arXiv subjects
Publications and source records attributed to Naihuan Jing.
Imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated, and several results are generalized from the affine Lie algebras. In particular, imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.
Based on the shifted Schensted correspondence and the shifted Knuth equivalence, a shifted analog of the Poirier-Reutenauer algebra as a higher lift of Schur's P-functions and a right coideal subalgebra of the Poirier-Reutenauer algebra is constructed. Its close relations with the peak subalgebra and the Stembridge algebra of peak functions are also uncovered.
Based on a generalized Newton's identity, we construct a family of symmetric functions which deform the modular Hall-Littlewood functions. We also give a determinant formula for the Macdonald functions.
Vertex operator realizations of symplectic and orthogonal Schur functions are studied and expanded. New proofs of determinant identities of irreducible characters for the symplectic and orthogonal groups are given. We also give a new proof of the duality between the universal orthogonal and symplectic Schur functions using vertex operators.
Based on the loop-algebraic presentation of 2-toroidal Lie superalgebras, free field representation of toroidal Lie superalgebras of type $A(m, n)$ is constructed using both vertex operators and bosonic fields.
A notion of Drinfeld polynomials is introduced for modules of two-parameter quantum affine algebras. Finite dimensional representations are then characterized by sets of $l$-tuples of pairs of Drinfeld polynomials with certain conditions.
We construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.
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.
We present computable criterion for completely classifying multi-qubit quantum states under local unitary operations. The criterion can be used to detect whether two quantum states in multi-qubit systems are local unitary equivalent or not. Once obtaining the positive answer, we are further able to compute the corresponding unitary operators precisely. Since the scheme is based on the mean values of some quantum mechanical observables, it supplies an experimental way to judge the local equivalence of quantum states.
We provide a Faddeev-Reshetikhin-Takhtajan's RTT approach to the quantum group Fun(GL_{r,s}(n)) and the quantum enveloping algebra U_{r,s}(gl_n) corresponding to the two-parameter R-matrix. We prove that the quantum determinant det_{r,s}T is a quasi-central element in Fun(GL_{r,s}(n)) generalizing earlier results of Dipper-Donkin and Du-Parshall-Wang. The explicit formulation provides an interpretation of the deforming parameters, and the quantized algebra U_{r,s}(R) is identified to U_{r,s}(gl_n) as the dual algebra. We then construct n-1 quasi-central elements in U_{r,s}(R) which are analogues of higher Casimir elements in U_q(gl_n).
A realization of the affine Lie algebra ${A^{(1)}_1}$ and the relevant $Z$-algebra at negative level $-k$ is given in terms of parafermions. This generalizes the recent work on realization of the affine Lie algebra at the critical level.
We construct a nontrivial set of invariants for any multipartite mixed states under the SLOCC symmetry. These invariants are given by hyperdeterminants and independent from basis change. In particular, a family of d^2 invariants for arbitrary d-dimensional even partite mixed states are explicitly given.
A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method.
The irreducible character values of the spin wreath products of the symmetric group and an arbitrary finite group are completely determined.
A loop-algebraic presentation is given for toroidal Lie superalgebras of classical types. Based on the loop superalgebra presentation free field realizations of toroidal Lie superalgebras are constructed for types $A(m,n)$, $B(m,n)$, C(n) and $D(m,n)$.
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.
An explicit realization of the affine Lie algebra \hat{sl}_2(C) at the critical level is constructed using a mixture of bosons and parafermions. Subsequently a representation of the associated Lepowsky-Wilson Z-algebra is given on a space of the tensor product of bosonic fields and certain semi-infinite wedge products.
We study certain quantum states for which the PPT criterion is both sufficient and necessary for separability. A class of $n\times n$ 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.