On quantum toroidal algebra of type A1

In this paper we introduce a new quantum algebra which specializes to the 2-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization.     

On vertex operator realizations of Jack functions

On the vertex operator algebra associated with a rank one lattice we derive a general formula for products of vertex operators in terms of generalized homogeneous symmetric functions. As an application we realize Jack symmetric functions of rectangular shapes as well as marked rectangular shapes.     

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 Pfaffan of the (p _ij , u)-quantum group is also introduced and the transformation under the congruent action is given. Generalization to the multiparameter hyper-Pfaffan and relationship with the quantum minors are also investigated.     

Quantum Permanents and Hafnians via Pfaffians

Quantum determinants and Pfaffians or permanents and Hafnians are introduced on the two-parameter quantum general linear group. Fundamental identities among quantum Pf, Hf, and det are proved in the general setting. We show that there are two special quantum algebras among the quantum groups, where the quantum Pfaffians have integral Laurent polynomials as coefficients. As a consequence, the quantum Hafnian is computed by a closely related quantum permanent and identical to the quantum Pfaffian on this special quantum algebra.     

A note on state decomposition independent local invariants

We present a method to change the fluorescence contrast between two dyes with high fluorescence anisotropy in the same solution under different polarization directions by using polarization-shaped pulses. First a theoretical relationship between contrast, anisotropy and polarization is derived for this case. Then the wavelength dependent polarization anisotropy and the change of contrast by phase-shaping for a solution of Rhodamine B and Stilbene 3 in glycerol is shown. Finally, the application of the polarization-shaped pulses is demonstrated.     

Separable State Decompositions for a Class of Mixed States

We study certain quantum states for which the PPT criterion is both sufficient and necessary for separability. A class of n×n bipartite mixed states is presented and the conditions of PPT for these states are derived. The separable pure state decompositions of these states are explicitly constructed when they are PPT.     

Principal Realization of Twisted Yangian {Y(\mathfrak{g}_{N})}

We give the principal realization of the twisted Yangians of orthogonal and symplectic types. The new bases are interpreted in terms of discrete Fourier transform over the cyclic group ${\mathbb Z_N}$ .