q-Hypergeometric Series and Macdonald Functions

We derive a duality formula for two-row Macdonald functions by studying their relation with basic hypergeometric functions. We introduce two parameter vertex operators to construct a family of symmetric functions generalizing Hall-Littlewood functions. Their relation with Macdonald functions is governed by a very well-poised q-hypergeometric functions of type ₄₃, for which we obtain linear transformation formulas in terms of the Jacobi theta function and the q-Gamma function. The transformation formulas are then used to give the duality formula and a new formula for two-row Macdonald functions in terms of the vertex operators. The Jack polynomials are also treated accordingly.     

A new quantum group associated with a ‘nonstandard’ braid group representation

A new quantum group is derived from a nonstandard braid group representation by employing the Faddeev-Reshetikhin-Takhtajan constructive method. The classical limit is not a Lie superalgebra, despite relations like x ²–y ²=0. We classify all finite-dimensional irreducible representations of the new Hopf algebra and find only one- and two-dimensional ones.     

Quantum Pfaffians and hyper-Pfaffians

The concept of the quantum Pfaffian is rigorously examined and refurbished using the new method of quantum exterior algebras. We derive a complete family of Plücker relations for the quantum linear transformations, and then use them to give an optimal set of relations required for the quantum Pfaffian. We then give the formula between the quantum determinant and the quantum Pfaffian and prove that any quantum determinant can be expressed as a quantum Pfaffian. Finally the quantum hyper-Pfaffian is introduced, and we prove a similar result of expressing quantum determinants in terms of quantum hyper-Pfaffians at modular cases.     

BRST Resolution for the Principally Graded Wakimoto Module of $$\widehat{\mathfrak{s}\mathfrak{l}}_2 $$

BRST resolution is studied for the principally graded Wakimoto module of recently found in math.QA/0005203. The submodule structure is completely determined and irreducible representations can be obtained as the zero-th cohomology group.     

Spin characters of generalized symmetric groups

In 1911 Schur computed the spin character values of the symmetric group using two important ingredients: the first one later became famously known as the Schur Q-functions and the second one was certain creative construction of the projective characters on Clifford algebras. In the context of the McKay correspondence and affine Lie algebras, the first part was generalized to all wreath products by the vertex operator calculus in Frenkel et al. (Duke Math J 111:51–96, 2002) where a large part of the character table was produced. The current paper generalizes the second part and provides the missing projective character values for the wreath product of the symmetric group with a finite abelian group. Our approach relies on Mackey–Wigner’s little groups to construct irreducible modules. In particular, projective modules and spin character values of all classical Weyl groups are obtained.     

Addendum to “Drinfeld Realization of Twisted Quantum Affine Algebras”

We provide necessary details to several arguments that appeared in our previous paper “Drinfeld Realization of Twisted Quantum Affine Algebras,” Commun. Algebra 35 (2007) 3683–3698.     

Higher Sugawara Operators for the Quantum Affine Algebras of Type A

We give explicit formulas for the elements of the center of the completed quantum affine algebra in type A at the critical level that are associated with the fundamental representations. We calculate the images of these elements under a Harish-Chandra-type homomorphism. These images coincide with those in the free field realization of the quantum affine algebra and reproduce generators of the q-deformed classical \({{\mathcal W}}\)-algebra of Frenkel and Reshetikhin.     

A twisted quantum toroidal algebra

As an analog of the quantum TKK algebra, a twisted quantum toroidal algebra of type A ₁ is introduced. Explicit realization of the new quantum TKK algebra is constructed with the help of twisted quantum vertex operators over a Fock space.     

Multipartite Separability of Density Matrices of Graphs

A new layers method is presented for multipartite separability of density matrices from simple graphs. Full separability of tripartite states is studied for graphs on degree symmetric premise. The models are generalized to multipartite systems by presenting a class of fully separable states arising from partially symmetric graphs.     

Vertex operators of level-one U q (B n (1) )-modules

We construct explicitly the level-one vertex operators for the fundamental modulesV(₁) (i=0, 1,n) of the quantum affine algebra of typeB using free boson and fermion fields.