Related papers: On double quantum affinization: 1. Type $\mathfrak…

In this work, we develop systematically the ``Dirichlet Hopf algebra of arithmetics'' by dualizing addition and multiplication maps. We study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra…

Let g be a simple complex finite dimensional Lie algebra and let U_q^+(g) be the positive part of the quantum enveloping algebra of g. If g is of type A_2, the group of algebra automorphisms of U_q^+(g) is a semidirect product of a…

We give a rigorous proof that the (codimension one) Connes-Moscovici Hopf algebra H_CM is isomorphic to a bicrossproduct Hopf algebra linked to a group factorisation of the group of positively-oriented diffeomorphisms of the real line. We…

Let $k$ be a field and suppose $p, q\in k$. We prove that the two affine Hecke algebras $H_q$ and $H_p$ of type $A_n$ are isomorphic as $k$-algebras if and only if $p=q^{\pm 1}$.

Let $U$ be a connected, simply connected compact Lie group with complexification $G$. Let $\mathfrak{u}$ and $\mathfrak{g}$ be the associated Lie algebras. Let $\Gamma$ be the Dynkin diagram of $\mathfrak{g}$ with underlying set $I$, and…

A Fock representation of the quantum affine algebra $U_q(\widehat{\sl}_2)$ is constructed by three bosonic fields for an arbitrary level with the help of the Drinfeld realization.

The renormalization of quantum field theory twists the antipode of a noncocommutative Hopf algebra of rooted trees, decorated by an infinite set of primitive divergences. The Hopf algebra of undecorated rooted trees, ${\cal H}_R$, generated…

We define an integral form of shifted quantum affine algebras of type $A$ and construct Poincar\'e-Birkhoff-Witt-Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these…

We establish a three-parameter Schur duality between the affine Hecke algebra of type C and a coideal subalgebra of quantum affine $\mathfrak{sl}_n$. At the equal parameter specializations, we obtain Schur dualities of types BCD.

Some duality properties for induced representations of enveloping algebras involve the character $Trad_{\goth g}$. We extend them to deformation Hopf algebras $A_{h}$ of a noetherian Hopf $k$-algebra $A_{0}$ satistying $Ext^{i}_{A_{0}}(k,…

The nonsemisimple quantum Cayley-Klein groups $ Fun(SU_{q}(2;\bf j}) $ are realized as Hopf algebra of the noncommutative functions with the dual (or Study) variables. The {\it dual} quantum algebras $ su_q(2;{\bf j}) $ are constructed and…

This is a review article on the quantum toroidal algebras, focusing on their roles in various solvable structures of 2d conformal field theory, supersymmetric gauge theory, and string theory. Using $\mathcal{W}$-algebras as our starting…

We compute the Hochschild homology of the crossed product $\Bbb C[S_n]\ltimes A^{\otimes n}$ in terms of the Hochschild homology of the associative algebra $A$ (over $\Bbb C$). It allows us to compute the Hochschild (co)homology of $\Bbb…

We introduce and define the quantum affine $(m|n)$-superspace (or say quantum Manin superspace) $A_q^{m|n}$ and its dual object, the quantum Grassmann superalgebra $\Omega_q(m|n)$. Correspondingly, a quantum Weyl algebra $\mathcal…

Quantum duality principle is applied to study classical limits of quantum algebras and groups. For a certain type of Hopf algebras the explicit procedure to construct both classical limits is presented. The canonical forms of quantized…

Let H be a connected Hopf k-algebra of finite Gel'fand-Kirillov dimension over an algebraically closed field k of characteristic 0. The objects of study in this paper are the left or right coideal subalgebras T of H. They are shown to be…

The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown…

A condition is identified which guarantees that the coinvariants of a coaction of a Hopf algebra on an algebra form a subalgebra, even though the coaction may fail to be an algebra homomorphism. A Hilbert Theorem (finite generation of the…

Let $\mathfrak{g}$ be a Borcherds-Bozec algebra, $U(\mathfrak{g})$ be its universal enveloping algebra and $U_{q}(\mathfrak{g})$ be the corresponding quantum Borcherds-Bozec algebra. We show that the classical limit of $U_{q}(\mathfrak{g})$…

We establish a degeneration isomorphism between quantum toroidal algebras and untwisted affine Yangians, valid for all untwisted affine Kac-Moody Lie algebras. Specifically, we prove that the affine Yangian $Y_\hbar(\mathfrak{g})$ is…