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Even Temperley-Lieb algebras and the dga of planar loops

Rachael Boyd, Guy Boyde, Oscar Randal-Williams, Robin J. Sroka

TL;DR

The paper resolves the homology of Temperley–Lieb algebras on an even number of strands by anchoring it to a differential graded algebra of planar loops, L(2n). It constructs a small, explicit quasi-free model M(2n) for L(2n) with generators x_{2i-1} and a differential mirroring a cobar-type coalgebra; Φ plays a central role with d(Φ)=a and Φ mapping to a distinguished loop in L(2n). The authors derive concrete homology descriptions across coefficient regimes (a=0, a not zerodivisor, universal Z[a]), connect L(2n)’s homology to Ext over truncated divided power algebras, and develop a suite of cup-based complexes (innermost, submaximal, outermost) to build derived resolutions. A key result is the derived complex of outermost cups, which yields an acyclic resolution and clarifies the shift between planar-loop homology and TL-homology, thereby extending known vanishing and structure theorems for TL_2n to the even-strand setting. The framework also reveals Massey-product structures and torsion phenomena, and provides a platform for future Ext-algebra computations and broader analogues in related algebras.

Abstract

We show that the homology of a Temperley-Lieb algebra on an even number of strands has a rich algebraic structure and is highly nontrivial in general. This is achieved by proving that it is entirely governed by a differential graded algebra: the differential graded algebra of planar loops. We provide a small model for this dga, and use it to obtain consequences on homology.

Even Temperley-Lieb algebras and the dga of planar loops

TL;DR

The paper resolves the homology of Temperley–Lieb algebras on an even number of strands by anchoring it to a differential graded algebra of planar loops, L(2n). It constructs a small, explicit quasi-free model M(2n) for L(2n) with generators x_{2i-1} and a differential mirroring a cobar-type coalgebra; Φ plays a central role with d(Φ)=a and Φ mapping to a distinguished loop in L(2n). The authors derive concrete homology descriptions across coefficient regimes (a=0, a not zerodivisor, universal Z[a]), connect L(2n)’s homology to Ext over truncated divided power algebras, and develop a suite of cup-based complexes (innermost, submaximal, outermost) to build derived resolutions. A key result is the derived complex of outermost cups, which yields an acyclic resolution and clarifies the shift between planar-loop homology and TL-homology, thereby extending known vanishing and structure theorems for TL_2n to the even-strand setting. The framework also reveals Massey-product structures and torsion phenomena, and provides a platform for future Ext-algebra computations and broader analogues in related algebras.

Abstract

We show that the homology of a Temperley-Lieb algebra on an even number of strands has a rich algebraic structure and is highly nontrivial in general. This is achieved by proving that it is entirely governed by a differential graded algebra: the differential graded algebra of planar loops. We provide a small model for this dga, and use it to obtain consequences on homology.

Paper Structure

This paper contains 34 sections, 36 theorems, 153 equations, 12 figures.

Table of Contents

  1. Introduction
  2. The differential graded algebra of planar loops
  3. Relation to the homology of Temperley--Lieb algebras
  4. Connection to previous works
  5. Remaining questions
  6. Overview
  7. Acknowledgments
  8. The Temperley--Lieb category and the dga of planar loops
  9. The Temperley--Lieb category and Temperley--Lieb algebras
  10. The algebra of planar loops
  11. The element $\Phi$
  12. Extra structure on the algebra: weight grading and hook maps
  13. Change of coefficients
  14. Small models
  15. The argument for $(\mathbb{Z}, 0)$
  16. ...and 19 more sections

Key Result

Theorem 1

There is a weak equivalence of differential graded $R$-algebras, where the left-hand side is the tensor algebra on the generators $x_i$ of degree $i$ equipped with the differential given by Under this equivalence $x_1$ is mapped to $\Phi$.

Figures (12)

  • Figure 1: With $2n=4$: $\mathrm{(a)}$ a typical basis element of $L(4)_3$, with the differential $d$ applied ('barwise') to that element, and $\mathrm{(b)}$ the ('juxtaposition') product.
  • Figure 2: The element $\Phi \in L(2n)_1$, shown (left-to-right) for $2n=2,4,6,8$.
  • Figure 3: Multiplication in the Temperley--Lieb algebra $\mathop{\mathrm{TL}}\nolimits_4$.
  • Figure 4: A planar diagram in $\mathop{\mathrm{TL}}\nolimits(6,4)$. In future diagrams we will sometimes suppress the node labels.
  • Figure 5: Composition of morphisms $\mathop{\mathrm{TL}}\nolimits(2, 6) \otimes_R \mathop{\mathrm{TL}}\nolimits(6,4) \to \mathop{\mathrm{TL}}\nolimits(2,4)$.
  • ...and 7 more figures

Theorems & Definitions (104)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • ...and 94 more