Semiinvariants for hopf algebra actions

LetH be a Hopf algebra andM a representation or a corepresentation ofH. In this paper we study semiinvariants ofM. This notion generalizes the known concept of weight spaces in the context of representations of Lie algebras. Our best results are attained for pointed Hopf algebras, and semiinvariants which are related to the coradical filtration ofH.     

Smashed extensions for hopf algebraSL q (2)

The structure of the smashed extensions forSL ₂(2) over a fixed algebraC is discussed, and the isomorphic classes of smashed exrersions forSL _q(2) overC are determined. Project supported by the National Natural Science Foundation of China (Grant No. 19671070).     

The Banach algebra generated by a contraction

Let be a contraction on a Banach space and the Banach algebra generated by . Let be the unitary spectrum (i.e., the intersection of with the unit circle) of . We prove the following theorem of Katznelson-Tzafriri type: If is at most countable, then the Gelfand transform of vanishes on if and only if

     

The isomorphism problem for varieties generated by a two-element algebra

This paper presents a complete classification of the isomorphism problem for varieties and quasivarieties generated by a two-element algebra.Presented by S. Burris.     

The Banach algebra generated by a C0-semigroup

Let $\mathbf{T}={\{T(t)\}}_{t?0}$ be a bounded $C_0$-semigroup on a Banach space with generator A. We define $A_{\mathbf{T}}$ as the closure with respect to the operator-norm topology of the set $\{\stackrel{?}{f}(\mathbf{T}):f\in L^1({\mathbb{R}}_{+})\}$, where $\stackrel{?}{f}(\mathbf{T})={\int }_0^{\infty }f(t)T(t)\mathrm{d}t$ is the Laplace transform of $f\in L^1({\mathbb{R}}_{+})$ with respect to the semigroup T. Then $A_{\mathbf{T}}$ is a commutative Banach algebra. It is shown that if the unitary spectrum $\sigma (A)\cap \mathrm{i}\mathbb{R}$ of A is at most countable, then the Gelfand transform of $S\in A_{\mathbf{T}}$ vanishes on $\sigma (A)\cap \mathrm{i}\mathbb{R}$ if and only if, ${\lim }_{t\to \infty }6T(t)S6=0$. Some applications to the semisimplicity problem are given. To cite this article: H. Mustafayev, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Invertibility symbol for a Banach algebra generated by two idempotents and a shift

A Banach algebra generated by two idempotentsp, r, identitye and a shiftv which satisfy the conditionspv=vp andrv=v(e–r) is investigated. It is proved that all irreducible representations of algebraA are two- and four-dimensional. The explicit form of these representations is obtained. An Invertibility Symbol is constructed. Some examples are considered.     

Cleft extensions of a family of braided Hopf algebras

We introduce a family of braided Hopf algebras that (in characteristic zero) generalizes the rank 1 Hopf algebras introduced by Krop and Radford and we study its cleft extensions.     

Commutative Abelian Galois extensions of a Banach algebra

Let A be a commutative unital Banach algebra with connected maximal ideal space X. We show that the Gelfand transform induces an isomorphism between the group of commutative Galois extensions of A with given finite Abelian Galois group, and the corresponding group of extensions of C(X). This result is applied, when X is sufficiently nice, to construct a separable projective finitely generated faithful Banach A-algebra whose maximal ideal space is a given finitely fibered covering space of X.