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The first four finite Temperley–Lieb algebras

Example

Let $R$ be a commutative ring and fix any loop value $δ \in R$ . For the ranks $n = 0, 1, 2, 3$ , the diagram and presentation algebras have the following $R$ -bases:

$\begin{array}{ccc} n & basis & rank \\ 0 & {1} & 1 \\ 1 & {1} & 1 \\ 2 & {1, u_{1}} & 2 \\ 3 & {1, u_{1}, u_{2}, u_{1} u_{2}, u_{2} u_{1}} & 5 \end{array}$

The rank column is $C_{n}$ for these four indices. In rank $3$ , writing $a = u_{1}$ and $b = u_{2}$ , the product table is

$\begin{array}{cccccc} \cdot & 1 & a & b & a b & b a \\ 1 & 1 & a & b & a b & b a \\ a & a & δ a & a b & δ a b & a \\ b & b & b a & δ b & b & δ b a \\ a b & a b & a & δ a b & a b & δ a \\ b a & b a & δ b a & b & δ b & b a \end{array}$

Facts & Assumptions

Given: A commutative ring $R$ , a loop value $δ \in R$ , and $n \in {0, 1, 2, 3}$ .

[F1]

$P_{n} (R, δ)$ is generated by $u_{1}, \dots, u_{n - 1}$ with $u_{i}^{2} = δ u_{i}$ , $u_{i} u_{i \pm 1} u_{i} = u_{i}$ , and distant generators commuting. (The abstract Temperley–Lieb generator presentation)

[F2]

The Jones normal forms use blocks $(u_{j_{1}} \dots u_{k_{1}}) \dots (u_{j_{r}} \dots u_{k_{r}})$ with $1 \leq k_{i} \leq j_{i} < n$ and both endpoint sequences strictly increasing. (Jones normal forms span and are counted by Catalan paths)

[F3]

The Jones normal forms are an $R$ -basis of $P_{n} (R, δ)$ . (The generator presentation is the planar diagram algebra over every commutative ring)

[F4]

The empty generator set gives $P_{0} = P_{1} = R$ . (The abstract Temperley–Lieb generator presentation)

Proof

technique · direct

1.1F2F3F4construct

For $n = 0, 1$ there are no generators, so the presentation is $R$ and the unique normal form is $1$ . For $n = 2$ , the admissible lists in [F2] give $1, u_{1}$ ; for $n = 3$ , they give $1, u_{1}, u_{2}, u_{2} u_{1}, u_{1} u_{2}$ . These words are bases by [F3], so the ranks are $1, 1, 2, 5$ ; the A-page path bijections identify those counts with $C_{0}, C_{1}, C_{2}, C_{3}$ .

2.1F1algebra∎

Set $a = u_{1}$ , $b = u_{2}$ in rank $3$ . From [F1], $a^{2} = δ a$ , $b^{2} = δ b$ , $a b a = a$ , and $b a b = b$ . The row of $a$ is $a, δ a, a b, a (a b) = a^{2} b = δ a b, a (b a) = a b a = a$ . The row of $b$ is $b, b a, δ b, b (a b) = b a b = b, b (b a) = b^{2} a = δ b a$ . The row of $a b$ is $a b, a b a = a, a b b = δ a b, a b a b = (a b a) b = a b, a b b a = a b^{2} a = δ a$ . The row of $b a$ is $b a, b a a = δ b a, b a b = b, b a a b = b a^{2} b = δ b, b a b a = (b a b) a = b a$ . These are the displayed entries and use no division by $δ$ .

Remarks

The rank is the number of basis diagrams, not the number of generators. The table remains valid at $δ = 0$ and over rings with zero divisors.

Depends on

Used by

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