Quantum groups based on spatial partitions (Q6630429)

The notion of \textit{easy quantum group}, introduced in [\textit{T. Banica} and \textit{R. Speicher}, Adv. Math. 222, No. 4, 1461--1501 (2009; Zbl 1247.46064)], provides a powerful combinatorial tool for studying subgroups of the quantum group \(O_{n}^{+}\). In particular, the intertwiner maps of an easy quantum group correspond (via Tannaka-Krein duality) to a \textit{category of partitions}, and are therefore amenable to combinatorial methods. Such intertwiner maps are spanned by maps of the form \(T_p : (\mathbb{C}^n)^{\otimes k} \to (\mathbb{C}^n)^{\otimes l}\) where \(p \in P(k,l)\) is a partition of a fixed set of size \(k + l\). These can be represented by 2-dimensional diagrams with \(k\) `upper' nodes and \(l\) `lower' nodes.\N\NThe present paper extends the above by defining new classes of such maps \(T_p\) whenever the dimension \(n\) admits a factorisation \(n = n_1 n_2 \ldots n_m\). In this case the isomorphism \(\mathbb{C}^{n} \cong \mathbb{C}^{n_1} \otimes \mathbb{C}^{n_2} \otimes \cdots \otimes \mathbb{C}^{n_m}\) allows one to obtain maps \(T_p : (\mathbb{C}^{n})^{\otimes k} \to (\mathbb{C}^{n})^{\otimes l}\) where now \(p \in P(mk,ml)\). This larger class of linear maps can now be represented by \(3\)-dimensional diagrams, consisting of \(m\) `levels' of 2-dimensional diagrams, and these are the spatial partitions in the paper's title. In analogy with easy quantum groups, then, a spatial partition quantum group is defined as a quantum subgroup of \(O_{n}^{+}\) whose intertwiner maps are spanned by such maps \(T_{p}\) as \(p\) varies over a category of spatial partitions.\N\NThis extended theory allows the authors to address several questions at once: Firstly, a category of partitions must always contain two particular base-case partitions (the pair and the identity); spatial partition admit more general base elements. Next, the authors show that product constructions with categories of spatial partitions correspond to glued direct products (with amalgamation) in compact matrix quantum groups, and in particular, such products of two spatial partition quantum groups are again spatial partition quantum groups (this can fail for easy quantum groups). Third, the authors show that quantum subgroups corresponding to spatial partitions are much more general than easy quantum groups. For instance, Theorem B shows that a spatial partition quantum subgroup of \(O_{n^{m}}^{+}\) need only contain a symmetric subgroup \(S_{n}\), as compared with easy quantum subgroups of \(O_{n^{m}}^{+}\), which contain the significantly larger group \(S_{n^m}\).\N\NAlong the way the authors develop some general theory for categories of spatial partitions. They give canonical generators for categories of spatial partitions (and their so-called \(\pi\)-graded versions), with particularly detailed descriptions given in the case \(m = 2\).