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The generator presentation is the planar diagram algebra over every commutative ring

Statement

Let $R$ be a commutative ring with unit, $δ \in R$ , and $n \geq 0$ . The homomorphism $π_{n} : P_{n} (R, δ) \to {T L}_{n} (R, δ)$ sending $u_{i}$ to $d_{i}$ is an isomorphism. Consequently the Jones normal forms of Jones normal forms span and are counted by Catalan paths are an $R$ -basis of the presented algebra, and both algebras are free of rank $C_{n}$ . This holds without a field assumption and without requiring $δ$ to be nonzero or invertible.

Facts & Assumptions

Given: $R, δ, n$ as above; $P_{n} (R, δ)$ and ${T L}_{n} (R, δ)$ ; their diagram generators, relations, normal forms, and Catalan counts.

[F1]

The assignment $u_{i} \mapsto d_{i}$ extends to a unital $R$ -algebra homomorphism $π_{n} : P_{n} (R, δ) \to {T L}_{n} (R, δ)$ . (The elementary diagram generators satisfy the local Temperley–Lieb relations)

[F2]

Every basis diagram of ${T L}_{n} (R, δ)$ is a product of the $d_{i}$ ; hence $π_{n}$ is surjective. (Elementary cup-cap diagrams generate every finite planar pairing endomorphism diagram)

[F3]

The images of the finite set $N_{n}$ of Jones normal forms span $P_{n} (R, δ)$ , and $N_{n}$ is in bijection with Dyck paths of semilength $n$ . (Jones normal forms span and are counted by Catalan paths)

[F4]

The diagram basis $D (n, n)$ is in bijection with Dyck paths of semilength $n$ , so $∣ D (n, n) ∣ = C_{n}$ . (The rank-n diagram basis is counted by the nth Catalan number)

[F5]

For a set $X$ , the free module $R^{(X)}$ has basis $(e_{x})_{x \in X}$ ; any map on this basis extends uniquely to an $R$ -linear map. (The free module on a set and its standard basis, Universal property of the free module on a set)

[F6]

A matrix records the coordinates of a linear map on the chosen finite bases; the product formula $(A B)_{i j} = \sum_{k} a_{i k} b_{k j}$ makes composition correspond to matrix multiplication, and $I_{n}$ has diagonal entries $1$ and off-diagonal entries $0$ . (The free module on a set and its standard basis, Universal property of the free module on a set, Finite rectangular matrices over a commutative ring, their entries, rows and columns, Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose, Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products)

[F7]

For positive-sized square matrices of the same size over a commutative ring, $\det (A B) = \det (A) \det (B)$ . (For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ )

[F8]

A positive-sized square matrix over a commutative ring is invertible if its determinant is a unit. (A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit)

[F9]

The determinant is given by the Leibniz formula, so $\det (I_{N}) = 1$ for every $N \geq 1$ ; a unit is an element with a two-sided multiplicative inverse. (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix, Left inverse, right inverse, and invertible element of a monoid)

[F10]

The vertical matching $1_{n}$ belongs to $D (n, n)$ , so this diagram set is nonempty. (Temperley–Lieb planar diagram data and endomorphism modules)

Proof

technique · direct

1.1F1F2

The homomorphism $π_{n}$ is well defined by [F1], and it is surjective by [F2], because its image contains every diagram basis vector.

2.1F3F4F5step 1.1

Let $q : R^{(N_{n})} \to P_{n} (R, δ)$ send the standard basis vector indexed by $N$ to the class of $N$ . It is $R$ -linear by [F5] and surjective by [F3]. Hence $S : = π_{n} \circ q : R^{(N_{n})} \to R^{(D (n, n))}$ is surjective, using the diagram basis in [F4].

3.1F3F4F5F6F10step 2.1construct

The explicit path encodings in [F3] and [F4] give a bijection between $N_{n}$ and $D (n, n)$ by composing the first with the inverse of the second. Both finite free modules in step 2.1 therefore have rank $C_{n}$ . This rank is positive: the identity matching belongs to $D (n, n)$ by [F10], so [F4] implies $C_{n} \geq 1$ . Order the standard bases by this bijection. For each basis vector $e_{N}$ , the coordinates of $S (e_{N})$ are the $N$ th column of a matrix $A$ ; [F5] gives the finite-basis map representation, and [F6] identifies composition with matrix multiplication. Thus $S$ is represented by a square matrix $A$ of size $C_{n}$ .

4.1F4F5F6step 2.1step 3.1choose

Step 2.1 established that $S$ is surjective. For each member of the finite basis indexed by $D (n, n)$ , choose one preimage under $S$ ; only finitely many choices are made. By [F5] these assignments extend linearly to a right inverse $B$ of $S$ . With the bases fixed in step 3.1, $B$ is represented by a square matrix and $S \circ B = I_{C_{n}}$ , so $A B = I_{C_{n}}$ .

5.1F6F7F9step 4.1algebra

Taking determinants and using [F7] gives $\det (A) \det (B) = \det (I_{C_{n}}) = 1$ by [F9]. Since $R$ is commutative, $\det (B)$ is a two-sided inverse of $\det (A)$ , so $\det (A)$ is a unit.

6.1F1F2F3F4F8step 2.1step 4.1step 5.1∎

By [F8], $A$ is invertible, hence $S$ is an isomorphism. If $q (x) = 0$ , then $S (x) = 0$ and injectivity of $S$ gives $x = 0$ ; thus the already-surjective $q$ is an isomorphism. For any $y \in P_{n} (R, δ)$ with $π_{n} (y) = 0$ , choose $x$ with $q (x) = y$ using surjectivity of $q$ ; then $S (x) = 0$ , whence $x = 0$ and $y = 0$ . Thus $π_{n}$ is injective as well as surjective, proving the claimed isomorphism and the basis assertion.

Remarks

The ring-theoretic independence argument is what extends the normal-form count to arbitrary commutative rings: a surjective square matrix has a right inverse, so its determinant is a unit. No specialization argument, flatness assumption, division by $δ$ , or semisimplicity hypothesis is used.

Depends on

Used by

Dependency tree · two levels

56 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

David Ridout and Yvan Saint-Aubin, Standard Modules, Induction and the Structure of the Temperley-Lieb Algebra, §2 (standard reference, not scraped)
Dana C. Ernst, Michael G. Hastings, and Sarah K. Salmon, Factorization of Temperley–Lieb diagrams (standard reference, not scraped)
David Guichard, Combinatorics and Graph Theory, Section 3.5: Catalan Numbers (standard reference, not scraped)