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A skew group ring of $\mathbb Z/2\mathbb Z$ over $U(\mathfrak{sl}_2)$, Leonard triples and odd graphs

Hau-Wen Huang, Chin-Yen Lee

TL;DR

This paper builds a bridge between the representation theory of twisted universal enveloping algebras and the combinatorics of Terwilliger algebras for odd graphs. By using a skew group ring of $\

Abstract

We employ a skew group ring of $\mathbb Z/2\mathbb Z$ over $U(\mathfrak{sl}_2)$ to construct modules over the universal Bannai--Ito algebra. In addition, we give the conditions under which the defining generators act as Leonard triples on the resulting modules. As a combinatorial realization, we establish an algebra homomorphism from the universal Bannai--Ito algebra onto the Terwilliger algebra of an odd graph. This homomorphism provides a unified description of Leonard triples on all irreducible modules over the Terwilliger algebra.

A skew group ring of $\mathbb Z/2\mathbb Z$ over $U(\mathfrak{sl}_2)$, Leonard triples and odd graphs

TL;DR

This paper builds a bridge between the representation theory of twisted universal enveloping algebras and the combinatorics of Terwilliger algebras for odd graphs. By using a skew group ring of $\

Abstract

We employ a skew group ring of over to construct modules over the universal Bannai--Ito algebra. In addition, we give the conditions under which the defining generators act as Leonard triples on the resulting modules. As a combinatorial realization, we establish an algebra homomorphism from the universal Bannai--Ito algebra onto the Terwilliger algebra of an odd graph. This homomorphism provides a unified description of Leonard triples on all irreducible modules over the Terwilliger algebra.
Paper Structure (10 sections, 44 theorems, 97 equations, 1 figure, 3 tables)

This paper contains 10 sections, 44 theorems, 97 equations, 1 figure, 3 tables.

Table of Contents

  1. Preliminaries on skew group rings and Hopf algebras
  2. Introduction
  3. A presentation for $U(\mathfrak{sl}_2)_{\mathbb Z/2\mathbb Z}$
  4. Proof for Theorem \ref{['thm:V(1)']}
  5. Finite-dimensional $U(\mathfrak{sl}_2)_{\mathbb Z/2\mathbb Z}$-modules
  6. Finite-dimensional irreducible $\mathfrak{BI}$-modules
  7. Leonard triples on finite-dimensional irreducible $\mathfrak{BI}$-modules
  8. Proof for Theorem \ref{['thm:V(1)&LT']}
  9. Proof for Theorem \ref{['thm:BI->T']}
  10. The Clebsch--Gordan rule for $U(\mathfrak{sl}_2)_{\mathbb Z/2\mathbb Z}$

Key Result

Theorem 2.1

Suppose that $V$ is a $U(\mathfrak{sl}_2)_{\mathbb Z/2\mathbb Z}^{\otimes 2}$-module. Then $V(1)$ is a $\mathfrak{BI}$-module given by

Figures (1)

  • Figure :

Theorems & Definitions (81)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • ...and 71 more