Quantum Group Research Articles

ON THE QUANTUM STUCTURE OF THE UNIVERSAL ENVELOPING ALGEBRA OF THE LIE ALGEBRA ST (2)

The structure of Hopf co-Poisson algebra on the universal enveloping algebra U(S7(2)) of Lie algebra S7(2) is determined with the help of a solution of the Yang-Baxter equation. Using this solution, a bracket on the dual space of Lie algebra ST(2) is also determined. This cobracket on ST(2) induces a deformation of the universal enveloping algebra U(S7(2)) which has a Hopf algebra structure, as we shall verify. This Hopf algebra is called the quantum group associated to a universal enveloping algebra.

Full runner removal theorem for Ariki-Koike algebras

We consider the representation theory of the Ariki-Koike algebra, a q-deformation of the group algebra of the complex reflection group Cr≀Sn. We define the addition of a runner full of beads for the abacus display of a multipartition and investigate some combinatorial properties of this operation. We focus our attention on the v-decomposition numbers, i.e. the polynomials arising from the Fock space representation of the quantum group Uv(slˆe). Using Fayers' LLT-type algorithm for Ariki-Koike algebras, we relate v-decomposition numbers for different values of e for the class of e-multiregular multipartitions, by adding a full runner of beads to each component of the abacus displays for the labelling multipartitions.

Functional Bethe Ansatz for a sinh-Gordon Model with Real q

Recently, Bazhanov and Sergeev have described an Ising-type integrable model which can be identified as a sinh-Gordon-type model with an infinite number of states but with a real parameter q. This model is the subject of Sklyanin’s Functional Bethe Ansatz. We develop in this paper the whole technique of the FBA which includes: (1) Construction of eigenstates of an off-diagonal element of a monodromy matrix. The most important ingredients of these eigenstates are the Clebsh-Gordan coefficients of the corresponding representation. (2) Separately, we discuss the Clebsh-Gordan coefficients, as well as the Wigner’s 6j symbols, in details. The later are rather well known in the theory of 3D indices. Thus, the Sklyanin basis of the quantum separation of variables is constructed. The matrix elements of an eigenstate of the auxiliary transfer matrix in this basis are products of functions satisfying the Baxter equation. Such functions are called usually the Q-operators. We investigate the Baxter equation and Q-operators from two points of view. (3) In the model considered the most convenient Bethe-type variables are the zeros of a Wronskian of two well defined particular solutions of the Baxter equation. This approach works perfectly in the thermodynamic limit. We calculate the distribution of these roots in the thermodynamic limit, and so we reproduce in this way the partition function of the model. (4) The real parameter q, which is the standard quantum group parameter, plays the role of the absolute temperature in the model considered. Expansion with respect to q (tropical expansion) gives an alternative way to establish the structure of the eigenstates. In this way we classify the elementary excitations over the ground state.

Quantum group intertwiner space from quantum curved tetrahedron

Quantum group intertwiner space from quantum curved tetrahedron

Topological boundaries of representations and coideals

For a locally compact quantum group G, a (left) coideal is a (left) G-invariant von Neumann subalgebra of L∞(G). We introduce and analyze various generalizations of amenability and coamenability to coideals of discrete and compact quantum groups. We focus on a particular class of coideals found in the category of compact quantum groups, which are associated with a compact quasi-subgroup. This class includes all coideals of the quotient type. We also introduce the notion of a Furstenberg-Hamana boundary for representations of discrete quantum groups and use it to study amenability and coamenability properties of coideals. We then prove that a coideal of a compact quantum group that is associated with a compact quasi-subgroup is coamenable if and only if its codual coideal is G-injective. If G is a unimodular or an exact discrete quantum group, we can replace G-injectivity in the latter statement with the weaker condition of relative amenability. This result leads to a complete characterization of the unique trace property. Specifically, a unimodular discrete quantum group G has the unique trace property if and only if the action of G on its noncommutative Furstenberg boundary is faithful. We also demonstrated that if a unimodular discrete quantum group G is C⁎-simple then it has the unique trace property. These findings are the quantum analogs of the groundbreaking results of Breuillard, Kalantar, Kennedy, and Ozawa and they provide answers to questions posted by Kalantar, Kasprzak, Skalski, and Vergnioux.

A Fock space approach to socle series of Grothendieck group over Lie superalgebra o s p ( 2 | 2 )

We explore two modules F and K over a subalgebra U of the quantum group U q ( g l ( ∞ ) ) . 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 o s p ( 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 .

On Borel subalgebras of quantum groups

For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra. Besides the positive part of the quantum group and its reflections, we find new unfamiliar Borel subalgebras, for example, ones containing copies of the quantum Weyl algebra. Given a Borel subalgebra, we study its induced (Verma-)modules and prove among others that they have all irreducible finite-dimensional modules as quotients. We give two structural conjectures involving the associated graded right coideal subalgebra, which we prove in certain cases. In particular, they predict the shape of all triangular Borel subalgebras. As examples, we determine all Borel subalgebras of [Formula: see text] and [Formula: see text] and discuss the induced modules.

Hofstadter-Toda spectral duality and quantum groups

The Hofstadter model allows to describe and understand several phenomena in condensed matter such as the quantum Hall effect, Anderson localization, charge pumping, and flat-bands in quasiperiodic structures, and is a rare example of fractality in the quantum world. An apparently unrelated system, the relativistic Toda lattice, has been extensively studied in the context of complex nonlinear dynamics, and more recently for its connection to supersymmetric Yang-Mills theories and topological string theories on Calabi-Yau manifolds in high-energy physics. Here we discuss a recently discovered spectral relationship between the Hofstadter model and the relativistic Toda lattice which has been later conjectured to be related to the Langlands duality of quantum groups. Moreover, by employing similarity transformations compatible with the quantum group structure, we establish a formula parametrizing the energy spectrum of the Hofstadter model in terms of elementary symmetric polynomials and Chebyshev polynomials. The main tools used are the spectral duality of tridiagonal matrices and the representation theory of the elementary quantum group.

Quantization of the Rank Two Heisenberg–Virasoro Algebra

Quantum groups occupy a significant position in both mathematics and physics, contributing to progress in these fields. It is interesting to obtain new quantum groups by the quantization of Lie bialgebras. In this paper, the quantization of the rank two Heisenberg–Virasoro algebra by Drinfel’d twists is presented, Lie bialgebra structures of which have been investigated by the authors recently.

Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates

We investigate the relations between the (completely bounded) local Coulhon-Varopoulos dimension and the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) Lp-spaces associated to finite measure spaces. More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension d exceeds the spectral dimension, i.e. that the associated Hodge-Dirac operator is d+-summable. We explore different settings to compare these two values: compact Riemannian manifolds, compact Lie groups, sublaplacians, metric measure spaces, noncommutative tori and quantum groups. Specifically, we prove that, while very often equal in smooth compact settings, these dimensions can diverge. Finally, we show that the existence of a symmetric sub-Markovian semigroup on a von Neumann algebra with finite completely bounded local Coulhon-Varopoulos dimension implies that the von Neumann algebra is necessarily injective.

Quantum $\operatorname{SL}(2)$ and logarithmic vertex operator algebras at $(p,1)$-central charge

We provide a ribbon tensor equivalence between the representation category of small quantum \operatorname{SL}(2) , at parameter q=e^{\pi i/p} , and the representation category of the triplet vertex operator algebra at integral parameter p>1 . We provide similar quantum group equivalences for representation categories associated to the Virasoro and singlet vertex operator algebras at central charge c=1-6(p-1)^{2}/p . These results resolve a number of fundamental conjectures coming from studies of logarithmic CFTs in type A_{1} .

Unrestricted Quantum Moduli Algebras, II: Noetherianity and Simple Fraction Rings at Roots of 1

We prove that the quantum graph algebra and the quantum moduli algebra associated to a punctured sphere and complex semisimple Lie algebra $\mathfrak{g}$ are Noetherian rings and finitely generated rings over $\mathbb{C}(q)$. Moreover, we show that these two properties still hold on $\mathbb{C}\big[q,q^{-1}\big]$ for the integral version of the quantum graph algebra. We also study the specializations $\mathcal{L}_{0,n}^\epsilon$ of the quantum graph algebra at a root of unity $\epsilon$ of odd order, and show that $\mathcal{L}_{0,n}^\epsilon$ and its invariant algebra under the quantum group $U_\epsilon(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degrees that we compute.

Analyzing the impact of acupuncture in conjunction with a Quantum lipid-lowering device on hyperlipidemic rats: insights into adenosine monophosphate-activated protein kinase signaling, blood lipids, and gut microbiota.

We explored the impact of acupuncture (ACUP) in conjunction with a quantum lipid-lowering device (Quantum) on the blood lipids and gut microbiota in hyperlipidemic rats, focusing on the adenosine monophosphate- (AMP)-activated protein kinase (AMPK) signaling pathway. Fifty Sprague-Dawley rats were randomly allocated into five groups: Normal, Model, Acup + Quantum, Acup, and Quantum. Hyperlipidemic models were established in all groups except Normal. The Model group did not receive any intervention after modeling. The Acup + Quantum group received both treatments, the Acup group received only acupuncture, and the Quantum group received only the quantum lipid-lowering device. We used ELISA to measure serum lipid and liver enzyme levels, hematoxylin and eosin (HE) staining for liver pathology, Western blot for protein expression, and 16S rRNA sequencing to analyze intestinal microbiota diversity in rats. Elisa results showed that compared with the model group, Acup + Quantum group could reduce the levels of total cholesterol (TC), triglycerides (TG), low-density lipoprotein-cholesterol (LDL-C), aspartate transaminase (AST) and aspartate transaminase (ALT) in rats with hyperlipidemia (P<0.01), and increase the level of high-density lipoprotein-cholesterol (HDL-C) (P<0.01). HE staining results showed that compared with the model group, the hepatocytes of rats in the Acup + Quantum group looked round and full, the liver plates were arranged regularly and neatly, and there was no obvious abnormality in the liver sinusoids. Western blot results showed that compared with the model group, the Acup + Quantum group inhibited AMPK activation, increased P-AMPK/AMPK protein expression (P<0.05), and decreased phospho-acetyl-CoA carboxylases (P-ACC/ACC), Sterol regulatory element-binding transcription factor-1C (SREBP-1C), and FAS protein expression (P<0.05; P<0.01; P<0.01), which resulted in lipid-lowering effect. The results of intestinal flora showed that Acup + Quantum group improved the intestinal microbial microenvironment of hyperlipidemic rats by regulating the structure of intestinal microflora, increasing the abundance of Firmicutes flora, and decreasing the abundance of harmful bacteria, such as Bacteroidetes and Proteobacteria. Acupuncture combined with quantum lipid-lowering device can improve the blood lipid and liver function levels and regulate the intestinal microbial microenvironment of hyperlipidemic rats. This therapeutic outcome is likely achieved through the activation of the AMPK pathway and the beneficial modulation of the intestinal microbiota of rats.

Actions of Compact and Discrete Quantum Groups on Operator Systems

Abstract We introduce the notion of an action of a discrete or compact quantum group on an operator system, and study equivariant operator system injectivity. We then prove a duality result that relates equivariant injectivity with dual injectivity of associated crossed products. As an application, we give a description of the equivariant injective envelope of the reduced crossed product built from an action of a discrete quantum group on an operator system.

A new quantum group blind signature scheme based on GHZ states

With the development of quantum computers, traditional communication security based on classical computers can no longer guarantee unconditional security. At the same time, a series of encrypted communication solutions that rely on quantum mechanical properties have been spawned. This article focuses on the field of quantum signatures, based on the entangled Greenberger–Horne–Zeilinger state, combining group signatures (any member of the group can sign on behalf of the group) and blind signatures (the signer cannot know the accuracy of the signature content) proposed a new quantum group blind signature scheme that utilizes quantum key distribution to achieve unconditional security. By splitting the original message, the scheme strengthens the relevance of the signature participants and makes the signature to have stronger binding. Then, the security of the scheme against internal and external attacks is analyzed. In this scheme, a pleasing efficiency has been achieved, and the non-repudiation of signatures has been enhanced. At the same time, disputed signatures can be traced back to the signer.

Permanence of the torsion-freeness property for divisible discrete quantum subgroups

We prove that torsion-freeness in the sense of Meyer-Nest is preserved under divisible discrete quantum subgroups. As a consequence, we obtain some stability results of the torsion-freeness property for relevant constructions of quantum groups (quantum (semi-)direct products, compact bicrossed products and quantum free products). We improve some stability results concerning the Baum-Connes conjecture appearing already in a previous work of the author. For instance, we show that the (resp. strong) Baum-Connes conjecture is preserved by discrete quantum subgroups (without any torsion-freeness or divisibility assumption).

Multi-attribute quantum group interference decision model considering minimum constraint and information entropy

About multi-attribute group decision making process, the opinions of different decisions will interfere with each other, or be affected by the decision-making environment, resulting in an unobserved interference effect, which is regarded as quantum entanglement in quantum theory. In this paper, an interference effect model based on minimum constraint and information entropy is proposed to measure the interference value in quantum decision models. Firstly, considering the interference effects of decision makers’ opinions, the evaluation opinions of decision makers are aggregated using Quantum-like Bayesian Network (QLBN), which are regarded as wave functions superimposed in the quantum model. Secondly, due to the uncertainty of decision process, information entropy and minimum constraint are combined to solve the interference effect. The minimum constraint is used to limit the range of interference values and the information entropy represents the uncertainty in the decision process. The grey relational degree represents the conditional probability of the decision maker about the alternatives. Thirdly, the weights of attributes and the probabilities of the alternatives are aggregated about decision makers by the proposed interference effect model, and then the alternatives are ranked by the aggregated probabilities. Subsequently, a numerical example is used to verify the validity and rationality of the model. Comparative analyses and sensitivity analyses are also implemented to demonstrate the stability and reliability of our method.

A remark on the Hochschild dimension of liberated quantum groups

Let A be a Hopf algebra equipped with a projection onto the coordinate Hopf algebra \mathcal{O}(G) of a semisimple algebraic group G . It is shown that if A admits a suitably non-degenerate comodule V and the induced G -module structure of V is non-trivial, then the third Hochschild homology group of A is non-trivial.

Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group

Let $G$ be a reductive group, and let $\check {G}$ be its Langlands dual group. Arkhipov and Bezrukavnikov proved that the Whittaker category on the affine flags ${\operatorname {Fl}}_G$ is equivalent to the category of $\check {G}$ -equivariant quasi-coherent sheaves on the Springer resolution of the nilpotent cone. This paper proves this theorem in the quantum case. We show that the twisted Whittaker category on ${\operatorname {Fl}}_G$ and the category of representations of the mixed quantum group are equivalent. In particular, we prove that the quantum category $\mathsf {O}$ is equivalent to the twisted Whittaker category on ${\operatorname {Fl}}_G$ in the generic case. The strong version of our main theorem claims a motivic equivalence between the Whittaker category on ${\operatorname {Fl}}_G$ and a factorization module category, which holds in the de Rham setting, the Betti setting, and the $\ell$ -adic setting.

A quantum group decision model for meteorological disaster emergency response based on D-S evidence theory and Choquet integral

In addressing complex and dynamic meteorological disaster decision-making environment, the traditional multi-attribute group decision-making domain model is often unable to effectively deal with the correlation between attributes and the mutual influence of group opinions. To overcome this challenge, this paper proposes a novel quantum framework for group decision-making, which is used to deal with the emergency situation of meteorological disaster. The model initially characterizes attribute correlations using 2-additive Choquet integrals and employs Dempster-Shafer evidence theory to both integrate information and ascertain attribute weights, and decision makers' weights are calculated based on grey relative correlation. On this basis, a quantum-like Bayesian network is developed to capture the interference among decision-makers' opinions. The alternatives are ranked by quantum probabilities computed based on Bayesian principle. Finally, a case study on meteorological disaster emergency scenario assessment is conducted to validate the proposed model's effectiveness and superiority. Additionally, its stability and practicality are confirmed through sensitivity analysis and comparative analysis.