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Cohomology of dilute Temperley--Lieb algebras

Andrew Fisher, Daniel Graves

Abstract

Dilute Temperley--Lieb algebras are variants of Temperley--Lieb algebras arising in statistical mechanics in the study of solvable lattice models. In this paper we prove that the (co)homology of dilute Temperley--Lieb algebras vanishes in all positive degrees.

Cohomology of dilute Temperley--Lieb algebras

Abstract

Dilute Temperley--Lieb algebras are variants of Temperley--Lieb algebras arising in statistical mechanics in the study of solvable lattice models. In this paper we prove that the (co)homology of dilute Temperley--Lieb algebras vanishes in all positive degrees.
Paper Structure (7 sections, 11 theorems, 15 equations)

This paper contains 7 sections, 11 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Dilute Temperley--Lieb algebras
  3. Mayer--Vietoris complex
  4. Link states and technical lemmas
  5. Link states
  6. Technical lemmas
  7. An idempotent cover: proving Theorem \ref{['main-thm']}

Key Result

Theorem 1.1

For any commutative ring $k$, any $\delta \in k$ and any $n\in \mathbb{N}$, the (co)homology of $dTL_n(\delta)$ is concentrated in degree zero, where it is isomorphic to $k$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Remark 2.6
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 29 more