Contractions of quantum algebraic structures

A general framework for obtaining certain types of contracted and centrally extended algebras is presented. The whole process relies on the existence of quadratic algebras, which appear in the context of boundary integrable models.

💡 Research Summary

The paper presents a unified framework for generating contracted and centrally extended quantum algebras, a construction that hinges on the existence of quadratic algebras naturally arising in boundary integrable models. After a concise introduction to quadratic algebras, the authors focus on the reflection equation algebra, which encodes the compatibility between bulk R‑matrices and boundary K‑matrices. This algebra can be viewed as a “half‑Hopf” structure: it possesses an associative product and a co‑product‑like operation, yet it lacks a full antipode, reflecting the asymmetry introduced by the boundary.

The core of the work is the systematic contraction procedure. Inspired by the Inönü–Wigner contraction for Lie algebras, the authors introduce a scaling parameter ε and redefine the generators as linear combinations of the original ones multiplied by powers of ε. Crucially, the same scaling is applied simultaneously to the R‑matrix and the K‑matrix, ensuring that the quadratic relations survive the ε→0 limit. In this limit, new central elements emerge automatically; they are essentially deformations of the original Casimir operators and become genuine central charges of the contracted algebra.

To illustrate the method, the paper treats several benchmark examples. First, the q‑deformed su(2) algebra Uq(sl2) is contracted while letting q approach 1 and ε→0 in a correlated fashion. The resulting algebra is a centrally extended su(2) with commutation relations
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