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The Motzkin subproduct system

Valeriano Aiello, Simone Del Vecchio, Stefano Rossi

TL;DR

The paper constructs a Motzkin subproduct system by employing Motzkin algebras and their Jones–Wenzl idempotents, extending the Temperley–Lieb subproduct systems to a broader planar-algebraic setting. It provides a concrete realization of the associated Toeplitz and Cuntz–Pimsner C*-algebras as universal C*-algebras defined by generators and relations, and analyzes their representation theory via a gauge action and conditional expectations. Key results include faithfulness of Motzkin-pair representations for suitable parameters, the standard subproduct-system structure generated by $g_k$, and the identification of $\mathcal{O}_P$ as the compact-quotient of $\mathcal{T}_P$, with universality and irreducibility results guiding the representation theory. The work connects noncommutative polynomial ideals, planar algebras, and C*-algebraic invariants, offering a pathway to new symmetries and potential connections to quantum group actions and Thompson-like groups.

Abstract

We introduce a subproduct system of finite-dimensional Hilbert spaces by using the Motzkin planar algebra and its Motzkin Jones-Wenzl idempotents, which generalizes the Temperley-Lieb subproduct system of Habbestad and Neshveyev. We provide a description of the corresponding Toeplitz and Cuntz-Pimsner C$^*$-algebras as universal C$^*$-algebras, defined in terms of generators and relations, and we highlight properties of their representation theory.

The Motzkin subproduct system

TL;DR

The paper constructs a Motzkin subproduct system by employing Motzkin algebras and their Jones–Wenzl idempotents, extending the Temperley–Lieb subproduct systems to a broader planar-algebraic setting. It provides a concrete realization of the associated Toeplitz and Cuntz–Pimsner C*-algebras as universal C*-algebras defined by generators and relations, and analyzes their representation theory via a gauge action and conditional expectations. Key results include faithfulness of Motzkin-pair representations for suitable parameters, the standard subproduct-system structure generated by , and the identification of as the compact-quotient of , with universality and irreducibility results guiding the representation theory. The work connects noncommutative polynomial ideals, planar algebras, and C*-algebraic invariants, offering a pathway to new symmetries and potential connections to quantum group actions and Thompson-like groups.

Abstract

We introduce a subproduct system of finite-dimensional Hilbert spaces by using the Motzkin planar algebra and its Motzkin Jones-Wenzl idempotents, which generalizes the Temperley-Lieb subproduct system of Habbestad and Neshveyev. We provide a description of the corresponding Toeplitz and Cuntz-Pimsner C-algebras as universal C-algebras, defined in terms of generators and relations, and we highlight properties of their representation theory.

Paper Structure

This paper contains 3 sections, 18 theorems, 101 equations, 1 figure.

Table of Contents

  1. The Motzkin algebra
  2. The Motzkin subproduct system
  3. The Toeplitz and Cuntz-Pimsner C$^*$-algebras

Key Result

Lemma 1.2

For any Temperley-Lieb vector $v_A$, one has where $\{e_{kh}\}_{h,k=1}^n$ are the matrix units.

Figures (1)

  • Figure 1: The elements $l_2$, $r_2=l_2^*$, $t_1$, $p_2$, and the identity $\operatorname{id}_5$ of $M_5(\lambda^{-1})$.

Theorems & Definitions (50)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Proposition 1.3
  • proof
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Definition 1.6
  • Example 1.7
  • ...and 40 more