We begin with some terminology and background, in particular we define the notions of Hopf
algebras and quantum groups. Let H be the algebra of functions of a Lie group G. Then the …

This paper improves several previously known results. First, the results describing the R-
skewsymmetric algebra and the quadratic dual of the R-symmetric algebra as Frobenius …

The universal (co) acting bi/Hopf algebras introduced by Yu.\, I.~ Manin, M.~ Sweedler and
D.~ Tambara, the universal Hopf algebra of a given (co) module structure, as well as the …

Let H be a Hopf algebra that is ℤ-graded as an algebra. We provide sufficient conditions for
a 2-cocycle twist of H to be a Zhang twist of H. In particular, we introduce the notion of a …

We introduce the concept of a universal quantum linear semigroupoid (UQSGd), which is a
weak bialgebra that coacts on a (not necessarily connected) graded algebra A universally …

We introduce the notion of quantum-symmetric equivalence of two connected graded
algebras, based on Morita–Takeuchi equivalences of their universal quantum groups, in the …

Let g and h be two Lie algebras with h finite dimensional and consider A= A (h, g) to be the
corresponding universal algebra as introduced in [4]. Given an A-module U and a Lie h …

We show that if two $ m $-homogeneous algebras have Morita equivalent graded module
categories, then they are quantum-symmetrically equivalent, that is, there is a monoidal …

We introduce the universal algebra of two Poisson algebras P and Q as a commutative
algebra A:= P (P, Q) satisfying a certain universal property. The universal algebra is shown …

We construct a family of cogroupoids associated to preregular forms and recover the Morita–
Takeuchi equivalence for Artin–Schelter regular algebras of dimension two, observed by …