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Normalized Temperley–Lieb trace from diagram closure
Definition
Let . For a basis diagram in Temperley–Lieb planar diagram data and endomorphism modules, join each top boundary point to the corresponding bottom boundary point outside the rectangle. Let be the number of closed components of this finite matching graph. Define the normalized closure trace on the diagram basis by
and extend -linearly to . In particular, .
Remarks
This is a trace defined directly on the diagram algebra by closing its diagrams. It is not obtained by assuming that the Ocneanu trace on a Hecke algebra descends through a quotient.
Depends on
Used by
- Finite diagram tower inclusion and normalized right closure Definition
- Unnormalized diagram closure and partial trace Definition
- At q squared of order three, TL2 is semisimple with a degenerate Jones trace Example
- Right closure on the five rank-three diagrams Example
- The rank-two Jones–Wenzl idempotent at loop value two Example
- Normalized right closure preserves the diagram trace radical Lemma
- Right closure is a trace-preserving diagram-tower expectation Lemma
- The normalized closure trace is cyclic and satisfies the Markov rules Lemma
- At the first root boundary the Jones trace degenerates before semisimplicity fails Theorem
Dependency tree · two levels
3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Joshua Chen, The Temperley-Lieb Categories and Skein Modules (standard reference, not scraped)
- K. Iohara, G. I. Lehrer, and R. B. Zhang, The Jones Quotients of the Temperley-Lieb Algebras (standard reference, not scraped)