量子代数

Nondegenerate cycle sets were introduced by Rump as an algebraic framework for nondegenerate, involutive solutions to the Yang--Baxter equation. Nondegenerate cycle set structures on abelian groups, such as translation-invariant and…

In this paper, we construct braiding structures on the multi-interval Jones-Wassermann subfactor planar algebra associated with any unitary modular fusion category. Utilizing this construction, we provide a new proof of the self-duality of…

We discuss noncommutative differential geometry from a vector field centric point of view. This is based on the notion of first order vector field calculus (FOVC), which has been previously introduced by Borowiec under the name Cartan pair.…

We study cocentral split abelian Hopf algebra extensions over an algebraically closed field of characteristic zero. The kernel is $k^V$ and the quotient is $k\Gamma$, where $V$ is finite abelian and $\Gamma$ acts on $V$. For a fixed action,…

We construct a $\frac{\mathbb{Z}}{2}$-graded meromorphic open-string vertex algebra from a finite-dimensional vector space with a nondegenerate symmetric bilinear form, together with its canonically twisted module. This algebra is generated…

We develop the representation theory of vertex algebras over arbitrary rings using higher Zhu algebras and mode transition algebras. Among our results, we give several equivalent conditions for rationality of M\"obius vertex algebras over a…

We introduce the left, right, and full center of an algebra in a monoidal 2-category and prove their Morita invariance. This categorifies Davydov's theory of centers of algebras in monoidal categories, and specializes to give a uniform,…

In this paper, we continue the study of the reflection theory of Nichols algebras over coquasi-Hopf algebras with bijective antipode. We prove that for a tuple of finite-dimensional simple Yetter-Drinfeld modules admitting all reflections,…

In this paper, the universal KZB connection on the configuration space of points on a closed Riemann surface of an arbitrary genus, introduced by Enriquez, is lifted to the configuration space of points with tangent vectors. This lifted…

The universal R-matrix of the quantum affine algebra associated to a finite-dimensional simple complex Lie algebra admits a Gauss decomposition into an uper unitriangular part, an abelian part, and a lower unitriangular part. In this paper,…

We review the classical theory of principal bundles, with particular emphasis on frame bundles and $G$-structures. We then develop the noncommutative framework by introducing the necessary notions of differential calculi, Hopf algebras,…

The recently introduced formalism of chiral cluster seeds replaces quantum cluster variables with deformed vertex operators. In this framework, a decorated quiver associated with a seed encodes the operator product expansions of the…

We study the double affine Hecke algebra (DAHA) of type $(C_n^\vee,C_n)$ from the perspective of deformation theory. First, we provide a zeros-and-residues realization of this algebra, extending the construction of Ginzburg, Kapranov, and…

We identify the zonal and character spherical functions for quantum symmetric pairs with the symmetric Koornwinder--Macdonald polynomials. To this end, the methods of Letzter's 2004 paper are translated to modern conventions and right…

The primary aim of this paper is to provide an explicit construction of Kontsevich graphs whose integrals give certain multiple zeta values. Furthermore, by using this construction, we explicitly determine the weights of $\mathbb{Z}$-linear…

Given a non-zero polynomial $P(x)$, we study Fuchsian differential operators of the form $L=\partial_x^2-u(x)$ such that for all $\lambda\in\mathbb{C}$ the operator $L+\lambda P(x)$ is monodromy free. We prove that all such operators are…

We study spaces of conformal blocks associated with line bundles over elliptic curves, with coefficients in a vertex algebra. For vertex algebras satisfying suitable finiteness and semisimplicity conditions, which are met by all admissible…

We develop a graphical calculus for monoidal categories equipped with twisted pivotal structures, which are a generalization of pivotal structures originating from the study of orientation structures in the context of the Cobordism…

We develop the theory of $\hbar$-vertex algebras, algebraic structures closely related to vertex algebras but with a deformed translation covariance axiom. We establish their structure theory, including analogues of Goddard's Uniqueness…

In this paper, we introduce and study shifted twisted quantum affine algebras which provide a twisted counterpart of the theory of shifted quantum affine algebras. The shifted twisted quantum affine algebra $\U_q^{\mu_+,\mu_-}(\hgs)$ is…