The equitable presentation of the quantum algebra $U_q(\widehat{sl_2})$ is considered. This was originally introduced by T. Ito and P. Terwilliger. In this paper, following Terwilliger's recent works (nonstandard) positive part type $U_q^{IT,+}$ its second realization (current algebra) $U_q^{T,+}$ are studied. A for given in terms a K-operator satisfying Freidel-Maillet equation condition on de...

Connection matrices were introduced in [1], where they were used to characterize graph homomorphism functions. The goal of this note is to determine the exact rank of these matrices. The result can be rephrased in terms of graph algebras, also introduced in [1]. Yet another version proves that if two k-tuples of nodes behave the same way from the point of view of graph homomorphisms, then they ...

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T (T (B)+), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)+), the double symmetric algebra of B, into a comm...

A class of C*-algebras is described for which the homomorphism from C0(0, 1] to the algebra may be classified by means of the Cuntz semigroup functor. Examples are given of algebras—simple and non-simple—for which this classification fails. It is shown that a suitable suspension of the Cuntz semigroup functor deals successfully with some of these counterexamples.

For a graph H, we compare two notions of uniquely H-colourable graphs, where one is defined via automorphisms, the second by vertex partitions. We prove that the two notions of uniquely H-colourable are not identical for all H, and we give a condition for when they are identical. The condition is related to the first homomorphism theorem from algebra.

For a graph H, we compare two notions of uniquely H-colourable graphs, where one is defined via automorphisms, the second by vertex partitions. We prove that the two notions of uniquely H-colourable are not identical for all H, and we give a condition for when they are identical. The condition is related to the first homomorphism theorem from algebra.

Let Q be a finite acyclic valued quiver. We define bialgebra structure and an integration map on the Hall algebra associated to morphism category of projective representations Q. As application, we recover surjective homomorphism defined in [12], which realizes principal coefficient quantum cluster $$\mathcal {A}\,_q(Q)$$ as sub-quotient morphisms. Moreover, also Caldero–Chapoton formula, well ...

The Lie algebra of a nuclear group is a locally convex nuclear vector space. Mathematics Subject Classification (2000): 22B05, 22E65, 46Axx 1. The Lie Algebra of Abelian Topological Groups Let G be an abelian topological group. The set of all one–parameter–subgroups L(G) := {λ : R→ G : λ is a continuous homomorphism} is called the Lie algebra of G . We define addition pointwise and scalar multi...

We use the results on the minimal basis of the centre of an Iwahori-Hecke algebra from our earlier work, as well as some additional results on the minimal basis, to describe the image and kernel of the Brauer homomorphism defined by L. Jones for Iwahori-Hecke algebras of type A.