Related papers: Path subcoalgebras, finiteness properties and quan…
Semigroup algebras admit certain `coherent' deformations which, in the special case of a path algebra, may associate a periodic function to an evolving path; for a particle moving freely on a straight line after an initial impulse, the wave…
Our main purpose is to introduce the notion of trans-datum for quivers, and apply it to the study of automorphism groups of path coalgebras and algebras. We observe that any homomorphism of path coalgebras is uniquely determined by a…
Right coideal subalgebras are interesting substructures of Hopf algebras such as quantum groups. Examples of right coideal subalgebras are the quantum Borel part as well as quantum symmetric pairs. Classifying right coideal subalgebras is a…
Each quiver corresponds to a path semigroup, and such a path semigroup also corresponds to an associative K-algebra over an algebraically closed field K. Let Q be a quiver and S_Q, KQ be its path semigroup, path algebra, respectively. In…
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 introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hecke operators. At the intuitive level, these are algebras of…
For any k-coalgebra C it is shown that similar quasi-finite C-comodules have strongly equivalent coendomorphism coalgebras; (the converse is in general not true). As an application we give a general result about codepth two coalgebra…
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 use braided groups to introduce a theory of $*$-structures on general inhomogeneous quantum groups, which we formulate as {\em quasi-$*$} Hopf algebras. This allows the construction of the tensor product of unitary representations up to…
We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear…
For a set-endofunctor $F$, we extend the notion of universal $F$-coalgebras to $F$-graphs. These generalized coalgebras are models for various types of graphs, such as (un)directed (hyper)graphs, relational structures or fuzzy graphs. The…
We provide a construction of minimal injective resolutions of simple comodules of path coalgebras of quivers with relations. Dual to Calabi-Yau condition of algebras, we introduce the Calabi-Yau condition to coalgebras. Then we give some…
The paper presents applications of Toeplitz algebras in Noncommutative Geometry. As an example, a quantum Hopf fibration is given by gluing trivial U(1) bundles over quantum discs (or, synonymously, Toeplitz algebras) along their…
It is proved that the entire multi-parameter (small-)quantum groups of symmetrizable Kac-Moody algebras can be realized as certain subquotients of the cotensor Hopf algebras. This is an axiomatic construction. Hopf 2-cocycle deformations…
Lead by examples we introduce the notions of Hopf algebra and quantum group. We study their geometry and in particular their Lie algebra (of left invariant vectorfields). The examples of the quantum sl(2) Lie algebra and of the quantum…
We try to classify Hopf algebras with the dual Chevalley property of discrete corepresentation type over an algebraically closed field $\Bbb{k}$ with characteristic 0. For such Hopf algebra $H$, we characterize the link quiver of $H$ and…
We study the restriction of representations of Cayley-Hamilton algebras to subalgebras. This theory is applied to determine tensor products and branching rules for representations of quantum groups at roots of 1.
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum groups, we…
Adapting a recent work of Brannan et al., on extending graph $C^*$-algebras to Quantum graphs, we introduce "Quantum Quivers" as an analogue of quivers where the edge and vertex set has been replaced by a $C^*$-algebra and the maps between…
L-infinity morphisms are studied from the point of view of perturbative quantum field theory, as generalizations of Feynman expansions. The connection with the Hopf algebra approach to renormalization is exploited. Using the coalgebra…