Vertex Representations of Toroidal Special Linear Lie Superalgebras

Based on the loop-algebraic presentation of 2-toroidal Lie superalgebras, a free field representation of toroidal Lie superalgebras of type A(m, n) is constructed using both vertex operators and bosonic fields.     

Addendum to “The Structure of Groups of Order 2^3p^2

Tbe present paper is a complement to the paper [1]. It is proved that groups of order 2^37^2 have 44 types.     

On multivariable Zassenhaus formula

We give a recursive algorithm to compute the multivariable Zassenhaus formula e^X1+X2+…+Xn=e^X1eX2…e^Xn∏∞k=2e^Wk and derive an effective recursion formula of Wk.     

Quantum Circuit Realization of Morphological Gradient for Quantum Grayscale Image

In this paper, based on the principle of classical morphology operations, the flat grayscale dilation and erosion operations are proposed for NEQR quantum image model. Furthermore, through combining these two morphology operations, we further realize the morphological gradient operation. As the basis of designing of grayscale morphology operations, a series of quantum circuit designs arepresented, which includes special add one operation UA₁(n) and special subtract one operation US₁(n) both for an n-length qubits sequence, quantum unitary operation U_C, parallel subtractor (PS) module, quantum comparator output the large QCOL and quantum comparator output the small QCOS modules. When designsthe concrete quantum circuit, a sequence of UA₁(n) and US₁(n) modules are used to obtain the quantum image sets based on the shape of specific structuring element. Then, the searching for maximaor minima in a certain space is involved, which can be solved by cascading a series of QCOL and QCOS modules in certain order. Finally, the PS module can be used to calculate the difference of the maxima and minima for producing the morphological gradient. The circuit’s complexity analysis illustrate that our scheme is very lower to the classical morphology operations.

     

q-Wedge Modules for Quantized Enveloping Algebras of Classical Type

We use the fusion construction in twisted quantum affine algebras to obtain a unified method to deform the wedge product for classical Lie algebras. As a by-product we uniformly realize all non-spin fundamental modules for quantized enveloping algebras of classical types, and show that they admit natural crystal bases as modules for the (derived) twisted quantum affine algebra. These crystal bases are parametrized in terms of the q-wedge products.     

Quantum Vertex Representations via Finite Groups¶and the McKay Correspondence

We establish a q-analog of our recent work on vertex representations and the McKay correspondence. For each finite group Γ we construct a Fock space and associated vertex operators in terms of wreath products of $Γ×ℂ^× and the symmetric groups. An important special case is obtained when Γ is a finite subgroup of SU ₂, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of ADE type. Received: 17 August 1999 / Accepted: 2 December 1999     

Sharing quantum nonlocality in star network scenarios

The Bell nonlocality is closely related to the foundations of quantum physics and has significant applications to security questions in quantum key distributions. In recent years, the sharing ability of the Bell nonlocality has been extensively studied. The nonlocality of quantum network states is more complex. We first discuss the sharing ability of the simplest bilocality under unilateral or bilateral POVM measurements, and show that the nonlocality sharing ability of network quantum states under unilateral measurements is similar to the Bell nonlocality sharing ability, but different under bilateral measurements. For the star network scenarios, we present for the first time comprehensive results on the nonlocality sharing properties of quantum network states, for which the quantum nonlocality of the network quantum states has a stronger sharing ability than the Bell nonlocality.