量子代数

Among simple Z-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C_{2}: these are the Lie superalgebra k^L(1|6) of vector fields on the…

We introduce and study a family of power series, which we call Theta series, whose coefficients are in the tensor square of a quantum loop algebra. They arise from a coproduct factorization of the T-series of Frenkel--Hernandez, which are…

We define the notion of quantized function algebras at $q=0$ or crystallization of the $q$ deformations of the type $A_{n}$ compact Lie groups at the $C^*$-algebra level. The $C^{*}$-algebra $A_{n}(0)$ is defined as a universal…

We prove a general mirror duality theorem for a subalgebra $U$ of a simple conformal vertex algebra $A$ and its commutant $V=\mathrm{Com}_A(U)$. Specifically, we assume that $A\cong\bigoplus_{i\in I} U_i\otimes V_i$ as a $U\otimes…

We derive explicit closed formulas for the Kirillov-Kostant-Souriau (KKS) coaction maps of open path regularized holonomies of the Knizhnik-Zamolodchikov (KZ) equation, and the corresponding Poisson brackets for the Lie algebra ${\rm gl}(N,…

Let $H$ be a weak Hopf algebra with a bijective antipode and $A$ an $H$-comodule Poisson algebra. In this paper, we mainly generalize the fundamental theorem of Poisson Hopf modules to the case of weak Hopf algebras. Besides we will deduce…

The first quantum group cohomology with trivial coefficients of the discrete dual of any unitary easy quantum group is computed. That includes those potential quantum groups whose associated categories of two-colored partitions have not yet…

In 1999, Fomin-Kirillov introduced the quadratic algebras $\mathcal{E}_m$ in terms of generators and relations which are the universal quadratic cover of the algebra generated by divided difference operators $\partial_{ij}$ acting on the…

We establish a structure theorem, analogous to the classical result of Milnor and Moore, for differential graded Hopf algebras: any differential Hopf algebra $H$ that is free as a coalgebra carries an underlying $B_\infty$ algebra structure…

We describe a proof of the following folklore theorem: If $\cX = G/K$ is the homogeneous space of a simply connected compact semisimple Lie group with Poisson-Lie stabilizers, then the $q$-deformed algebras of regular functions $\CC[\cX_q]$…

We analyze the $G$-skein theory invariants of the 3-torus $T^3$ and the two-torus $T^2$, for the groups $G = GL_N, SL_N$ and for generic quantum parameter. We obtain formulas for the dimension of the skein module of $T^3$, and we describe…

We construct a quantum Dolbeault double complex $\oplus_{p,q}\Omega^{p,q}$ on the quantum plane $\Bbb C_q^2$. This solves the long-standing problem that the standard differential calculus on the quantum plane is not a $*$-calculus, by…

We elucidate the comment in (Kapranov-Vasserot, Adv.\ Math., 2011, Remark 5.3.4) that the $1|1$-dimensional factorization structure of the formal superloop space of a smooth algebraic variety $X$ induces the $N_K=1$ SUSY vertex algebra…

We consider the restricted Jordan plane in characteristic $2$, a finite-dimensional Nichols algebra quotient of the Jordan plane that was introduced by Cibils, Lauve and Witherspoon. We extend results from \texttt{arXiv:2002.02514} on the…

We study the Yang-Baxter algebras $A(K,X,r)$ associated to finite set-theoretic solutions $(X,r)$ of the braid relations. We introduce an equivalent set of quadratic relations $\Re\subseteq G$, where $G$ is the reduced Gr\"obner basis of…

After recalling the notion of higher roots (or hyper-roots) associated with "quantum modules" of type $(G, k)$, for $G$ a semi-simple Lie group and $k$ a positive integer, following the definition given by A. Ocneanu in 2000, we study the…

In arXiv:1811.04649, we extended the Dong-Mason theorem on irreducibility of modules for cyclic orbifold vertex algebras to the entire category weak modules and applied this result to Whittaker modules. In this paper we present further…

We construct a generalization of the $C^\vee C_1$-type double affine Hecke algebra for the skein algebra on the twice-punctured torus $\Sigma_{1,2}$ using the Heegaard dual of the Iwahori--Hecke operator recently introduced in our previous…

We construct Sugawara operators for the quantum affine algebra of type $A$ in an explicit form. The operators are associated with primitive idempotents of the Hecke algebra and parameterized by Young diagrams. This generalizes a previous…

The copointed liftings of the Fomin-Kirillov algebra $\mathcal{FK}_3$ over the algebra of functions on the symmetric group $\mathbb{S}_3$ were classified by Andruskiewitsch and the author. We demonstrate here that those associated to a…