## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Integrable Systems in Quantum Field Theory and Statistical Mechanics, M. Jimbo, T. Miwa and A. Tsuchiya, eds. (Tokyo: Mathematical Society of Japan, 1989), 567 - 639

### Yang–Baxter Algebras, Conformal Invariant Models and Quantum Groups

#### Abstract

The Yang–Baxter algebras (YBA) are introduced and formulated in a general way stressing graphical methods. Their various physical applications are then exposed: lattice statistical models, integrable field theories and factorizable S-matrices. The Bethe Ansatz (BA) and its generalizations provide the explicit solutions of all these models using the appropiate YBA. The six-vertex model solution is exposed. YB algebras and their associated physical models are classified in terms of simple Lie algebras.

It is exposed how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach.

The method of finite-size calculations from the BA is exposed as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YB algebras.

To conclude braid and quantum groups are derived from the YBA in the limit of infinite spectral parameter.

#### Article information

**Dates**

Received: 16 December 1988

First available in Project Euclid:
17 June 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1529259763

**Digital Object Identifier**

doi:10.2969/aspm/01910567

**Mathematical Reviews number (MathSciNet)**

MR1048606

**Zentralblatt MATH identifier**

0695.17007

#### Citation

Vega, H. J. de. Yang–Baxter Algebras, Conformal Invariant Models and Quantum Groups. Integrable Systems in Quantum Field Theory and Statistical Mechanics, 567--639, Mathematical Society of Japan, Tokyo, Japan, 1989. doi:10.2969/aspm/01910567. https://projecteuclid.org/euclid.aspm/1529259763