International Journal of Mathematics

There is a long-standing belief that the modular tensor categories ${𝒞}(𝔤,k)$, for $k\in {\mathbb{Z}}_{\ge 1}$ and finite-dimensional simple complex Lie algebras $𝔤$, contain exceptional connected étale algebras (sometimes called quantum subgroups) at only finitely many levels $k$. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here, we confirm this conjecture when $𝔤$ has rank 2, contributing proofs and explicit bounds when $𝔤$ is of type $B_2$ or $G_2$, adding to the previously known positive results for types $A_1$ and $A_2$.

We describe all rigid algebras and all irreducible components in the variety of four-dimensional Leibniz algebras ${𝔏𝔢𝔦𝔟}_4$ over $ℂ.$ In particular, we prove that the Grunewald–O'Halloran conjecture is not valid and the Vergne conjecture is valid for ${𝔏𝔢𝔦𝔟}_4.$

We compute the group of Morita auto-equivalences of the even parts of the $ADE$ subfactors, and Galois conjugates. To achieve this, we study the braided auto-equivalences of the Drinfeld centers of these categories. We give planar algebra presentations for each of these Drinfeld centers, which we leverage to obtain information about the braided auto-equivalences of the corresponding categories. We also perform the same calculations for the fusion categories constructed from the full $ADE$ subfactors.

Of particular interest, the even part of the $D_{10}$ subfactor is shown to have Brauer–Picard group $S_3\times S_3$. We develop combinatorial arguments to compute the underlying algebra objects of these invertible bimodules.

We investigate the uniqueness problems for meromorphic functions with higher multiplicities of zeros and poles. As a consequence, we present a class of strong uniqueness polynomials for meromorphic functions (SUPM) of degree $6$. We also proved that there exist the sets $S$ of seven elements such that for arbitrary two meromorphic functions $f$ and $g$, the condition $E_{f^d}(S)=E_{g^d}(S),$ with $d\ge 2,$ implies $f=\xi g,$ where $\xi$ is a root of unity of degree $d.$

Let $n\in \mathbb{Z}$ with $n\ge 3$. Let $S_n$ and $A_n$ denote, respectively, the symmetric group and alternating group on $n$ letters. Let $m$ be an indeterminate, and define

In this paper, we prove the existence of coupled Kähler–Einstein metrics on complex manifolds whose canonical bundle is ample. These metrics were introduced and their existence in the said case was proven by Hultgren and Nyström using calculus of variations. We prove the result using the method of continuity. In the process of proving estimates, akin to the usual Kähler–Einstein metrics, we reduce existence in the Fano case to a $C^0$ estimate.