In this paper we show that a commutative semisimple ring is always a Smarandache ring. We will also give a necessary and sufficient condition for group algebra to be a Smarandache ring. Examples are provided for justification.

In this paper we relate two constructions of representations of semisimple Lie groups constructions that appear quite different at first glance. Homogeneous vector bundles are one source of representations: if a real semisimple Lie group Go acts on a vector bundle E --* M over a quotient space M=Go/Ho, then Go acts also on the space of sections C~(M, E), and on any subspace VcC~(M, E) defined b...

1. Notation. The object of this note is to announce some results on representations of complex semisimple Lie groups and Lie algebras. © is a semisimple Lie algebra over C, the field of complex numbers. ®, considered over i?, the field of real numbers, is denoted by ®0. ^ is a Cartan subalgebra of ®, W, the Weyl group of (®, Ï)). We use the standard terminology in the theory of semisimple Lie a...

The theory of minimal types for representations of complex semisimple Lie groups [K. R. Parthasarathy, R. Ranga Rao and V. S. Varadarajan, Ann. of Math. (2) 85 (1967), 383-429, Chapters 1, 2 and 3] is reformulated so that it can be generalized, at least partially, to real semisimple Lie groups. A rather complete extension of the complex theory is obtained for the semisimple Lie groups of real r...

Introduction. In this paper we give a sufficient condition that an algebra have a minimal left (or right) ideal. Specifically, we prove that if A is a complex semisimple Banach algebra with the property that the spectrum of every element in A is at most countable, then A has a minimal left ideal. If A is an ^4*-algebra, we prove that A has a minimal left ideal if the spectrum of every self-adjo...

We exhibit the primitive central idempotents of a semisimple group algebra of a finite nilpotent group over an arbitrary field (without using group characters), examining the abelian case separately. Our result extends and improves the main result in [1].

The Lie algebra of a compact semisimple Lie group G is determined by the degrees of the irreducible representations of G. However, two different groups can have the same representation degrees.

We give a soft geometric proof of the classical result due to Conn stating that a Poisson structure is linearizable around a singular point (zero) at which the isotropy Lie algebra is compact and semisimple.

Let g be a complex semisimple Lie algebra, and let G be a complex semisimple group with trivial center whose root system is dual to that of g. We establish a graded algebra isomorphism H q (Xλ,C) ∼= Sg e/Iλ, where Xλ is an arbitrary spherical Schubert variety in the loop Grassmannian for G, and Iλ is an appropriate ideal in the symmetric algebra of g, the centralizer of a principal nilpotent in...

We give a coadjoint orbit's diffeomorphic deformation between the classical semisimple case and semi-direct product given by Cartan decomposition. The two structures admit Hermitian symplectic form defined in complex Lie algebra. provide some applications such as constructions of Lagrangian submanifolds.