相关论文: FRT-duals as Quantum Enveloping Algebras

We find the Hopf algebra $U_{g,h}$ dual to the Jordanian matrix quantum group $GL_{g,h}(2)$. As an algebra it depends only on the sum of the two parameters and is split in two subalgebras: $U'_{g,h}$ (with three generators) and $U(Z)$ (with…

An algebraic quantum group is a multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication…

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…

Starting with a given generalized boson algebra U_<q>(h(1)) known as the bosonized version of the quantum super-Hopf U_q[osp(1/2)] algebra, we employ the Hopf duality arguments to provide the dually conjugate function algebra Fun_<q>(H(1)).…

In this paper we establish a duality between etale Lie groupoids and a class of non-necessarily commutative algebras with a Hopf algebroid structure. For any etale Lie groupoid G over a manifold M, the groupoid algebra C_c(G) of smooth…

For any simple Lie algebra g and any complex number q which is not zero or a nontrivial root of unity, we construct a dynamical quantum group (Hopf algebroid), whose representation theory is essentially the same as the representation theory…

We study non-counital coalgebras and their dual non-unital algebras, and introduce the finite dual of a non-unital algebra. We show that a theory that parallels in good part the duality in the unital case can be constructed. Using this, we…

Let $k_q[x, x^{-1}, y]$ be the localization of the quantum plane $k_q[x, y]$ over a field $k$, where $0\neq q\in k$. Then $k_q[x, x^{-1}, y]$ is a graded Hopf algebra, which can be regarded as the non-negative part of the quantum enveloping…

We study topological Hopf algebras that are holomorphically finitely generated (HFG) as Fr\'echet Arens--Micheal algebras in the sense of Pirkovskii. Some of them, but not all, can be obtained from affine Hopf algebras by applying the…

"Co-Frobenius" coalgebras were introduced as dualizations of Frobenius algebras. Recently, it was shown in \cite{I} that they admit left-right symmetric characterizations analogue to those of Frobenius algebras: a coalgebra $C$ is…

We build on our construction of Hopf algebroids from noncommutative calculi under the further assumption of surjectivity for the calculus. We also introduce the notions of Hopf ideals and isotopy quotients for arbitrary Hopf algebroids.…

By using cocycle deformation, we construct a certain class of Hopf algebras, containing the quantized enveloping algebras and their analogues, from what we call pre-Nichols algebras. Our construction generalizes in some sense the known…

The Heisenberg double $D_q(E_2)$ of the quantum Euclidean group $\mathcal{O}_q(E_2)$ is the smash product of $\mathcal{O}_q(E_2)$ with its Hopf dual $U_q(\mathfrak{e}_2)$. For the algebra $D_q(E_2)$, explicit descriptions of its prime,…

A general theory of matrix-spherical functions for dual Hopf algebras and right coideal subalgebras is developed. We establish their existence and define their orthogonality relations. When specialized to Kolb and Letzter's quantum…

Let $B$ and $C$ be non-degenerate idempotent algebras and assume that $E$ is a regular separability idempotent in $M(B\otimes C)$. Define $A=C\otimes B$ and $\Delta:A\to M(A\otimes A)$ by $\Delta(c\otimes b)=c\otimes E\otimes b$. The pair…

Our first collection of results parametrize (filtered) actions of a quantum Borel $U_q(\mathfrak{b}) \subset U_q(\mathfrak{sl}_2)$ on the path algebra of an arbitrary (finite) quiver. When $q$ is a root of unity, we give necessary and…

We introduce the notions of Hopf quasigroup and Hopf coquasigroup $H$ generalising the classical notion of an inverse property quasigroup $G$ expressed respectively as a quasigroup algebra $k G$ and an algebraic quasigroup $k[G]$. We prove…

Suppose that we have a semisimple, connected, simply connected algebraic group $G$ with corresponding Lie algebra $\mathfrak{g}$. There is a Hopf pairing between the universal enveloping algebra $U(\mathfrak{g})$ and the coordinate ring…

Let $A$ and $B$ be two algebraic quantum groups (i.e. multiplier Hopf algebras with integrals). Assume that $B$ is a right $A$-module algebra and that $A$ is a left $B$-comodule coalgebra. If the action and coaction are matched, it is…

The relationship between comodules of a coring and flat connections is reviewed. In particular we specialise to corings which are built on a tensor product of algebra and a coalgebra. Such corings are in one-to-one correspondence with…