Quantum Group Research Articles

Shapovalov elements of classical and quantum groups

Shapovalov elements θβ,m of the classical or quantized universal enveloping algebra of a simple Lie algebra g are parameterized by a positive root β and a positive integer m. They relate the highest vector of a reducible Verma module with highest vectors of its submodules. We obtain a factorization of θβ,m to a product of θβ,1 and calculate θβ,1 as a residue of a matrix element of the inverse Shapovalov form via a generalized Nagel-Moshinsky algorithm. This way we explicitly express θβ,m of a classical simple Lie algebra through the Cartan-Weyl basis in g. In the case of quantum groups, we give an analogous formulation through the entries of the R-matrix (quantum L-operator) in fundamental representations.

Dynamical actions and q-representation theory for double-scaled SYK

We show that DSSYK amplitudes are reproduced by considering the quantum mechanics of a constrained particle on the quantum group SUq(1, 1). We construct its left-and right-regular representations, and show that the representation matrices reproduce two-sided wavefunctions and correlation functions of DSSYK. We then construct a dynamical action and path integral for a particle on SUq(1, 1), whose quantization reproduces the aforementioned representation theory. By imposing boundary conditions or constraining the system we find the q-analog of the Schwarzian and Liouville boundary path integral descriptions. This lays the technical groundwork for identifying the gravitational bulk description of DSSYK. We find evidence the theory in question is a sine dilaton gravity, which interestingly is capable of describing both AdS and dS quantum gravity.

Quantum Euler angles and agency-dependent space-time

Abstract Quantum gravity is expected to introduce quantum aspects into the description of reference frames. Here we begin exploring how quantum gravity induced deformations of classical symmetries could modify the transformation laws among reference frames in an effective regime. We invoke the quantum group SUq(2) as a description of deformed spatial rotations and interpret states of a representation of its algebra as describing the relative orientation between two reference frames. This leads to a quantization of one of the Euler angles and to an aspect of agency dependence: space is reconstructed as a collection of fuzzy points, exclusive to each agent, which depends on their choice of reference frame. Each agent can choose only one direction in which points can be sharp, while points in all other directions become fuzzy in a way that depends on this choice. Two agents making different choices will thus observe the same points with different degrees of fuzziness.

A public-key quantum group blind signature scheme based on single-qubit rotations

With the continuous development of quantum technology, the quantum signature as an application of quantum cryptography has received great attention. In this paper, we propose a public-key quantum group blind signature scheme based on single-qubit rotations. In this scheme, the group manager generates a public key. Each group member randomly generates his own private key according to the public key. The signer uses his private key and random sequence to generate the signature. The verifier uses the public key to verify the correctness of the quantum signature. The public and private keys can be reused, which simplifies the key management of the signature system. In this scheme, the random sequence is used to enhance the security of the scheme. At the same time, the quantum efficiency is improved by using single-qubit rotations. The security analysis shows that our scheme can ensure the security of the keys, the unforgeability and the non-deniability of the signature.

Hopf PBW-deformations of a new type quantum group Uq(𝔰𝔩2∗) and deformed preprojective algebras

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

A type C study of Braverman–Gaitsgory–Ginzburg’s construction of sln representations

In [Z. Fan, H. Ma and H. Xiao, Equivariant K-theory approach to [Formula: see text]-quantum groups, Publ. Res. Inst. Math. Sci. 58 (2022), no.3, 635–668], it is proved that the convolution algebra of the top Borel–Moore homology on the Steinberg variety of type [Formula: see text] realizes [Formula: see text], where [Formula: see text] is the fixed point subalgebra of involution on [Formula: see text]. So, the top Borel–Moore homology of the partial Springer fibers gives the representations of [Formula: see text]. In this paper, we study these representations using Schur–Weyl duality and Springer theory.

Duplex Hecke algebras of type B

As a sequel to [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.], in this article we first introduce a so-called duplex Hecke algebras of type [Formula: see text] which is a [Formula: see text]-algebra associated with the Weyl group [Formula: see text] of type [Formula: see text], and symmetric groups [Formula: see text] for [Formula: see text], satisfying some Hecke relations (see Definition 3.1). This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur–Weyl duality of Levi-type (see [B. Shu and Y. Yao, On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate duplex Hecke algebras, with an appendix by B. Liu, submitted (2023)]), extending the duplex Hecke algebra of type [Formula: see text] arising from the related [Formula: see text]-Schur–Weyl duality of Levi-type (see [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.]). A duplex Hecke algebra of type [Formula: see text] admits natural representations on certain tensor spaces. We then establish a Levi-type [Formula: see text]-Schur–Weyl duality of type [Formula: see text], which reveals the double centralizer property between such duplex Hecke algebras and [Formula: see text]quantum groups studied by Bao and Wang in [H. Bao and W. Wang, A new approach to Kazhdan–Lusztig theory of type B via quantum symmetric pairs, Astérisque 402 (2018)].

Tensor ideals for quantum groups via minimal tilting complexes

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.

A Wilson line realization of quantum groups

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 Uħ(g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_\\hbar ({\\mathfrak {g}})$$\\end{document}. We outline a connection of this theory with the moduli spaces of local systems defined by Goncharov and Shen.

Examples of compact quantum groups with [formula omitted] a factor

For each λ∈]0,1] we exhibit an uncountable family of compact quantum groups G such that the von Neumann algebra L∞(G) is the injective factor of type IIIλ with separable predual. We also show that uncountably many injective factors of type III0 arise as L∞(G) for some compact quantum group G. We introduce natural invariants of quantum groups related to the scaling group, modeled on the Connes invariant T for von Neumann algebras. We compute the values of these invariants in our examples, which allows us to distinguish between them. We also investigate their connection to the Connes invariants T(L∞(G)), S(L∞(G)).In the final section we show that for a compact quantum group G the von Neumann algebra L∞(G) cannot have a direct summand of the form B(H) with dim⁡H=∞. In particular, factors of type I (of any dimension) cannot be obtained as L∞(G) for a non-trivial compact quantum group G.

The approximation property for locally compact quantum groups

We study the Haagerup–Kraus approximation property for locally compact quantum groups, generalising and unifying previous work by Kraus–Ruan and Crann. Along the way we discuss how multipliers of quantum groups interact with the C⁎-algebraic theory of locally compact quantum groups. Several inheritance properties of the approximation property are established in this setting, including passage to quantum subgroups, free products of discrete quantum groups, and duals of double crossed products. We also discuss a relation to the weak⁎ operator approximation property. For discrete quantum groups, we introduce a central variant of the approximation property, and relate this to a version of the approximation property for rigid C⁎-tensor categories, building on work of Arano–De Laat–Wahl.

Holomorphically Finitely Generated Hopf Algebras and Quantum Lie Groups

Holomorphically Finitely Generated Hopf Algebras and Quantum Lie Groups

Non‐semisimple Levin–Wen models and Hermitian TQFTs from quantum (super)groups

AbstractWe develop the categorical context for defining Hermitian non‐semisimple topological quantum field theories (TQFTs). We prove that relative Hermitian modular categories give rise to modified Hermitian Witten–Reshetikhin–Turaev TQFTs and provide numerous examples of these structures coming from the representation theory of quantum groups and quantum superalgebras. The Hermitian theory developed here for the modified Turaev–Viro TQFT is applied to define new pseudo‐Hermitian topological phases that can be considered as non‐semisimple analogs of Levin–Wen models.

Tracial states and $\mathbb G$-invariant states of discrete quantum groups

Tracial states and $\mathbb G$-invariant states of discrete quantum groups

A connection between Uq(sl(3)) and Z2×Z2-graded special linear Lie colour algebras via Klein operators

We provide an explicit connection between the quantum group Uq(sl(3)) in the limit q → −1 and a Z2×Z2-graded Lie colour algebra. This is realised through an algebra embedding assisted by a Klein operator. This provides a proof of concept of an explicit method that could lead to constructing more general Z2×Z2-graded Lie colour algebras, and their representations.

U0-free quantum group representations

Let g be a finite dimensional complex simple Lie algebra, and let U=Uq(g) be its quantized enveloping algebra with a triangular decomposition U=U−U0U+. We classify all U-module structures on U0 with the regular action of U0 on itself, which are shown to have direct connections with the q-Weyl algebra and the bounded weight representations. We obtain that the necessary and sufficient condition for the existence of such modules is that U has to be of type An(n≥1), Bn(n≥2) or Cn(n≥3). Moreover, we study their module structures and associated weight modules for each type.

QSPR analysis to predict some quantum chemical properties of 2‐phenylindol derivatives as anticancer drugs using molecular descriptor and genetic algorithm multiple linear regression

AbstractThermodynamic properties of molecules provide fundamental information for binding in fragment‐based drug discovery programs and for understanding complex interactions. Experimental measurements of some thermodynamic properties are not always feasible. Quantum mechanical methods and group contribution methods are the major toolboxes to obtain some theoretical quantities. In drug discovery and development, computational studies can reduce the costs and risks of bringing a new medicine to market. Computational methods have been widely used to optimize promising new ligands by estimating their binding affinity to proteins using the thermodynamic ligand binding parameters (Gibbs energy, enthalpy, entropy, and heat capacity). In this article, the relationship between molecular descriptors and quantum chemical properties, such as the Gibbs free energy (ΔG kJ mol−1) and enthalpy of formation (ΔHf kJ mol−1) of 2‐phenylindole derivatives as anticancer drugs is studied. These properties were calculated at the DFT‐B3LYP/6‐311G (d,p) level of quantum chemistry by Gaussian 09 software. Molecular descriptors were calculated by Dragon 5.4 software, and the stepwise multiple linear regression (MLR) and the Genetic algorithm (GA) techniques were used to select the best descriptors and build QSPR models. To evaluate the predictive ability of developed quantitative structure‐property relationship (QSPR) models, different internal and external validation methods were adopted. The predictive powers of the models were found to be satisfactory. The models revealed that 3D matrix‐based descriptors (H3D, SEig) are useful to predict the Gibbs free energy and enthalpy of formation of the investigated compounds respectively.

Equivariant -correspondences and compact quantum group actions on Pimsner algebras

Abstract Let G be a compact quantum group. We show that given a G-equivariant $\textrm {C}^*$ -correspondence E, the Pimsner algebra $\mathcal {O}_E$ can be naturally made into a G- $\textrm {C}^*$ -algebra. We also provide sufficient conditions under which it is guaranteed that a G-action on the Pimsner algebra $\mathcal {O}_E$ arises in this way, in a suitable precise sense. When G is of Kac type, a KMS state on the Pimsner algebra, arising from a quasi-free dynamics, is G-equivariant if and only if the tracial state obtained from restricting it to the coefficient algebra is G-equivariant, under a natural condition. We apply these results to the situation when the $\textrm {C}^*$ -correspondence is obtained from a finite, directed graph and draw various conclusions on the quantum automorphism groups of such graphs, both in the sense of Banica and Bichon.

Noncommutative spacetimes versus noncommutative spaces of geodesics

The aim of this contribution is twofold. First, we show that when two (or more) different quantum groups share the same noncommutative spacetime, such an ‘ambiguity’ can be resolved by considering together their corresponding noncommutative spaces of geodesics. In any case, the latter play a mathematical/physical role by themselves and, in some cases, they can be interpreted as deformed phase spaces. Second, we explicitly show that noncommutative spacetimes can be reproduced from ‘extended’ noncommutative spaces of geodesics which are those enlarged by the time translation generator. These general ideas are described in detail for the κ-Poincaré and κ-Galilei algebras.

Hopf-Galois extensions and twisted Hopf algebroids

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 Lς 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#σH is a cleft extension and the cocycle-action (▹,σ) is of associative type. We also show in the latter case that L(B#σH,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 Uq(sl2)ˆ and the quantum Weyl group of uq(sl2) as examples of associative type.