Preface These notes are based on a course given at the Tata Institute of Fundamental Research in the beginning of 1990. The aim of the course was to describe the solution by O. Mathieu of some conjectures in the representation theory of semi-simple algebraic groups. These conjectures concern the inner structure of dual Weyl modules and some of their analogues. Recall that if G is a (connected, ...

Let G be an affine algebraic group over an algebraically closed field k and let H be a closed subgroup of G. If V is a rational H-module (a comodule for the coordinate ring of H) there is a now well-known notion of an induced module VI G for G, defined as the space Morph/~(G, V) of all H-equivariant morphisms from G to a finite dimensional subspace of V, with obvious G-action. The question aris...

Let $G$ be a group with identity $e$. $R$ $G$-graded commutative ring and $M$ graded $R$-module. In this paper, we introduce the concept of $WAG2$-absorbing submodule. A number results concerning these classes submodules their homogeneous components are given.
 $N=\bigoplus _{h\in G}N_{h}$ submodule $h\in G.$ We say that $N_{h}$ is $h$-$WAG2$-absorbing $R_{e}$-module $M_{h}$ if $N_{h}\neq ...

Let H(S) be the Hardy space on the unit sphere S in C, n ≥ 2. Then H(S) is a natural Hilbert module over the ball algebra A(B). Let Mz1 , ..., Mzn be the module operators corresponding to the multiplication by the coordinated functions. Each submodule M ⊂ H(S) gives rise to the module operators ZM,j = Mzj |M, j = 1, ..., n, on M. In this paper we establish the following commonly believed, but n...

Let G be a connected reductive group. We find a necessary and sufficient condition for a quasiaffine homogeneous space of G/H to be embeddable into an irreducible G-module. If H is reductive we also find a necessary and sufficient condition for a closed embedding of G/H into an irreducible module to exist. These conditions are stated in terms of the group of central automorphisms of G/H .

In this paper, we present a dataow based approa h to testing aspe ts. Our approa h onsiders three levels of testing for aspe ts, i.e., intra-module testing, inter-module testing, and intra-aspe t testing. For an individual module (i.e., a pie e of advi e, an introdu tion, or a method), or a publi module along with other modules it alls in an aspe t, we perform intra-module testing or inter-modu...

Motivated by the study of von Neumann regular skew groups as carried out by Alfaro, Ara and del Rio in [1] we investigate regular and biregular Hopf module algebras. If A is an algebra with an action by an affine Hopf algebra H, then any H-stable left ideal of A is a direct summand if and only if A is regular and the invariance functor (−) induces an equivalence of A -Mod to the Wisbauer catego...

The characterization of H-prime radical is given in many ways. Meantime, the relations between the radical of smash product R#H and the H-radical of Hopf module algebra R are obtained. 0 Introduction and Preliminaries In this paper, let k be a commutative associative ring with unit, H be an algebra with unit and comultiplication △ ( i.e. ∆ is a linear map: H → H ⊗H), R be an algebra over k (R m...

Let H be a Hopf algebra over a commutative ring k with unity and σ : H ⊗ H −→ k be a cocycle on H. In this paper, we show that the Yetter-Drinfeld module category of the cocycle deformation Hopf algebra H is equivalent to the Yetter-Drinfeld module category of H. As a result of the equivalence, the “quantum Brauer” group BQ(k,H) is isomorphic to BQ(k,H). Moreover, the group Gal(HR) constructed ...

in this paper we proved that every g-riesz basis for hilbert space $h$ with respect to $k$ by adding a condition is a riesz basis for hilbert $b(k)$-module $b(h,k)$. this is an extension of [a. askarizadeh,m. a. dehghan, {em g-frames as special frames}, turk. j. math., 35, (2011) 1-11]. also, we derived similar results for g-orthonormal and orthogonal bases. some relationships between dual fram...