A descent class, in the symmetric group S,, is the collection of permutations with a given descent set. It was shown by L. Solomon (J. Algebra 41 (1976), 255-264) that the product (in the group algebra Q(S,)) of two descent classes is a linear combination of descent classes. Thus descent classes generate a subalgebra of Q(.S,). We refer to it here as Solomon’s descenr algebra and denote it by C...

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue...

Contrary to the classical methods of quantum mechanics, the deformation quan-tization can be carried out on phase spaces which are not even topological manifolds. In particular, the Moyal star product gives rise to a canonical functor F from the category of affine analytic spaces to the category of associative (in general, non-commutative) C-algebras. Curiously, if X is the n-tuple point, x n =...

Theorem 1.1. Let A be a semisimple complex Banach algebra with the following properties: (i) for every closed right ideal R in A, there exists a closed left ideal L such that R∩L= {0} and R+ L= A (each a∈ A can be written in the form a= a1 + a2 with a1 ∈ R, a2 ∈ L); (ii) if a,b in A are such that ab = ba= 0, then ‖a+ b‖2 = ‖a‖2 +‖b‖2. Then A is a commutative proper H∗-algebra [1]. It is easy to...

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue...

1. Basics 2 1.1. Commutants 2 1.2. Opposite Rings 3 1.3. Units 3 1.4. Ideals 4 1.5. Modules 4 1.6. Division rings 5 1.7. Endomorphism rings 7 1.8. Monoid Rings 8 1.9. Noetherian and Artinian Rings 11 1.10. Simple rings 15 1.11. Prime ideals and prime rings 17 1.12. Notes 18 2. Wedderburn-Artin Theory 18 2.1. Semisimple modules and rings 18 2.2. Wedderburn-Artin I: Semisimplicity of Mn(D) 20 2.3...

In [28], for any real non associative algebra of dimension \(m\geq2\), having \(k\) linearly independent nilpotent elements \(n_{1}\), \(n_{2}\), …, \(n_{k},\) \(1\leq k\leq m-1\), Mencinger and Zalar defined near idempotents nilpotents associated to \(n_{k}\). Assuming \(\mathcal{N}_{k}\mathcal{N}_{k}=\left\{ 0\right\}\), where \(\mathcal{N} _{k}=\operatorname*{span}\left\{ n_{1},n_{2},\ldots,...

Let S(RG) be a normed Sylow p-subgroup in a group ring RG of an abelian group G with p-component Gp and a p-basic subgroup B over a commutative unitary ring R with prime characteristic p. The first central result is that 1 + I(RG;Bp) + I(R(p)G;G) is basic in S(RG) and B[1 + I(RG;Bp) + I(R(p )G;G)] is p-basic in V (RG), and [1 + I(RG;Bp) + I(R(p )G;G)]Gp/Gp is basic in S(RG)/Gp and [1 + I(RG;Bp)...