An algebra homomorphism ψ from the q-deformed algebra U q (iso 2) with generating elements I, T 1 , T 2 and defining relations [I, T 2 ] q = T 1 , [T 1 , I] q = T 2 , [T 2 , T 1 ] q = 0 (where [A, B] q = q 1/2 AB − q −1/2 BA) to the extensionˆU q (m 2) of the Hopf algebra U q (m 2) is constructed. The algebra U q (iso 2) at q = 1 leads to the Lie algebra iso 2 ∼ m 2 of the group ISO(2) of motio...

Let CK∗ denote the C ∗-algebra defined by the direct sum of all Cuntz-Krieger algebras. We introduce a comultiplication ∆φ and a counit ε on CK∗ such that ∆φ is a nondegenerate ∗-homomorphism from CK∗ to CK∗ ⊗ CK∗ and ε is a ∗-homomorphism from CK∗ to C. From this, CK∗ is a counital non-commutative non-cocommutative C ∗bialgebra. Furthermore, C∗-bialgebra automorphisms, a tensor product of repr...

In this paper we introduce the notion of $varphi$-commutativity for a Banach algebra $A$, where $varphi$ is a continuous homomorphism on $A$ and study the concept of $varphi$-weak amenability for $varphi$-commutative Banach algebras. We give an example to show that the class of $varphi$-weakly amenable Banach algebras is larger than that of weakly amenable commutative Banach algebras. We charac...

In this paper, we introduce an algebra H from a subspace lattice with respect to a fixed flag which contains its incidence algebra as a proper subalgebra. We then establish a relation between the algebra H and the quantum affine algebra Uq1/2(ŝl2), where q denotes the cardinality of the base field. It is an extension of the well-known relation between the incidence algebra of a subspace lattice...

Forest algebras are defined for investigating languages of forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors [9]. We show that several parameters on forests can be realized as forest algebra homomorphisms from the free forest algebra into algebras which retain the equational axioms of forest algebras. This includes the number of nodes, the n...

We investigate how a C*-algebra could consist of functions on a noncommutative set: a discretization of a C*-algebra A is a ∗-homomorphism A → M that factors through the canonical inclusion C(X) ⊆ `∞(X) when restricted to a commutative C*-subalgebra. Any C*-algebra admits an injective but nonfunctorial discretization, as well as a possibly noninjective functorial discretization, where M is a C*...

Every finite dimensional Hopf algebra is a Frobenius algebra, with Frobenius homomorphism given by an integral. The Nakayama automorphism determined by it yields a decomposition with degrees in a cyclic group. For a family of pointed Hopf algebras, we determine necessary and sufficient conditions for this decomposition to be strongly graded.

Definition 1.1. A Poisson algebra is an associative algebra A over a field K (fixed, of characteristic zero), equipped with a Lie bracket {−,−} such that {x,−} is a derivation for any x ∈ A, i.e. {x, yz} = {x, y}z + y{x, z}. Definition 1.2. A Poisson structure on a manifold M is a Poisson bracket {−,−} on the algebra C∞(M). Example 1.3. On T ∗Rn with position coordinates q1, ..., qn and momentu...

Let CK∗ denote the C ∗-algebra defined by the direct sum of all Cuntz-Krieger algebras. We introduce a comultiplication ∆φ and a counit ε on CK∗ such that ∆φ is a nondegenerate ∗-homomorphism from CK∗ to CK∗ ⊗ CK∗ and ε is a ∗-homomorphism from CK∗ to C. From this, CK∗ is a counital non-commutative non-cocommutative C ∗bialgebra. Furthermore, C∗-bialgebra automorphisms, a tensor product of repr...

For finitary set functors preserving inverse images several concepts of coalgebras A are proved to be equivalent: (i) A has a homomorphism into the initial algebra, (ii) A is recursive, i.e., A has a unique coalgebra-to-algebra morphism into any algebra, and (iii) A is parametrically recursive. And all these properties mean that the system described by A always halts in finitely many steps.