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The ghost algebra and the dilute ghost algebra

Authors
  • Nurcombe, Madeline
Type
Published Article
Journal
Journal of Statistical Mechanics: Theory and Experiment
Publisher
IOP Publishing
Publication Date
Feb 13, 2024
Volume
2024
Issue
2
Identifiers
DOI: 10.1088/1742-5468/ad1be6
Source
ioppublishing
Keywords
Disciplines
  • Paper section: Quantum statistical physics, condensed matter, integrable systems
License
Unknown

Abstract

We introduce the ghost algebra, a two-boundary generalisation of the Temperley–Lieb (TL) algebra, using a diagrammatic presentation. The existing two-boundary TL algebra has a basis of string diagrams with two boundaries, and the number of strings connected to each boundary must be even; in the ghost algebra, this number may be odd. To preserve associativity while allowing boundary-to-boundary strings to have distinct parameters according to the parity of their endpoints, as seen in the one-boundary TL algebra, we decorate the boundaries with bookkeeping dots called ghosts. We also introduce the dilute ghost algebra, an analogous two-boundary generalisation of the dilute TL algebra. We then present loop models associated with these algebras, and classify solutions to their boundary Yang–Baxter equations, given existing solutions to the Yang–Baxter equations for the TL and dilute TL models. This facilitates the construction of a one-parameter family of commuting transfer tangles, making these models Yang–Baxter integrable.

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