‘az + b’ is the group of affine transformations of complex plane C. The coefficients a, b ∈ C. In quantum version a, b are normal operators such that ab = q2ba, where q is the deformation parameter. We shall assume that q is a root of unity. More precisely q = e 2πi N , where N is an even natural number. To construct the group we write an explicit formula for the Kac Takesaki operator W . It is...

let r be a ring, m a right r-module and (s,≤) a strictly ordered monoid. in this paper we will show that if (s,≤) is a strictly ordered monoid satisfying the condition that 0 ≤ s for all s ∈ s, then the module [[ms,≤]] of generalized power series is a uniserial right [[rs,≤]] ]]-module if and only if m is a simple right r-module and s is a chain monoid.

Let $A$ be a Banach algebra and $E$ be a Banach $A$-bimodule then $S = A oplus E$, the $l^1$-direct sum of $A$ and $E$ becomes a module extension Banach algebra when equipped with the algebras product $(a,x).(a^prime,x^prime)= (aa^prime, a.x^prime+ x.a^prime)$. In this paper, we investigate $triangle$-amenability for these Banach algebras and we show that for discrete inverse semigroup $S$ with...

the concepts of free modules, projective modules, injective modules and the likeform an important area in module theory. the notion of free fuzzy modules was introducedby muganda as an extension of free modules in the fuzzy context. zahedi and ameriintroduced the concept of projective and injective l-modules. in this paper we give analternate definition for projective l-modules. we prove that e...

 Let $R$ be a commutative ring with identity and $M$ be a finitely generated unital $R$-module. In this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. Moreover, we investigate some conditions which imply that the module $M$ is the direct sum of some cyclic modules and free modules. Then some properties of Fitting ideals o...

Infinitesimal bialgebras were introduced by Joni and Rota [J-R]. An infinitesimal bialgebra is at the same time an algebra and a coalgebra, in such a way that the comultiplication is a derivation. In this paper we define infinitesimal Hopf algebras, develop their basic theory and present several examples. It turns out that many properties of ordinary Hopf algebras possess an infinitesimal versi...

We show that a tensor product among representation of certain C∗-algebras induces a bialgebra. Let Õ∗ be the smallest unitization of the direct sum of Cuntz algebras O∗ ≡ C⊕O2 ⊕O3 ⊕O4 ⊕ · · · . We show that there exists a non-cocommutative comultiplication ∆ and a counit ε of Õ∗. From ∆, ε and the standard algebraic structure, Õ∗ is a C ∗-bialgebra. Furthermore we show the following: (i) The an...

The algebra of diffeomorphisms derived from general coordinate transformations on commuting coordinates is represented by differential operators on noncommutative spaces. The algebra remains unchanged, the comultiplication however is deformed, that way we have found a deformed bialgebra of diffeomorphisms. Scalar, vector and tensor fields are defined with appropriate transformation laws under t...

Suppose that S is a surface with boundary and that g and h are diffeomorphisms of S which restrict to the identity on the boundary. Let Yg, Yh, and Yh◦g be the threemanifolds with open book decompositions given by (S, g), (S, h), and (S, h ◦ g), respectively. We show that the Ozsváth-Szabó contact invariant is natural under a comultiplication map μ̃ : d HF (−Yh◦g) → d HF (−Yg)⊗ d HF (−Yh). It fo...

$r$-module. in this paper, we explore more properties of $max$-injective modules and we study some conditions under which the maximal spectrum of $m$ is a $max$-spectral space for its zariski topology.