Intertwiner Spaces of Quantum Group Subrepresentations

Abstract

We consider compact matrix quantum groups whose N -dimensional fundamental representation decomposes into an (N − 1)-dimensional and a onedimensional subrepresentation. Even if we know that the compact matrix quantum group associated to this (N − 1)-dimensional subrepresentation is isomorphic to the given N -dimensional one, it is a priori not clear how the intertwiner spaces transform under this isomorphism. In the context of so-called easy and non-easy quantum groups, we are able to define a transformation of linear combinations of partitions and we explicitly describe the transformation of intertwiner spaces. As a side effect, this enables us to produce many new examples of non-easy quantum groups being isomorphic to easy quantum groups as compact quantum groups but not as compact matrix quantum groups.

Gluing Compact Matrix Quantum Groups

“…Regarding the original definition of categories of partitions as defined in [7], we refer to the survey [27]. See also [23] for the definition of two-coloured categories of partitions and [16] for the definition of linear categories of partitions. Also see [15] for the definition of the two-coloured category Alt C associated to a given non-coloured category C .…”

Quantum symmetries of Cayley graphs of abelian groups

“…On this collection of linear spaces, we may define the structure of a monoidal * -category in terms of simple pictorial manipulations (see e.g. [GW20] for details) such that they respect the category structure in C Sn . In other words, the mapping T (N ) : Part n → C Sn is a monoidal unitary functor.…”

Generating linear categories of partitions

“…The non-easiness of the categories was proven in [GW19] for all δ ∈ N, δ ≥ 3. This already implies the non-easiness for every δ ∈ C. Indeed, all three generators can be understood as linear combinations of partitions with coefficients in the ring…”

“…The algorithm was useful first for providing the idea to solve such equations and secondly for checking (although not proving) that the categories remain non-easy even after more iterations of the tensor product. The categories we discovered this way were then studied by theoretical means in [GW19] (see [GW19, Example 1.1, 1.2]) within the theory of compact quantum groups which provided a proof of their non-easiness for δ being a sufficiently large natural number as mentioned above.…”

“…Our computer experiments indeed provided us with some results. Most of the new examples of partition categories were studied and interpreted within the theory of quantum groups in [GW19]. We study the remaining ones in Section 6 interpreting the corresponding quantum groups as some anticommutative twists.…”