量子代数

We develop a string-net construction of a modular functor whose algebraic input is a pivotal bicategory; this extends the standard construction based on a spherical fusion category. An essential ingredient in our construction is a graphical…

We use deformation sequences of (Hopf) algebras, extending the results of Negron and Pevtsova, to show that bosonizations of some suitable braided Hopf algebras by some suitable finite-dimensional Hopf algebras have finitely generated…

We use the $q$-characters to compute explicit expressions of the $R$-matrices for first fundamental representations of all types of twisted quantum affine algebras.

We discuss applications of the $q$-characters to the computation of the $R$-matrices. In particular, we describe the $R$-matrix acting in the tensor square of the first fundamental representation of E$_8$ and in a number of other cases,…

We consider algebras and Frobenius algebras, internal to a monoidal category, that are graded over a finite abelian group. For the case that A is a twisted group algebra in a linear abelian monoidal category we obtain a graded…

In this paper, we aim to study abelian extensions for some infinite group. We show that the Hopf algebra $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$ constructed through abelian extensions of $\Bbbk F$ by $\Bbbk^G$ for some (infinite) group $F$ and…

In this paper, we present explicit actions of braid group on the universal enveloping superalgebra ${\boldsymbol U}(\mathfrak{{q}}_n)$ and the quantum queer superalgebra ${\boldsymbol U}_{\!{v}}(\mathfrak{{q}}_{n})$. Then we provide a new…

We present several related examples of Hopf algebroids over a universal enveloping algebra which are of the scalar extension Hopf algebroid type and explain their origin in Lie and algebraic group theory.

A phenomenon of "algebraic self-similarity" on 3d cubic lattice, providing what can be called an algebraic analogue of Kadanoff--Wilson theory, is shown to possess a 4d version as well. Namely, if there is a $4\times 4$ matrix $A$ whose…

In this paper, we prove the equivalence between two ribbon tensor categories. On the one hand, we consider the category of modules of the Virasoro vertex operator algebra with generic central charge (generic Virasoro VOA) generated by those…

Let $V$ be a vertex algebra and $g$ an automorphism of $V$ of order $T$. We construct a sequence of associative algebras $\tilde{A}_{g,n}(V )$ for any $n\in(1/T)\mathbb{N}$, which are not depend on the conformal structure of $V$. We show…

Let ${\mathbf U}_q^-$ be the negative half of a quantum group of finite type. Let $P$ be the transition matrix between the canonical basis and a PBW basis of ${\mathbf U}_q^-$. In the case ${\mathbf U}_q^-$ is symmetric, Antor gave a simple…

We study the (extended) orthogonal Yangians associated to the Lie algebras types $B$ and $D$ over a field of positive characteristic. We define the $p$-center for the Yangians and obtain an explicit description of the center in terms of…

We study tensor products of two-dimensional evaluation $U_q\widehat{\mathfrak{sl}}_2$-modules at generic values of $q$, $U_q\widehat{\mathfrak{sl}}_2$ homomorphisms between them, and closely related subjects.

We investigate the splitting property of quasitriangular Hopf algebras through the lens of twisted tensor products. Specifically, we demonstrate that an infinite-dimensional quasitriangular Hopf algebra possesses the splitting property if…

We introduce a new family of graded 2-categories generalizing the 2-quantum groups introduced by Khovanov, Lauda and Rouquier. We use them to categorify quasi-split iquantum groups in all symmetric types.

Let $\C$ be a self-dual fusion category of rank $4$ which has a nontrivial proper fusion subcategory. We identify three new families of Grothendieck rings for $\C$: one of them is completely determined, the other two are parameterized by…

We revisit the classical Goddard-Kent-Olive coset construction. We find the formulas for the highest weight vectors in coset decomposition and calculate their norms. We also derive formulas for matrix elements of natural vertex operators…

Building on the work of Nenciu we provide examples of non-factorizable ribbon Hopf algebras, and introduce a stronger notion of non-factorizability. These algebras are designed to provide invariants of $4$-dimensional $2$-handlebodies up to…

The $q$-Schur category is a $\mathbb{Z}[q,q^{-1}]$-linear monoidal category closely related to the $q$-Schur algebra. We explain how to construct it from coordinate algebras of quantum $GL_n$ for all $n \geq 0$. Then we use Donkin's work on…