Equivalent Notions of Normal Quantum Subgroups, Compact Quantum Groups with Properties F and FD, and Other Applications

TL;DR

This paper establishes the equivalence of different notions of normal quantum subgroups in compact quantum groups, and explores their structural properties and applications to quantum group theory.

Contribution

It proves the equivalence of two definitions of normal quantum subgroups and derives new results on the structure of quantum groups with properties F and FD.

Findings

Equivalence of two notions of normal quantum subgroups.

Quantum analog of the third fundamental isomorphism theorem.

Results on normal quantum subgroups for tensor, free, and crossed products.

Abstract

The notion of normal quantum subgroup introduced in algebraic context by Parshall and Wang when applied to compact quantum groups is shown to be equivalent to the notion of normal quantum subgroup introduced by the author. As applications, a quantum analog of the third fundamental isomorphism theorem for groups is obtained, which is used along with the equivalence theorem to obtain results on structure of quantum groups with property F and quantum groups with property FD. Other results on normal quantum subgroups for tensor products, free products and crossed products are also proved.