Construction of new bornological quantum groups based on Galois objects

Let A be a bornological quantum group and R a bornological algebra. If R is an essential A-module, then there is a unique extension to M（A）-module with 1x = x. There is a one-to-one corresponding relationship between the actions of A and the coactions of . If R is a Galois object for A, then there exists a faithful δ-invariant functional on R. Moreover,the Galois objects also have modular properties such as algebraic quantum groups. By constructing the comultiplication Δ,counit ε, antipode S and invariant functional φ onR×R, R×R can be considered as a bornological quantum group.

Differential Operator Approach to■quantum Groups and Their Oscillator Representations

For a quasi-split Satake diagram,we define a modified q-Weyl algebra,and show that there is an algebra homomorphism between it and the corresponding■quantum group.In other words,we provide a differential operator approach to■quantum groups.Meanwhile,the oscillator representations of■quantum groups are obtained.The crystal basis of the irreducible subrepresentations of these oscillator representations are constructed.

Quantum Groups by Ore Extensions Associated withGroup Algebras

As a continuation of the work of Beattie on quantum groups constructed byOre extensions, in this paper, we characterize their centre and discuss the category ofquantum Yang-Baxter modules over them. In addition, we determine all finite dimen-sional irreducible representations over these quantum groups.

Categorification of Integrable Representations of Quantum Groups

We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac–Moody algebra. As a byproduct, we obtain a geometric realization of Lusztig＇s canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties.

Equitable Presentations for Multiparameter Quantum Groups

In this paper, we give an equitable presentation for the multiparameter quantum group associated to a symmetrizable Kac–Moody Lie algebra, which can be regarded as a natural generalization of the Terwilliger's equitable presentation for the one-parameter quantum group.

q-Deformation of W (2 , 2) Lie Algebra Associated with Quantum Groups

The q-deformation of W（2, 2） Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the q-deformation of W（2, 2） Lie algebra are completely determined. Finally, the 1-dimensional central extension of the q-deformed W（2, 2） Lie algebra is studied, which turns out to be coincided with the conventional W（2, 2） Lie algebra in the q → 1 limit.

On the centers of quantum groups of A_n-type

Let g be the finite dimensional simple Lie algebra of type A_n, and let U = U_q(g,Λ)and U= U_q(g,Q)be the quantum groups defined over the weight lattice and over the root lattice, respectively. In this paper, we find two algebraically independent central elements in U for all n ≥ 2 and give an explicit formula of the Casimir elements for the quantum group U = U_q(g,Λ), which corresponds to the Casimir element of the enveloping algebra U(g). Moreover, for n = 2 we give explicitly generators of the center subalgebras of the quantum groups U = U_q(g,Λ) and U = U_q(g,Q).

Representations of deformed preprojective algebras and quantum groups

Let (Г, I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group ? d . In this paper, we list all indecomposable representations of (θ, I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Пλ(Г, I). It is shown that those representations given by extending indecomposable representations of (Г, I) are all simple representations of Пλ(Г, I). Therefore, it is concluded that all simple representations of restricted quantum group ū q (sl 2) are realized in terms of deformed preprojective algebra.

Effects of quantum quench on entanglement dynamics in antiferromagnetic Ising model

We study the relationship between quench dynamics of entanglement and quantum phase transition in the antiferromagnetic Ising model with the Dzyaloshinskii–Moriya(DM)interaction by using the quantum renormalization-group method and the definition of negativity.Two types of quench protocols(i)adding the DM interaction suddenly and(ii)rotating the spins around x axis are considered to drive the dynamics of the system,respectively.By comparing the behaviors of entanglement in both types of quench protocols,the effects of quench on dynamics of entanglement are studied.It is found that there is the same characteristic time at which the negativity firstly reaches its maximum although the system shows different dynamical behaviors.Especially,the characteristic time can accurately reflect the quantum phase transition from antiferromagnetic to saturated chiral phases in the system.In addition,the correlation length exponent can be obtained by exploring the nonanalytic and scaling behaviors of the derivative of the characteristic time.

THE CANONICAL BASIS FOR THE QUANTUM GROUP OF TYPE B_2

We use the Ringel-Hall algebra approach to study the canonical basis elements for the quantum group of type B2 which are characterized in Xi [12]. However, our approach simplifies several computations there.

Multi-proxy quantum group signature scheme with threshold shared verification

A multi-proxy quantum group signature scheme with threshold shared verification is proposed. An original signer may authorize a proxy group as his proxy agent. Then only the cooperation of all the signers in the proxy group can generate the proxy signature on behalf of the original signer. In the scheme, any t or more of n receivers can verify the message and any t - 1 or fewer receivers cannot verify the validity of the proxy signature.

Cluster algebra structure on the finite dimensional representations of affine quantum group U_q(_3)

In this paper, we prove one case of conjecture given by Hemandez and Leclerc. We give a cluster algebra structuure on the Grothendieck ring of a full subcategory of the finite dimensional representations of affine quantum group Uq（A3）. As a conclusion, for every exchange relation of cluster algebra, there exists an exact sequence of the full subcategory corresponding to it.

Tunable emission of cadmium-free transition metal(Cu, Mn, Ag) co-doped ZnInS/ZnS core-shell quantum dots

Color tunable quantum dots（QDs） based on the Cu, Mn, Ag co-doped Zn In S core and Zn S outer-shell were synthesized by using an eco-friendly method. Core-shell doped QDs with the average size of 3.85 nm were obtained by using a one-pot synthesis followed by a hot injection with n-dodecanethiol（DDT） and oleylamine（OLA） as stabilizers in oil phase. Cu, Mn and Ag ions were introduced as single-dopant or co-dopants during the synthesis, providing an effective means to control the emission color of the QDs. The as-synthesized QDs showed photoluminescence emission ranging from green（530 nm） to near-red（613 nm）, adjusted by doping components, dopant concentration, and Zn/In ratio. Importantly, quasi-white emission has been achieved by controlling the concentration of co-doped metal ions（Mn, Cu and Ag）. The primary results demonstrated the promising potential of co-doped QDs as alternative materials for future high quality white LED applications.

Quantum Group Symmetry in Hofstadter Problem

We show that there is a quantum Sl_q(2) group symmetry in Hofstadter problem on square lattice.The cyclic representation of the quantum group is discussed and its application for computing the degeneracy density of the model is shown.

A chaos-based quantum group signature scheme in quantum cryptosystem

A quantum group signature(QGS) scheme is proposed on the basis of an improved quantum chaotic encryption algorithm using the quantum one-time pad with a chaotic operation string. It involves a small-scale quantum computation network in three phases, i.e. initializing phase, signing phase and verifying phase. In the scheme, a member of the group signs the message on behalf of the group while the receiver verifies the signature's validity with the aid of the trusty group manager who plays a crucial role when a possible dispute arises. Analysis result shows that the signature can neither be forged nor disavowed by any malicious attackers.

Doubled Hecke Algebras and Related Quantum Schur Duality

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.

Capelli identity on multiparameter quantum linear groups

A quantum Capelli identity is given on the multiparameter quantum general linear group based on the(p_(ij),u)-condition. The multiparameter quantum Pfaffian of the(p_(ij), u)-quantum group is also introduced and the transformation under the congruent action is given. Generalization to the multiparameter hyper-Pfaffian and relationship with the quantum minors are also investigated.

Convex PBW-type Lyndon Bases and Restricted Two-parameter Quantum Group of Type F_(4)

The convex PBW-type Lyndon basis for two-parameter quantum group U_(r,s)(F_(4))is given.Assume thatrs^(-1)is a primitive l-th root of unity with l odd,then the restricted quantum group u_(r,s)(F_(4))as a quotient of U_(r,s)(F_(4))is pointed,and of a Drinfel’d double structure under a certain condition.All of Hopf isomorphisms of u_(r,s)(F_(4))are determined,and the necessary and sufficient condition for u_(r,s)(F_(4))to be a ribbon Hopf algebra is singled out by describing the left and right integrals.

Explicit Generators of the Centre of the Quantum Group

A finite generating set of the centre of any quantum group is obtained,where the generators are given by an explicit formulae.For the slightly generalised version of the quantum group which we work with,we show that this set of generators is algebraically independent,thus the centre is isomorphic to a polynomial algebra.

The Heisenberg double of the quantum Euclidean group and its representations

The Heisenberg double D_(q)(E_(2))of the quantum Euclidean group O_(q)(E_(2))is the smash product of O_(q)(E_(2))with its Hopf dual U_(q)(e_(2)).For the algebra D_(q)(E_(2)),explicit descriptions of its prime,primitive and maximal spectra are obtained.All the prime factors of D_(q)(E_(2))are presented as generalized Weyl algebras.As a result,we obtain that the algebra D_(q)(E_(2))has no finite-dimensional representations,and D_(q)(E_(2))cannot have a Hopf algebra structure.The automorphism groups of the quantum Euclidean group and its Heisenberg double are determined.Some centralizers are explicitly described via generators and defining relations.This enables us to give a classification of simple weight modules and the so-called a-weight modules over the algebra D_(q)(E_(2)).