[Paper Review] Examples of inner linear Hopf algebras

Nicolás Andruskiewitsch, Julien Bichon|arXiv (Cornell University)|Nov 30, 2009

Algebraic structures and combinatorial models16 references3 citations

TL;DR

This paper establishes two foundational results for inner linear Hopf algebras: a characterization via the Hopf dual that proves Drinfeld-Jimbo quantum algebras $U_q(\mathfrak{g})$ and $\mathcal{O}_q(G)$ are inner linear when $q$ is not a root of unity, and a stability result under extensions that confirms inner linearity for quantum algebras at roots of unity and half-liberated orthogonal Hopf algebras. It further shows that $\mathcal{O}_q(K)$ for compact Lie groups is inner linear but not inner unitary when $q \in \mathbb{R}^*$, $q \neq \pm 1$, resolving a key distinction in $*$-algebra representations.

ABSTRACT

The notion of inner linear Hopf algebra is a generalization of the notion of discrete linear group. In this paper, we prove two general results that enable us to enlarge the class of Hopf algebras that are known to be inner linear: the first one is a characterization by using the Hopf dual, while the second one is a stability result under extensions. We also discuss the related notion of inner unitary Hopf *-algebra.

Motivation & Objective

• To extend the class of known inner linear Hopf algebras by providing general criteria for inner linearity.
• To clarify the distinction between inner linearity and inner unitarity in the context of $*$-Hopf algebras.
• To establish a stability result for inner linearity under algebraic extensions, applicable to quantum groups and half-liberated algebras.
• To investigate the representation-theoretic properties of compact quantum groups, particularly $\mathcal{O}_q(K)$, in relation to inner unitarity.

Proposed method

• Introduces a characterization of inner linearity using the Hopf dual $H^0$, showing that $H$ is inner linear iff $H^0$ contains a finitely generated Hopf subalgebra separating points of $H$.
• Applies this criterion to Drinfeld-Jimbo quantum algebras $U_q(\mathfrak{g})$ and $\mathcal{O}_q(G)$, proving inner linearity when $q$ is not a root of unity.
• Develops a stability result for inner linearity under extensions of Hopf algebras, showing that if $A$ is inner linear and $H$ is a faithfully flat extension of $A$, then $H$ is inner linear.
• Adapts the extension technique to $*$-Hopf algebras, proving that if $A \subset H$ is a normal, commutative, inner unitary Hopf $*$-subalgebra and $H$ is finitely generated as a right $A$-module, then $H$ is inner unitary.
• Uses induced representations on Hilbert spaces via conditional expectations and $C^*$-algebraic techniques to construct inner faithful $*$-representations.
• Applies the theory to compact quantum groups $\mathcal{O}_q(K)$ and half-liberated orthogonal algebras $A_o^*(n)$, proving inner unitarity for the latter.

Experimental results

Research questions

• RQ1 Under what conditions on $q$ are the Drinfeld-Jimbo quantum algebras $U_q(\mathfrak{g})$ and $\mathcal{O}_q(G)$ inner linear?
• RQ2 Can inner linearity be preserved under algebraic extensions of Hopf algebras, and if so, under what structural assumptions?
• RQ3 Is the compact quantum group $\mathcal{O}_q(K)$ inner unitary for $q \in \mathbb{R}^*$, $q \neq \pm 1$, despite being inner linear?
• RQ4 Does the Hopf $*$-algebra $A_o^*(n)$ admit an inner faithful $*$-representation on a finite-dimensional Hilbert space?
• RQ5 What is the relationship between inner linearity and inner unitarity in the context of $*$-Hopf algebras?

Key findings

• The Drinfeld-Jimbo quantum algebra $U_q(\mathfrak{g})$ is inner linear for all semisimple Lie algebras $\mathfrak{g}$ when $q$ is not a root of unity.
• The quantized function algebra $\mathcal{O}_q(G)$ is inner linear for $q$ not a root of unity, as established via the Hopf dual characterization.
• Quantum algebras $U_q(\mathfrak{g})$ and $\mathcal{O}_q(G)$ remain inner linear even when $q$ is a root of unity, due to the stability under extensions.
• The half-liberated orthogonal Hopf $*$-algebra $A_o^*(n)$ is inner unitary, as shown via the $*$-extension theorem.
• The compact quantum group $\mathcal{O}_q(K)$ is inner linear but not inner unitary for $q \in \mathbb{R}^*$, $q \neq \pm 1$, due to the existence of a nontrivial commutator ideal in its $*$-representations.
• The $*$-algebra $\mathcal{O}_{-1}(\mathrm{SU}_2)$ is inner unitary, providing a counterexample to the converse of the inner unitary implies inner linear implication.