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The real-form star involution on the loop-weight-two Temperley–Lieb algebra

Statement

For each integer $N \geq 0$ , let $P_{N} (C, 2)$ be the quotient of the free unital associative complex algebra on $u_{1}, \dots, u_{N - 1}$ by the relations $u_{i}^{2} = 2 u_{i}, u_{i} u_{j} = u_{j} u_{i} (∣ i - j ∣ > 1), u_{i} u_{i + 1} u_{i} = u_{i}, u_{i + 1} u_{i} u_{i + 1} = u_{i + 1} .$ This complex Temperley–Lieb algebra has a conjugate-linear anti-involution $*$ determined by $c^{*} = \overline{c} (c \in C), u_{i}^{*} = u_{i} (1 \leq i < N), (x y)^{*} = y^{*} x^{*} .$ For $N = 0$ or $N = 1$ this is complex conjugation on $C$ .

Statement references

The abstract Temperley–Lieb generator presentation

Facts & Assumptions

Given: $N \geq 0$ and the presentation $P_{N} (C, 2)$ .

[F1]

The abstract Temperley–Lieb generator presentation: the algebra is the quotient of the free unital associative $C$ -algebra by the two-sided ideal generated by $u_{i}^{2} - 2 u_{i}$ , $u_{i} u_{j} - u_{j} u_{i}$ for $∣ i - j ∣ > 1$ , and the two relations $u_{i} u_{i + 1} u_{i} - u_{i}$ , $u_{i + 1} u_{i} u_{i + 1} - u_{i + 1}$ .

[F2]

The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive: complex conjugation is additive, multiplicative, and involutive, and fixes the real scalar $2$ .

Proof

1.1F2constructalgebra

On a word $w = u_{i_{1}} \dots u_{i_{k}}$ define $w^{*} = u_{i_{k}} \dots u_{i_{1}}$ , extend by $(\sum_{w} c_{w} w)^{*} = \sum_{w} \overline{c_{w}} w^{*}$ , and set $1^{*} = 1$ . Reversal of words and [F2] give $(x y)^{*} = y^{*} x^{*}$ and $(x^{*})^{*} = x$ in the free algebra. Thus this is a conjugate-linear anti-involution before taking the quotient.

2.1F1F2step 1.1algebra

The star fixes each relation $u_{i}^{2} - 2 u_{i}$ and each palindromic relation $u_{i} u_{i + 1} u_{i} - u_{i}$ and $u_{i + 1} u_{i} u_{i + 1} - u_{i + 1}$ . It sends a distant commutator $u_{i} u_{j} - u_{j} u_{i}$ to its negative. Hence the star sends every generator of the defining ideal into that ideal.

3.1F1step 2.1algebra∎

Every element of the ideal is a finite sum of terms $x r y$ with $r$ a displayed defining relation; its star is $y^{*} r^{*} x^{*}$ , which is again in the ideal by step 2.1. The ideal is therefore star-invariant, so the operation descends to the quotient and remains conjugate-linear, anti-multiplicative, and involutive there. If $N \leq 1$ there are no generators or relations, and the same construction is just coefficient conjugation.

Depends on

Dependency tree · 2 levels

Sources

David Ridout and Yvan Saint-Aubin, Standard Modules, Induction and the Structure of the Temperley–Lieb Algebra, §2