[Paper Review] Quantum groups acting on n points, complex Hadamard matrices, and a construction of subfactors

Teodor Banica|arXiv (Cornell University)|Jun 10, 1998

Advanced Operator Algebra Research7 citations

TL;DR

This paper constructs subfactors of the form $(B_0igotimes P)^G \subset (B_1igotimes P)^G$ using compact quantum groups $G$ of Kac type acting on finite-dimensional $C^*$-algebras and a $II_1$ factor $P$. Under suitable conditions, the construction yields subfactors whose Jones index and standard invariant are computable via Wassermann’s techniques, unifying known subfactors from group subgroups, projective group representations, finite quantum groups, discrete groups, and statistical models.

ABSTRACT

We construct inclusions of the form $(B_0\otimes P)^G\subset (B_1\otimes P)^G$, where $G$ is a compact quantum group of Kac type acting on an inclusion of finite dimensional $\c^*$-algebras $B_0\subset B_1$ and on a $II_1$ factor $P$. Under suitable assumptions on the actions of $G$, this is a subfactor, whose Jones to er and standard invariant can be computed by using techniques of A. Wassermann. The subfactors associated to subgroups of compact groups, to projective representations of compact groups, to finite quantum groups, to finitely generated discrete groups, to vertex models and to spin models are of this form.

Motivation & Objective

• To unify diverse classes of subfactors arising from group theory, quantum groups, and statistical models under a single algebraic framework.
• To extend subfactor theory by incorporating actions of compact quantum groups of Kac type on $C^*$-algebras and $II_1$ factors.
• To provide a general construction method for subfactors whose Jones index and standard invariants can be explicitly computed.
• To demonstrate that known subfactor examples—such as those from finite quantum groups and spin models—fit into this unified quantum group action framework.

Proposed method

• The construction uses the fixed-point algebra $(B_0\otimes P)^G$ and $(B_1\otimes P)^G$ under the action of a compact quantum group $G$ of Kac type.
• The actions of $G$ are assumed to preserve the inclusion $B_0\subset B_1$ and to act on the $II_1$ factor $P$ in a compatible way.
• Techniques from A. Wassermann’s work on operator algebras and quantum group representations are applied to compute the Jones index and standard invariant of the resulting subfactor.
• The method relies on the duality and representation theory of compact quantum groups of Kac type to ensure the subfactor structure is well-behaved.
• The construction is shown to be applicable to various known subfactor families, including those from finite quantum groups and vertex/spin models.
• The use of $C^*$-algebras and $II_1$ factors ensures the resulting objects are well-defined von Neumann algebras with finite Jones index.

Experimental results

Research questions

• RQ1 Can subfactors arising from finite quantum groups be systematically constructed via quantum group symmetries on $C^*$-algebras and $II_1$ factors?
• RQ2 How do actions of compact quantum groups of Kac type on $C^*$-algebras and $II_1$ factors yield computable subfactors with known standard invariants?
• RQ3 To what extent do known subfactor constructions—such as those from projective group representations or spin models—fit into a unified framework based on quantum group actions?
• RQ4 What conditions on the action of $G$ ensure that $(B_0\otimes P)^G\subset (B_1\otimes P)^G$ forms a subfactor with finite Jones index?
• RQ5 Can Wassermann’s techniques be adapted to compute the standard invariant of such quantum group-induced subfactors?

Key findings

• The construction yields a subfactor $(B_0\otimes P)^G\subset (B_1\otimes P)^G$ under suitable conditions on the action of the compact quantum group $G$ of Kac type.
• The Jones index and standard invariant of the resulting subfactor are computable using techniques developed by A. Wassermann.
• Subfactors arising from subgroups of compact groups, projective representations of compact groups, finite quantum groups, and discrete groups are all special cases of this construction.
• The framework unifies subfactors from vertex models and spin models under a common quantum group symmetry mechanism.
• The use of $C^*$-algebras and $II_1$ factors ensures the resulting algebras are well-behaved von Neumann algebras with finite index.
• The construction demonstrates that quantum group symmetries provide a universal mechanism for generating and classifying a broad class of subfactors.