Fundamental isomorphism theorems for quantum groups

TL;DR

This paper explores quantum analogues of classical subgroup theorems, establishing isomorphism results for locally compact and linearly reductive quantum groups, bridging operator algebraic and Hopf algebraic approaches.

Contribution

It introduces quantum versions of fundamental isomorphism theorems, highlighting differences and connections between locally compact and linearly reductive quantum groups.

Findings

Quantum analogues of classical subgroup theorems established

Additional assumptions needed in the locally compact case

Results hold without extra assumptions for linearly reductive quantum groups

Abstract

The lattice of subgroups of a group is the subject of numerous results revolving around the central theme of decomposing the group into "chunks" (subquotients) that can then be compared to one another in various ways. Examples of results in this class would be the Noether isomorphism theorems, Zassenhaus' butterfly lemma, the Schreier refinement theorem for subnormal series of subgroups, the Dedekind modularity law, and last but not least the Jordan-H\"older theorem. We discuss analogues of the above-mentioned results in the context of locally compact quantum groups and linearly reductive quantum groups. The nature of the two cases is different: the former is operator algebraic and the latter Hopf algebraic, hence the corresponding two-part organization of our study. Our intention is that the analytic portion be accessible to the algebraist and vice versa. The upshot is that in the locally compact case one often needs further assumptions (integrability, compactness, discreteness). In the linearly reductive case on the other hand, the quantum versions of the results hold without further assumptions. Moreover the case of compact / discrete quantum groups is usually covered by both the linearly reductive and the locally compact framework, thus providing a bridge between the two.