Explicit finite-n lattice correlators for dense polymers on a cylinder are computed via Temperley-Lieb algebra and shown to match ratios of c=-2 CFT correlators involving boundary fields of dimensions -1/8 and 0, with non-abelian fusion.
Standard Modules, Induction and the Temperley-Lieb Algebra
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abstract
The basic properties of the Temperley-Lieb algebra $TL_n$ with parameter $\beta = q + q^{-1}$, for $q$ any non-zero complex number, are reviewed in a pedagogical way. The link and standard (cell) modules that appear in numerous physical applications are defined and a natural bilinear form on the standard modules is used to characterize their maximal submodules. When this bilinear form has a non-trivial radical, some of the standard modules are reducible and $TL_n$ is non-semisimple. This happens only when $q$ is a root of unity. Use of restriction and induction allows for a finer description of the structure of the standard modules. Finally, a particular central element $F_n$ of $TL_n$ is studied; its action is shown to be non-diagonalisable on certain indecomposable modules and this leads to a proof that the radicals of the standard modules are irreducible. Moreover, the space of homomorphisms between standard modules is completely determined. The principal indecomposable modules are then computed concretely in terms of standard modules and their inductions. Examples are provided throughout and the delicate case $\beta = 0$, that plays an important role in physical models, is studied systematically.
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UNVERDICTED 3representative citing papers
Introduces uncoiled affine and periodic Temperley-Lieb algebras as finite quotients and constructs explicit Wenzl-Jones idempotents projecting onto their one-dimensional modules, with Markov trace evaluations expressed via Chebyshev polynomials.
Sandwich cellularity is presented as a version of cell theory for algebras and applied to Hecke algebras plus monoid and diagram algebras.
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Logarithmic correlation functions for critical dense polymers on the cylinder
Explicit finite-n lattice correlators for dense polymers on a cylinder are computed via Temperley-Lieb algebra and shown to match ratios of c=-2 CFT correlators involving boundary fields of dimensions -1/8 and 0, with non-abelian fusion.
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Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors
Introduces uncoiled affine and periodic Temperley-Lieb algebras as finite quotients and constructs explicit Wenzl-Jones idempotents projecting onto their one-dimensional modules, with Markov trace evaluations expressed via Chebyshev polynomials.
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Sandwich cellularity and a version of cell theory
Sandwich cellularity is presented as a version of cell theory for algebras and applied to Hecke algebras plus monoid and diagram algebras.