver, the quantum isomorphisms of Atserias, Mancinska, Roberson, Šámal, Severini, and Varvitsiotis, and the quantum groups of Woronowicz. In each instance, we consider only those structures that are discrete in the sense that the underlying von Neumann algebra is hereditarily atomic. [ABSTRACT FROM AUTHOR]

We propose a new notion of antipode S for a left Hopf algebroid which does not assume antimultiplicativity. We also show that if L is a left Hopf algebroid then so is its cotwist L ς as an extension of a previous bialgebroid Drinfeld cotwist theory. We show that the Ehresmann-Schauenburg bialgebroid of a quantum principal bundle P or Hopf-Galois extension with structure quantum group H is in fact a left Hopf algebroid L (P , H) and that it has an antipode S at least if H is coquasitriangular or if the bundle is trivial so that P = B # σ H is a cleft extension and the cocycle-action (▹ , σ) is of associative type. We also show in the latter case that L (B # σ H , H) = L (B # H , H) σ ˜ for a Hopf algebroid cotwist ς = σ ˜. Thus, introducing a σ of associative type appears at the Hopf algebroid level as a Drinfeld cotwist. We view the affine quantum group U q (s l 2) ˆ and the quantum Weyl group of u q (s l 2) as examples of associative type. [ABSTRACT FROM AUTHOR]

We classify all the Hopf PBW-deformations of a new type quantum group U q ( 2 ∗) from which the classical Drinfeld–Jimbo quantum group U q ( 2) can arise as an almost unique nontrivial one. Different from the U q ( 2) case, the category of finite-dimensional U q ( 2 ∗) -modules is non-semisimple. We establish a block decomposition theorem for the category U q ( 2 ∗) - mod wt of finite-dimensional weight modules of U q ( 2 ∗). On the level of tensor category, we show that U q ( 2 ∗) - mod wt (respectively, the category U q ( 2) - mod of finite-dimensional U q ( 2) -modules) can be realized via (respectively, deformed) preprojective algebras of Dynkin type . [ABSTRACT FROM AUTHOR]

We use minimal tilting complexes to construct an explicit bijection between the set of thick tensor ideals with the two-out-of-three property in the category of finite-dimensional modules over a quantum group at a root of unity and the set of thick tensor ideals in the subcategory of tilting modules. We also explain why the analogous construction for rational representations of a reductive algebraic group over a field of positive characteristic does not give rise to a bijection. [ABSTRACT FROM AUTHOR]

We study Wilson line operators in three-dimensional Chern–Simons theory on a manifold with boundaries and prove to leading order, through a direct calculation of Feynman integrals, that the merging of parallel Wilson lines reproduces the coproduct on the quantum group U ħ (g) . We outline a connection of this theory with the moduli spaces of local systems defined by Goncharov and Shen. [ABSTRACT FROM AUTHOR]

For G a commutative group, we give a purely Hopf G -coalgebra construction of G -colored 3 -manifolds invariants using the notion of modified integral. [ABSTRACT FROM AUTHOR]

We explain the construction of minimal tilting complexes for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials. [ABSTRACT FROM AUTHOR]

As is well known, the classical nonsingular Whittaker modules over quantum groups cannot be defined for non A1-type. In this paper, by choosing different generators for the quantum group U q (g l n + 1) , we introduce and study the twisted Whittaker modules over U q (g l n + 1) . We classify all the simple twisted Whittaker modules with nonsingular Whittaker functions. This agrees Kostant's results on Whittaker modules for the simple complex Lie algebras s l n + 1 as q approaches 1. [ABSTRACT FROM AUTHOR]

We study the "geometric Ricci curvature lower bound", introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups O N + , q -deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on O N + admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the "geometric Ricci curvature lower bound" is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors. [ABSTRACT FROM AUTHOR]

We provide a new tableau model from which one can easily deduce the characters of finite-dimensional irreducible polynomial representations of the special orthogonal group S O n (C) . This model originates from the representation theory of the ı quantum group (also known as the quantum symmetric pair coideal subalgebra) of type A I and is equipped with a combinatorial structure, which we call A I -crystal structure. This structure enables us to describe combinatorially the tensor product of an S O n (C) -module and a G L n (C) -module, and the branching from G L n (C) to S O n (C) . [ABSTRACT FROM AUTHOR]

We prove a number of structural and representation-theoretic results on linearly reductive quantum groups, i.e., objects dual to those of cosemisimple Hopf algebras: (a) a closed normal quantum subgroup is automatically linearly reductive if its squared antipode leaves invariant each simple subcoalgebra of the underlying Hopf algebra; (b) for a normal embedding H ⊴ G there is a Clifford-style correspondence between two equivalence relations on irreducible G - and, respectively, H -representations; and (c) given an embedding H ≤ G of linearly reductive quantum groups, the Pontryagin dual of the relative center Z (G) ∩ H can be described by generators and relations, with one generator gVfor each irreducible G -representation V and one relation g U = g V g W whenever U and V ⊗ W are not disjoint over H. This latter center-reconstruction result generalizes and recovers Müger's compact-group analogue and the author's quantum-group version of that earlier result by setting H = G . [ABSTRACT FROM AUTHOR]

In this paper, we define the affine super Yangian ... ith a coproduct structure. We also obtain an evaluation homomorphism, that is, an algebra homomorphism from ... to the completion of the universal enveloping algebra of ... [ABSTRACT FROM AUTHOR]

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics. [ABSTRACT FROM AUTHOR]

We study geodesics flows on curved quantum Riemannian geometries using a recent formulation in terms of bimodule connections and completely positive maps. We complete this formalism with a canonical ∗ operation on noncommutative vector fields. We show on a classical manifold how the Ricci tensor arises naturally in our approach as a term in the convective derivative of the divergence of the geodesic velocity field and use this to propose a similar object in the noncommutative case. Examples include quantum geodesic flows on the algebra of 2 × 2 matrices, fuzzy spheres and the q-sphere. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the quantum groups (A , B) introduced by Mrozinski in [C. Mrozinski, Quantum groups of G L (2) representation type, J. Noncommut. Geom.8(1) (2014) 107–140] and their Hopf–Galois objects are twisted Calabi–Yau algebras, and give their Nakayama automorphisms explicitly. [ABSTRACT FROM AUTHOR]

We construct generic Gelfand-Tsetlin representations of the ı quantum groups U q tw ( 3) and U q tw ( 4). These representations are infinite-dimensional analogs to the finite-dimensional irreducible representations provided by Gavrilik and Klimyk in [ q -deformed orthogonal and pseudo-orthogonal algebras and their representations, Lett. Math. Phys.21 (1991) 215–220]. They are quantum analogs of generic Gelfand-Tsetlin representations constructed by Mazorchuk in [On Gelfand-Zetlin modules over orthogonal Lie algebras, Algebra Colloq.8 (2001) 345–360]. We give sufficient conditions for irreducibility and provide an upper bound for the length with the help of Casimir elements found by Molev, Ragoucy and Sorba. [ABSTRACT FROM AUTHOR]

Let U q + (A N) (resp. U q − (A N) ) be the (+) -part (resp. (−) -part) of the Drinfeld-Jimbo quantum group of type AN over a field K. With respect to Jimbo relations and the PBW K-basis B of U q + (A N) (resp. U q − (A N) ) established by Yamane, it is shown, by constructing an appropriate monomial ordering ≺ * − d on B , that U q + (A N) (resp. U q − (A N) ) is a solvable polynomial algebra. Consequently, further structural properties of U q + (A N) (resp. U q − (A N) ) and their modules may be established and realized in a constructive-computational way. [ABSTRACT FROM AUTHOR]

Let g be the complex simple Lie algebra of type B2. In this paper, we give the explicit generators of the center of the quantum group U q (g). We also study the twisted Whittaker modules over U q (g) , and all nonsingular twisted Whittaker modules are classified. [ABSTRACT FROM AUTHOR]

Let g be a finite dimensional complex simple Lie algebra with Cartan subalgebra h. Then C [ h ] has a g -module structure if and only if g is of type A or of type C ; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over U q (sl 2) have been classified. In this paper, we prove that the quantum group U q (sl 3) has no rank one polynomial module. [ABSTRACT FROM AUTHOR]

We study a class of super W-algebras whose even part is the Virasoro type Lie algebra W (2 , 2). We quantize the Lie superbialgebra of the super W-algebra by the Drinfeld twist quantization technique and obtain a class of noncommutative and noncocommutative Hopf superalgebras. [ABSTRACT FROM AUTHOR]

We study the semiclassical limit κ → ∞ of the generalized quantum Langlands kernel associated to a Lie algebra g and an integer level p. This vertex algebra acquires a big centre, containing the ring of functions over the space of g -connections. We conjecture that the fibre over the zero connection is the Feigin-Tipunin vertex algebra, whose category of representations should be equivalent to the small quantum group, and that the other fibres are precisely its twisted modules, and that the entire category of representations is related to the quantum group with a big centre. In this sense we present a generalized Kazhdan-Lusztig conjecture, involving deformations by any g -connection. We prove our conjectures in small cases (g , 1) and (sl 2 , 2) by explicitly computing all vertex algebras and categories involved. [ABSTRACT FROM AUTHOR]

The theory of groups has a twofold symmetry, sending a group to its opposite. Groups invariant under the symmetry are abelian. The theory of quasigroups has a richer, sixfold symmetry, obtained by permuting the multiplication with its two divisions. The Sixfold Way identifies the various classes of quasigroups which are invariant under the respective subgroups of the symmetry group of the theory. Quantum quasigroups provide a self-dual framework to unify the study of quasigroups and Hopf algebras. The goal of this paper is to classify the symmetry classes of quantum quasigroups. Corresponding to the Sixfold Way for classical quasigroups, we are able to identify a Sevenfold Way for general classes exhibiting a symmetry, and initiate a study of a fuller symmetry which holds for linear quantum quasigroups. [ABSTRACT FROM AUTHOR]

For an acyclic quiver, we establish a connection between the cohomology of quiver Grassmannians and the dual canonical bases of the algebra U q − (g) , where U q − (g) is the negative half of the quantized enveloping algebra associated with the quiver. In order to achieve this goal, we study the cohomology of quiver Grassmannians by Lusztig's category. As a consequence, we describe explicitly the Poincaré polynomials of rigid quiver Grassmannians in terms of the coefficients of dual canonical bases, which are viewed as elements of quantum shuffle algebras. By this result, we give another proof of the odd cohomology vanishing theorem for quiver Grassmanians. Meanwhile, for Dynkin quivers, we show that the Poincaré polynomials of rigid quiver Grassmannians are the coefficients of dual PBW bases of the algebra U q − (g) . [ABSTRACT FROM AUTHOR]

We give the first positive formulas for the weights of every simple highest weight module L (λ) over an arbitrary Kac–Moody algebra. Under a mild condition on the highest weight, we also express the weights of L (λ) as an alternating sum similar to the Weyl–Kac character formula. To obtain these results, we show the following data attached to a highest weight module are equivalent: (i) its integrability, (ii) the convex hull of its weights, (iii) the Weyl group symmetry of its character, and (iv) when a localization theorem is available, its behavior on certain codimension one Schubert cells. We further determine precisely when the above datum determines the weights themselves. Moreover, we use condition (iv) to relate localizations of the convex hull of the weights with the introduction of poles of the corresponding D -module on certain divisors, which answers a question of Brion. Many of these results are new even in finite type. We prove similar assertions for highest weight modules over a symmetrizable quantum group. [ABSTRACT FROM AUTHOR]

A solution to the reflection equation associated to a coideal subalgebra of U q (A 2 n − 1 (1)) of type AII in the symmetric tensor representations is presented. If parameters of the coideal subalgebra are suitably chosen, the K matrix does not depend on the quantum parameter q and still agrees with a solution in [8] at q = 0. [ABSTRACT FROM AUTHOR]

Let g be a hyperbolic Kac-Moody algebra of rank 2, and let λ be an arbitrary integral weight. We denote by B (λ) the crystal of all Lakshmibai-Seshadri paths of shape λ. Let V (λ) be the extremal weight module of extremal weight λ generated by the (cyclic) extremal weight vector v λ of weight λ, and let B (λ) be the crystal basis of V (λ) with u λ ∈ B (λ) the element corresponding to v λ. We prove that the connected component B 0 (λ) of B (λ) containing u λ is isomorphic, as a crystal, to the connected component B 0 (λ) of B (λ) containing the straight line π λ. Furthermore, we prove that if λ satisfies a special condition, then the crystal basis B (λ) is isomorphic, as a crystal, to the crystal B (λ). As an application of these results, we obtain an algorithm for computing the number of elements of weight μ in B (Λ 1 − Λ 2) , where Λ 1 , Λ 2 are the fundamental weights, in the case that g is symmetric. [ABSTRACT FROM AUTHOR]

Using Kuperberg’s web calculus (1996), and following Elias and Libedinsky, we describe a “light leaves” algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for sp4 (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when [2]q ≠ 0, the Karoubi envelope of the C2 web category is equivalent to the category of tilting modules for the divided powers quantum group UA q(sp4). [ABSTRACT FROM AUTHOR]

We apply quantum group methods to quantum computing, starting with the notion of interacting Frobenius Hopf algebras for ZX calculus with noncommutative algebra and noncocommutative coalgebra. We introduce the notion of *-structures in ZX calculus at this algebraic level and construct examples based on the quantum group u q (sl 2) at a root of unity. We provide an abstract formulation of the Hadamard gate related to Hopf algebra self-duality. We then solve the problem of extending the notion of interacting Hopf algebras and ZX calculus to take place in a braided tensor category. In the ribbon case, the Hadamard gate coming from braided self-duality obeys a modular identity. We give the example of b q (sl 2), the self-dual braided version of u q (sl 2). [ABSTRACT FROM AUTHOR]

We give a new combinatorial interpretation of Howe dual pairs of the form (g , Sp 2 ℓ) , where g is a Lie (super)algebra of classical type. This is done by establishing a symplectic analogue of the RSK algorithm associated to this pair, in a uniform way which does not depend on g. We introduce an analogue of jeu de taquin sliding for spinor model of irreducible characters of a Lie superalgebra g to define a P -tableau and show that the associated Q -tableau is given by a symplectic tableau due to King. [ABSTRACT FROM AUTHOR]

In this article, we classify all rank one polynomial modules over the quantum groups U q (s l 2) and U q (s l 2) for generic q. We also provide the equivalent conditions for the simplicity of these modules. [ABSTRACT FROM AUTHOR]

In this article, we classify all rank one polynomial modules over the quantum groups U q (s l 2) and U q (s l 2) for generic q. We also provide the equivalent conditions for the simplicity of these modules. [ABSTRACT FROM AUTHOR]

In 2023 we obtained a Q -polynomial structure for the projective geometry L N (q). In the present paper, we display a more general Q -polynomial structure for L N (q). Our new Q -polynomial structure is defined using a free parameter φ that takes any positive real value. For φ = 1 we recover the original Q -polynomial structure. We interpret the new Q -polynomial structure using the quantum group U q 1 / 2 (sl 2) in the equitable presentation. We use the new Q -polynomial structure to obtain analogs of the four split decompositions that appear in the theory of Q -polynomial distance-regular graphs. [ABSTRACT FROM AUTHOR]

We investigate the quantum deformation of the Wheeler–DeWitt equation of a Schwarzchild black hole. Specifically, the quantum deformed black hole is a quantized model constructed from the quantum Heisenberg–Weyl Uq(h4) group. We show that the event horizon area and the mass are quantized, degenerate, and bounded. The degeneracy of states indicates entangled quantum black hole/white hole states. Accordingly, quantum deformation provides a new framework to examine Einstein–Rosen wormhole solutions. Besides, we obtain the mass, the temperature, and the entropy of the q-deformed quantum Schwarzschild black hole. We find an upper bound on the mass of a black hole/white hole pair. Also, at the quantum deformation level, the entropy of the black hole contains three parts: the usual Bekenstein–Hawking entropy, the logarithmic term, and a Cube of usual black hole entropy.

Let G q be the Drinfeld-Jimbo deformation of a connected simply-connected compact Lie group and T be its maximal torus. We show that the induction functors from the categories of T -equivariant correspondences to the categories of G q -equivariant correspondences give equivalences of these categories under the existence of tracial states. As its application, we show some automatic imprimitivity results and classification results for extensions, subalgebras and finite index discrete quantum subgroups of G q ˆ. [ABSTRACT FROM AUTHOR]

We clarify the relation between noncommutative Poisson boundaries and Furstenberg–Hamana boundaries of quantum groups. Specifically, given a compact quantum group G , we show that in many cases where the Poisson boundary of the dual discrete quantum group G ˆ has been computed, the underlying topological boundary either coincides with the Furstenberg–Hamana boundary of the Drinfeld double D (G) of G or is a quotient of it. This includes the q -deformations of compact Lie groups, free orthogonal and free unitary quantum groups, quantum automorphism groups of finite dimensional C⁎-algebras. In particular, the boundary of D (G q) for the q -deformation of a compact connected semisimple Lie group G is G q / T (for q ≠ 1), in agreement with the classical results of Furstenberg and Moore on the Furstenberg boundary of G C. We show also that the construction of the Furstenberg–Hamana boundary of D (G) respects monoidal equivalence and, in fact, can be carried out entirely at the level of the representation category of G. This leads to a notion of the Furstenberg–Hamana boundary of a rigid C⁎-tensor category. [ABSTRACT FROM AUTHOR]

The semidirect product of a finitely generated group dual with the symmetric group can be described through so-called group-theoretical categories of partitions (covers only a special case; due to Raum–Weber, 2015) and skew categories of partitions (more general; due to Maaßen, 2018). We generalize these results to the case of graph categories, which allows to replace the symmetric group by the group of automorphisms of some graph. [ABSTRACT FROM AUTHOR]