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Right closure on the five rank-three diagrams

Statement

For $δ \in C^{\times}$ , use the five standard basis diagrams $1_{3}, d_{1}, d_{2}, d_{1} d_{2}, d_{2} d_{1}$ of ${T L}_{3} (C, δ)$ . Their right closures are $E_{2} (1_{3}) = 1_{2}, E_{2} (d_{1}) = d_{1}, E_{2} (d_{2}) = δ^{- 1} 1_{2}, E_{2} (d_{1} d_{2}) = E_{2} (d_{2} d_{1}) = δ^{- 1} d_{1} .$ The normalized trace satisfies $τ_{3} (D) = τ_{2} (E_{2} (D))$ for all five diagrams.

Facts & Assumptions

Given: $δ \neq 0$ , the tower map, closure, trace, and elementary generators.

[F1]

The five matchings are $1_{3}$ (all three through strands), $d_{1}$ (cap/cup at positions $1, 2$ ), $d_{2}$ (cap/cup at $2, 3$ ), $d_{1} d_{2}$ (top cap $1, 2$ , bottom cup $2, 3$ , through strand from top $3$ to bottom $1$ ), and $d_{2} d_{1}$ (top cap $2, 3$ , bottom cup $1, 2$ , through strand from top $1$ to bottom $3$ ); they are distinct members of the noncrossing diagram basis, whose rank-three Catalan count is five. (Temperley–Lieb planar diagram data and endomorphism modules, The rank-n diagram basis is counted by the nth Catalan number)

[F2]

$E_{2}$ is the normalized right closure, is bimodular and trace-preserving, and sends $ι_{2} (x) d_{2}$ to $δ^{- 1} x$ . (Finite diagram tower inclusion and normalized right closure, Right closure is a trace-preserving diagram-tower expectation)

[F3]

The normalized closure trace has $τ_{2} (1_{2}) = 1$ and $τ_{2} (d_{1}) = δ^{- 1}$ . (Normalized Temperley–Lieb trace from diagram closure)

Proof

Given: The five rank-three diagrams and [F1]–[F3].

1.1F2givenalgebra

Since $1_{3} = ι_{2} (1_{2})$ and $d_{1} = ι_{2} (d_{1})$ , the identity $E_{2} ι_{2} = i d$ gives the first two values. Also $d_{2} = ι_{2} (1_{2}) d_{2}$ , so the right-generator Markov rule in [F2] gives $E_{2} (d_{2}) = δ^{- 1} 1_{2}$ .

2.1F1F2step 1.1algebra

The tower inclusion sends $d_{1}$ to the first generator on three strands. Bimodularity and the last value therefore give $E_{2} (d_{1} d_{2}) = d_{1} E_{2} (d_{2}) = δ^{- 1} d_{1}$ and $E_{2} (d_{2} d_{1}) = E_{2} (d_{2}) d_{1} = δ^{- 1} d_{1}$ . This lists all five basis images by [F1].

3.1F2F3step 1.1step 2.1algebra∎

Trace preservation [F2] gives $τ_{2} (E_{2} (D)) = τ_{3} (D)$ for every basis element $D$ and hence for their linear combinations. Explicitly, the values are $1$ on $1_{3}$ , $δ^{- 1}$ on $d_{1}, d_{2}$ , and $δ^{- 2}$ on $d_{1} d_{2}, d_{2} d_{1}$ , by [F3] and the displayed closures.

Depends on

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Sources

David Penneys, Exercises in Quantum Algebra (standard reference, not scraped)