Kang Xue

Extended state model and group approach to new polynomials

The extended Kauffman state model is presented by introducing the annihilation diagrams. This new state model enables the authors to establish a connection between the Akutsu-Wadati type polynomials (1987) and the group approach of Witten (1988).

AKUTZU-WADATI LINK POLYNOMIALS FROM FEYNMAN-KAUFFMAN DIAGRAMS

By employing techniques familiar to particle physicists, we develop Kauffman’s state model for the Jones polynomial, which uses diagrams looking like Feynman diagrams for scattering, into a systematic, diagrammatic approach to new link polynomials. We systematize the ansatz for S matrix by symmetry considerations and find a natural interpretation for CPT symmetry in the context of knot theory. The invariance under Reidemeister moves of type III, II and I can be imposed diagrammatically step by step, and one obtains successively braid group representations, regular isotopy and ambient isotopy invariants from Kauffman’s bracket polynomials. This procedure is explicitiy carried out for the N=3 and 4 cases. N being the number of particle labels (or charges). With appropriate symmetry ansatz and with annihilation and creation included in the S matrix, we have obtained link polynomials which generalize the definition of the Akutzu-Wadati polynomials from closed braids to any oriented knots or links with explicit invariance under Reidemeister moves.

RTT RELATIONS AND REALIZATIONS OF A YANGIAN IN QUANTUM MECHANICS

Abstract Drinfelds commutation relations for Y ( sl (2)) are recast into a more practical form that coincides with the solution of the RTT relations with tr T ( u )-extension. In terms of this form we present the realizations of a Yangian in terms of elementary quantum mechanics. It sheds new light on the physical meaning of Y ( sl (2)).

Trigonometric Yang-Baxterization of coloured R-matrix

For a given coloured solution R( lambda , mu ) of the (quantum) Yang-Baxter equation in appropriate form, the authors present an explicit prescription to generate trigonometric solutions R( lambda , mu ;x) of the Yang-Baxter equation with spectral parameters x. When the given R( lambda , mu ) admits the Birman-Wenzl algebraic structure with colour, they find that two types of the R( lambda , mu ;x) are generated by such R( lambda , mu ). New explicit examples associated with the fundamental representations of An, Bn, Cn and Dn are given in terms of this prescription.

The braid group representations associated with some non-fundamental representations of Lie algebras

The braid group representations associated with the eight-dimensional representation of B3 and six-dimensional representation of A2, including the quantum Lie algebra connected with the latter, are explicitly calculated.

NEW SOLUTIONS OF YANG-BAXTER EQUATIONS - BIRMAN-WENZL ALGEBRA AND QUANTUM GROUP STRUCTURES

New braid group representations associated with the fundamental representations of Bn, Cn and Dn are derived. They are shown to satisfy Birman-Wenzl algebra. By Yang-Baxterization the authors obtain new solutions of Yang-Baxter equations, including the twisted extensions. Quantum group structures related to these new solutions are explicitly given.

Greater understanding of the hydrogen atom: RTT-integrability and Yangian symmetry

Abstract We show that the hydrogen atom possesses Yangian symmetry and integrability in terms of the RTT relation. A possible experimental test of the physical effect of Yangian is proposed.

New R-matrices for the Yang-Baxter equation associated with the representations of the quantum superalgebra Uqosp(1, 2) with q a root of unity☆

Abstract Both the highest weight representations and the cyclic representation of the quantum superalgebra Uqosp(1, 2)=A are explicitly constructed in this letter. By making use of the obtained finite dimensional representations with highest weights, the new R-matrices (solutions) of the Yang-Baxter equation are obtained through the universal R-matrix. In the main discussion of this paper emphasis is placed on the case that q is root of unity.

New solutions of the Yang-Baxter equation without additivity, and its coloured interpretation

General forms of 4*4 solutions of the Yang-Baxter equation, without the additivity of spectral parameters, are obtained. By making a parameterization of the colours in terms of the spectral parameters the authors shed new light on the coloured solution of the Yang-Baxter equation.

The Haldane - Shastry model and RTT relations

A new approach is presented to derive the Hamiltonian family of spin- Haldane - Shastry model based on the RTT relation and Yangian symmetry. We show that the first members of the Hamiltonian family are related to the quantum determinant of the transfer matrix T.