Prof. Makoto Yamashita

In 1993, Kazhdan and Wenzl classified the spherical tensor categories with the fusion rules of the Dynkin type A [5]. The classification tells that such categories are parametrized by two parameters. The first parameter q, which can be either generic or a root of unity, parametrizes the Drinfeld–Jimbo quantum groups Uq(slN), and arises intrinsically from the structure of Hecke algebras in a remarkable way. The second parameter τ, which is an N-th root of unity, is related to the extension theory of tensor categories later formalized by Etingof, Nikshych, and Ostrik in their work on homotopy theoretic structure of tensor categories. These categories are important because they emerge from an a priori different setting at the intersection of various fields related to Quantum Field Theory. To name a few: They give a prominent series of examples in Conformal Field Theory, where one deals with infinite-dimensional representations of infinite-dimensional structures in an algebraic and analytic setting, concretely realized by Vertex Operator Algebras or conformal nets of von Neumann algebras. The categories at positive q are related to the theory of compact quan- tum groups through the Tannaka–Krein type duality principle due to Woronowicz [9], and give rise to many examples of noncommutative homogeneous spaces in noncommutative geometry. Ever since [5], further studies have attempted to extend the Kazhdan–Wenzl classifica- tion to the other types. One prominent work is due to Tuba and Wenzl [7], who looked at the braided categories with fusion rules of the Dynkin type B, C, and D, with different methods, but the subject is considered far from complete. Recently, P. Grossman, S. Neshveyev, and M. Yamashita have advanced results by Tuba and Wenzl, but working in the setting of spherical tensor categories, and gave a similar classification for the BCD analogous to [5]. The proof is based on recent advances in the subfactor theory, and crucially uses a recent breakthrough on the structure of Lie type modular tensor categories by Schopieray [6] and Gannon [3]. After this development, the outstanding problem is to settle the case for exceptional series, namely, the Dynkin types EN for N = 6, 7, 8, F4, and G2. Among them, the most challenging case is the E series, and this was the subject of our discussions during the visit by M. Yamashita. One of the reasons why we are interested in this subject is due to its close connection with the study of the associator in a braided tensor category with applications to affine vertex operator algebras at a positive integer level [1]. Our starting point is a work by H. Wenzl [8], that analyzes the centralizer algebras EndG(V⊗n) for a suitable representation V of the Lie algebra associated to EN, N ̸= 9. These centralizer algebras are shown to be generated by the R-matrices and one suitable additional element. The discussions have been very fruitful, and we plan to make them useful to achieve contributions in the above-mentioned problems. 1