Fusion in the periodic Temperley–Lieb algebra: General definition of a bifunctor
Abstract
<jats:p> The periodic Temperley–Lieb category consists of connectivity diagrams drawn on a ring with <jats:inline-formula> <jats:alternatives> <jats:tex-math>N</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math>N'</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>N</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> nodes on the outer and inner boundary, respectively. We consider families of modules, namely sequences of modules <jats:inline-formula> <jats:alternatives> <jats:tex-math>\mathsf{M}(N)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi mathvariant="sans-serif">𝖬</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> over the enlarged periodic Temperley–Lieb algebra for varying values of <jats:inline-formula> <jats:alternatives> <jats:tex-math>N</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> , endowed with an action <jats:inline-formula> <jats:alternatives> <jats:tex-math>\mathsf{M}(N') \to \mathsf{M}(N)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi mathvariant="sans-serif">𝖬</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>→</mml:mo> <mml:mi mathvariant="sans-serif">𝖬</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of the diagrams. Examples of modules that can be organised into families are those arising in the RSOS model and in the XXZ spin- <jats:inline-formula> <jats:alternatives> <jats:tex-math>\frac12</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> </jats:alternatives> </jats:inline-formula> chain, as well as several others constructed from link states. We construct a fusion product which outputs a family of modules from any pair of families. Its definition is inspired from connectivity diagrams drawn on a disc with two holes. It is thus defined in a way to describe intermediate states in lattice correlation functions. We prove that this fusion product is a bifunctor, and that it is distributive, commutative, and associative. </jats:p>