# Quantum and super-quantum group related to the Alexander-Conway polynomial

@article{Majid1993QuantumAS, title={Quantum and super-quantum group related to the Alexander-Conway polynomial}, author={Shahn Majid and M. J. Rodr{\'i}guez‐Plaza}, journal={Journal of Geometry and Physics}, year={1993}, volume={11}, pages={437-443} }

We describe the quasitriangular structure (universal R-matrix) on the nonstandard quantum group Uq(H1, H2, X±) associated to the Alexander-Conway matrix solution of the Yang-Baxter equation. We show that this Hopf algebra is connected with the super-Hopf algebra Uqgl(1 | 1) by a general process of superization.

#### 4 Citations

Nonstandard quantum groups and superization

- Mathematics, Physics
- 1995

We obtain the universal R‐matrix of the nonstandard quantum group associated to the Alexander–Conway knot polynomial. We show further that this nonstandard quantum group is related to the… Expand

Quasidiagonal Solutions of the Yang–Baxter Equation, Quantum Groups and Quantum Super Groups

- Mathematics
- 1998

This paper answers a few questions about algebraic aspects of bialgebras, associated with the family of solutions of the quantum Yang–Baxter equation in Acta Appl. Math. 41 (1995), pp. 57–98. We… Expand

Spin chains viewed as the fermionic parts of supersymmetric quantum mechanical models

- Physics
- 1995

Abstract This paper shows that (anisotropic) spin chains of XY type arise by matrix representation of the fermionic terms of hamiltonians of quantum systems whose supersymmetry involves in general no… Expand

Algebras and Hopf Algebras in Braided Categories

- Mathematics
- 1995

This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise super-algebras and super-Hopf algebras, aswell as colour-Lie… Expand

#### References

SHOWING 1-10 OF 10 REFERENCES

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

- Mathematics
- 1991

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,… Expand

Quantum Groups

- Mathematics
- 1994

Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups… Expand

Quasitriangular Hopf Algebras and Yang-Baxter Equations

- Physics
- 1990

This is an informal introduction to the theory of quasitriangular Hopf algebras and its connections with physics. Basic properties and applications of Hopf algebras and Yang-Baxter equations are… Expand

Free fermions and the Alexander-Conway polynomial

- Mathematics
- 1991

AbstractWe show how the Conway Alexander polynomial arises from theq deformation of (Z2 graded)sl(n, n) algebras. In the simplestsl(1, 1) case we then establish connection between classical knot… Expand

Transmutation theory and rank for quantum braided groups

- Mathematics
- 1993

Let f: H 1 → H 2 be any pair of quasitriangular Hopf algebras over k with a Hopf algebra map f between them. We construct in this situation a quasitriangular Hopf algebra B ( H 1 , f , H 2 ) in the… Expand

Quantization of Lie Groups and Lie Algebras

- Mathematics
- 1987

Publisher Summary
This chapter focuses on the quantization of lie groups and lie algebras. The Algebraic Bethe Ansatz—the quantum inverse scattering method—emerges as a natural development of the… Expand

Quantum Groups

- 1993

This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions… Expand

New families of commuting transfer matrices in q state vertex models

- Physics
- 1981

Abstract New exactly solvable two-dimensional q -state vertex models are given which generalize the ( q = 2) eight-vertex model, using the method of commuting transfer matrices. Associated… Expand

Rodríguez-Plaza Universal R-matrix for a non-standard quantum group and superization

- Rodríguez-Plaza Universal R-matrix for a non-standard quantum group and superization
- 1991