TY - JOUR
T1 - Categories of two-colored pair partitions Part II: Categories indexed by semigroups
AU - Mang, Alexander
AU - Weber, Moritz
JO - Journal of Combinatorial Theory, Series A
VL - 180
SP - 105409
PY - 2021
DA - 2021/05/01/
SN - 0097-3165
DO - https://doi.org/10.1016/j.jcta.2021.105409
UR - https://www.sciencedirect.com/science/article/pii/S009731652100008X
KW - Quantum group
KW - Unitary easy quantum group
KW - Unitary group
KW - Half-liberation
KW - Tensor category
KW - Two-colored partition
KW - Partition of a set
KW - Category of partitions
KW - Brauer algebra
AB - Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial categories whose morphisms are formed by partitions of finite sets into disjoint subsets of cardinality two; the points of these sets are colored black or white. These categories correspond to “half-liberated easy” interpolations between the unitary group and Wang's quantum counterpart. We complete the classification of all such categories demonstrating that the subcategories of a certain natural halfway point are equivalent to additive subsemigroups of the natural numbers; the categories above this halfway point have been classified in a preceding article. We achieve this using combinatorial means exclusively. Our work reveals that the half-liberation procedure is quite different from what was previously known from the orthogonal case.
ER -