We show that the support of a simple weight module over the Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all non-trivial weight spaces of such module are infinite dimensional. As a corollary we obtain that every simple weight module over the Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra ...

Miriam Cohen raised the question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. In this paper we show that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. In particular we show that the central H-invariant elements of the Martindale ring of quotients of a module algebra for...

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true for Hopf algebras over rings. We show that over any commutative ring R tha...

Now choose a Borel subgroup B in G and a maximal torus T in B so that, if B is the opposite Borel subgroup, then B ∩ H and B ∩ H are a Borel subgroups in H and T ∩H is a maximal torus in H . We follow the convention that the roots of B are positive. If λ ∈ X(T ), then indGB(−λ) is the dual Weyl module ∇G(λ ) with highest weight λ = −w0λ and lowest weight −λ. Its dual is the Weyl module ∆G(λ). I...

We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) gradings. show that each class algebra structures on given A, there exists unique universal H together with an H-(co)module structure A such any other equivalent factors through action H. study and mentioned above group gradings, Hopf-Galois extensions, actions ...

We show that the support of a simple weight module over the Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all non-trivial weight spaces of such module are infinite dimensional. As a corollary we obtain that every simple weight module over the Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra ...

Factor H is a regulatory glycoprotein of the complement system. We expressed the three N-terminal complement control protein modules of human factor H (FH1-3) and confirmed FH1-3 to be the minimal unit with cofactor activity for C3b proteolysis by factor I. We reconstructed FH1-3 from NMR-derived structures of FH1-2 and FH2-3 revealing an approximately 105-A-long rod-like arrangement of the mod...

We introduce the notion of idempotent radical class of module coalgebras over a bialgebra B. We prove that if R is an idempotent radical class of B-module coalgebras, then every B-module coalgebra contains a unique maximal B-submodule coalgebra in R. Moreover, a B-module coalgebra C is a member of R if, and only if, DB is in R for every simple subcoalgebra D of C. The collection of B-cocleft co...