### Abstract

Quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts. Mapinduced coproducts are connected to the usual ones by similarity transformations U that may be expressed either in terms of Clebsch-Gordan coefficients, or in a universal operator form. The product of two such U matrices yields the R matrix for a fixed value of the spectral parameter, which bears on the Yang-Baxterization of U as well as R. All this is explicitly illustrated for the tensor product 1/2-j of SU(2)q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a twoparameter manifold. Some observations are made on the general structure of U and R matrices, and of coproduct manifolds, based on the solutions of hierarchies of partial difference equations. Applications of deforming maps and U matrices to the physics of spinchains are outlined.

Original language | English (US) |
---|---|

Pages (from-to) | 676-688 |

Number of pages | 13 |

Journal | Journal of Mathematical Physics |

Volume | 32 |

Issue number | 3 |

DOIs | |

State | Published - 1991 |

### Fingerprint

### Keywords

- ALGEBRAS
- CASIMIR OPERATORS
- CLEBSCHGORDAN COEFFICIENTS
- IRREDUCIBLE REPRESENTATIONS
- MAPPING
- QUANTUM MECHANICS
- SU2 GROUPS
- TRANSFORMATIONS

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*32*(3), 676-688. https://doi.org/10.1063/1.529410

**Quantum algebra deforming maps, Clebsch-Gordan coefficients, coproducts, R and U matrices.** / Curtright, Thomas; Ghandour, G. I.; Zachos, C. K.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 32, no. 3, pp. 676-688. https://doi.org/10.1063/1.529410

}

TY - JOUR

T1 - Quantum algebra deforming maps, Clebsch-Gordan coefficients, coproducts, R and U matrices

AU - Curtright, Thomas

AU - Ghandour, G. I.

AU - Zachos, C. K.

PY - 1991

Y1 - 1991

N2 - Quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts. Mapinduced coproducts are connected to the usual ones by similarity transformations U that may be expressed either in terms of Clebsch-Gordan coefficients, or in a universal operator form. The product of two such U matrices yields the R matrix for a fixed value of the spectral parameter, which bears on the Yang-Baxterization of U as well as R. All this is explicitly illustrated for the tensor product 1/2-j of SU(2)q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a twoparameter manifold. Some observations are made on the general structure of U and R matrices, and of coproduct manifolds, based on the solutions of hierarchies of partial difference equations. Applications of deforming maps and U matrices to the physics of spinchains are outlined.

AB - Quantum algebra deforming maps explicitly define comultiplications that differ from the usual noncocommutative coproducts. Mapinduced coproducts are connected to the usual ones by similarity transformations U that may be expressed either in terms of Clebsch-Gordan coefficients, or in a universal operator form. The product of two such U matrices yields the R matrix for a fixed value of the spectral parameter, which bears on the Yang-Baxterization of U as well as R. All this is explicitly illustrated for the tensor product 1/2-j of SU(2)q using several deforming maps whose coproducts are continuously connected by similarity transformations to form a twoparameter manifold. Some observations are made on the general structure of U and R matrices, and of coproduct manifolds, based on the solutions of hierarchies of partial difference equations. Applications of deforming maps and U matrices to the physics of spinchains are outlined.

KW - ALGEBRAS

KW - CASIMIR OPERATORS

KW - CLEBSCHGORDAN COEFFICIENTS

KW - IRREDUCIBLE REPRESENTATIONS

KW - MAPPING

KW - QUANTUM MECHANICS

KW - SU2 GROUPS

KW - TRANSFORMATIONS

UR - http://www.scopus.com/inward/record.url?scp=0000128197&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000128197&partnerID=8YFLogxK

U2 - 10.1063/1.529410

DO - 10.1063/1.529410

M3 - Article

VL - 32

SP - 676

EP - 688

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 3

ER -