Making non-trivially associated tensor categories from left coset representatives
Let \(G\) be a subgroup of a finite group \(X\), \(M\subset X\) a set of left coset representatives. From the algebraic structure on \(X\), the author first constructs a binary operation on \(M\) which has a left identity and right division. Conversely any set with such an operation can be realized as a set of left coset representatives for the quotient of two permutation groups. This operation is not associative, but the breakdown of associativity is given by a ``cocycle'' \(\tau: M\times M\to G\). Using this cocycle he constructs a tensor product with non-trivial associator for the category \({\mathcal C}\) of \(G\setminus X\)-graded right representations of \(G\). This category also can be given evaluation and coevaluation maps, depending on the choice of \(M\), making it into a rigid tensor category. Under the assumption that the binary operation on \(M\) has left division, he proves that the grading and group action can be combined into the action of an algebra \(A\) on the objects in the category. It turns out that A itself is in \({\mathcal C}\), and that the multiplication is associative by using the associator. Next the author constructs a double factorization from two copies of the original group \(X\), where one copy of \(X\) is given a different binary operation. From the double, similarly the author constructs a category \({\mathcal D}\) which is braided, as well as being non-trivially associated. Moreover, he again forms an algebra \(D\) in \({\mathcal D}\) whose action combines both the gradings and action in the definition of \({\mathcal D}\). Then using the braiding in \(\mathcal D\), he reconstructs a coproduct on \(D\) from the tensor product structure in \({\mathcal D}\), and explicitly calculates the braided Hopf algebra structure on \(D\).
scientific article; zbMATH DE number 1878446
scientific article