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Abstract: We consider a family of parabolic subalgebras of a simple Lie algebra of type and give a full description of the adjoint homology of the nilradicals of as a module over its Levi factor. All the nilradicals considered are 2-step nilpotent. [Copyright &y& Elsevier]

Abstract: The structure of a finitely presented monomial algebra over a field K is described. Here X is a finitely generated free monoid and I is a prime ideal of X that is finitely generated. As an application, a new structural proof of the recent result of Bell and Pekcagliyan [J. Bell, P. Pekcagliyan, Primitivity of finitely presented monomial algebras, preprint, arXiv: 0712.0815v1] on the primitivity of such algebras is presented, which yields a positive solution to the trichotomy problem, raised by Bell and Smoktunowicz [J. Bell, A. Smoktunowicz, The prime spectrum of algebras of quadratic growth, J. Algebra 319 (2008) 414–431], in the finitely presented case. Our approach is based on a new result on the form of prime Rees factors of semigroups satisfying the ascending chain condition on one-sided annihilators and on its refinement in the case of finitely presented factors of the form . [Copyright &y& Elsevier]

Abstract: Parabolic subalgebras of semi-simple Lie algebras decompose as where is a Levi factor and the corresponding nilradical. By Richardson''s theorem [R.W. Richardson, Bull. London Math. Soc. 6 (1974) 21–24], there exists an open orbit under the action of the adjoint group P on the nilradical. The elements of this dense orbits are known as Richardson elements. In this paper we describe a normal form for Richardson elements in the classical case. This generalizes a construction for of Brüstle et al. [Algebr. Represent. Theory 2 (1999) 295–312] to the other classical Lie algebra and it extends the authors normal forms of Richardson elements for nice parabolic subalgebras of simple Lie algebras to arbitrary parabolic subalgebras of the classical Lie algebras [K. Baur, Represent. Theory 9 (2005) 30–45]. As applications we obtain a description of the support of Richardson elements and we recover the Bala–Carter label of the orbit of Richardson elements. [Copyright &y& Elsevier]