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A diverse collection of fusion categories may be realized by the representation theory of quantum groups. There is substantial literature where one will find detailed constructions of quantum groups, and proofs of the…
The notion and theory of the quantum space of all maps from a quantum space pioneered by So{\l}tan have been mainly focused on finite-dimensional C*-algebras which are matrix algebra bundles over a finite set $S$. We propose a modification…
There is a long-standing belief that the modular tensor categories $\mathcal{C}(\mathfrak{g},k)$, for $k\in\mathbb{Z}_{\geq1}$ and finite-dimensional simple complex Lie algebras $\mathfrak{g}$, contain exceptional connected \'etale algebras…
An equivariant topological field theory is defined on a cobordism category of manifolds with principal fiber bundles for a fixed (finite) structure group. We provide a geometric construction which for any given morphism $G \to H$ of finite…
In this paper we construct two new fusion categories and many new subfactors related to the exceptional Extended Haagerup subfactor. The Extended Haagerup subfactor has two even parts EH1 and EH2. These fusion categories are mysterious and…
In this paper, we prove that there is a canonical homotopy $(n+1)$-algebra structure on the shifted operadic deformation complex $Def(e_n\to\mathcal{P})[-n]$ for any operad $\mathcal{P}$ and a map of operads $f\colon e_n\to\mathcal{P}$.…
There is a canonical derived Poisson structure on the universal enveloping algebra $\mathcal{U}\mathfrak{a}$ of a (DG) Lie algebra $\mathfrak{a}$ that is Koszul dual to a cyclic cocommutative (DG) coalgebra. Interesting special cases of…
The trace functions for the Parafermion vertex operator algebra associated to any finite dimensional simple Lie algebra $\g$ and any positive integer $k$ are studied and an explicit modular transformation formula of the trace functions is…
We introduce a rigid analytification of the quantized coordinate algebra of a semisimple algebraic group and a quantized Arens-Michael envelope of the enveloping algebra of the corresponding Lie algebra, working over a non-archimedean field…
In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do some structure theory for rack bialgebras…
We give formulae for the multiplicities of eigenvalues of generalized rotation operators in terms of generalized Frobenius-Schur indicators in a semisimple spherical tensor category $\mathcal{C}$. In particular, this implies that the entire…
We introduce the quantum determinant for the elliptic quantum algebra $\mathcal{A}_{q,p}(\mathfrak{\widehat{gl}}_N)$ and prove that it generates the center of this algebra.
The aim of this short note is to communicate an example of a finite-dimensional Hopf algebra that does not admit a modular pair in involution in the sense of Connes and Moscovici.
We show that the restriction functor from oriented factor planar algebras to subfactor planar algebras admits a left adjoint, which we call the free oriented extension functor. We show that for any subfactor planar algebra realized as the…
Starting from a finite-dimensional representation of the Yangian $Y(\mathfrak{g})$ for a simple Lie algebra $\mathfrak{g}$ in Drinfeld's original presentation, we construct a Hopf algebra $X_\mathcal{I}(\mathfrak{g})$, called the extended…
In this paper, we study a class of Leibniz conformal algebras called quadratic Leibniz conformal algebras. An equivalent characterization of a Leibniz conformal algebra $R=\mathbb{C}[\partial]V$ through three algebraic operations on $V$ are…
We study the first and second cohomology groups of the $^*$-algebras of the universal unitary and orthogonal quantum groups $U_F^+$ and $O_F^+$. This provides valuable information for constructing and classifying L\'evy processes on these…
Looking back over 55 years of higher homotopy structures, I reminisce as I recall the early days and ponder how they developed and how I now see them. From the history of $A_\infty$-structures and later of $L_\infty$ -structures and their…
We consider several vertex operator (super)algebras closely related to $V_{-1}(\frak{sl} (n) )$, $n \ge 3$ : (a) the parafermionic subalgebra $K(\frak{sl}(n),-1)$ for which we completely describe its inner structure, (b) the vacuum algebra…
We classify finite-dimensional pointed Hopf algebras with abelian coradical, up to isomorphism, and show that they are cocycle deformations of the associated graded Hopf algebra. More generally, for any braided vector space of diagonal type…