arXiv subjects

Masoud Sabzevari

Publications and source records attributed to Masoud Sabzevari.

21 records · Page 2Linked to original sources

arXiv2012

Modules Satisfying the Prime Radical Condition and a Sheaf Construction for Modules II

In this paper we continue our study of modules satisfying the prime radical condition ($\mathbb{P}$-radical modules), that was introduced in Part I (see \cite{BS}). Let $R$ be a commutative ring with identity. The purpose of this paper is to show that the theory of spectrum of $\mathbb{P}$-radical $R$-modules (with the Zariski topology) resembles to that of rings. First, we investigate the behavior of $\mathbb{P}$-radical modules under localization and direct sums. Finally, we describe the construction of a structure sheaf on the prime spectrum Spec$(M)$, which generalizes the classical structure sheaf of the ring $R$ in Algebraic Geometry to the module $M$.

math.AC↗

arXiv2011

A Groebner-bases algorithm for the computation of the cohomology of Lie (super) algebras

We present an effective algorithm for computing the standard cohomology spaces of finitely generated Lie (super) algebras over a commutative field K of characteristic zero. In order to reach explicit representatives of some generators of the quotient space Z^k/B^k of cocycles Z^k modulo coboundaries B^k, we apply Groebner bases techniques (in the appropriate linear setting) and take advantage of their strength. Moreover, when the considered Lie (super) algebras enjoy a grading -- a case which often happens both in representation theory and in differential geometry --, all cohomology spaces Z^k/B^k naturally split up as direct sums of smaller subspaces, and this enables us, for higher dimensional Lie (super) algebras, to improve the computer speed of calculations. Lastly, we implement our algorithm in the Maple software and evaluate its performances via some examples, most of which have several applications in the theory of Cartan-Tanaka connections.

math.AC↗

arXiv2011

Effective Cartan-Tanaka connections on C6-smooth strongly pseudoconvex hypersurfaces M3 of C2

In a recent expository article (Notices of the AMS, 58 (2011), no. 1, 20-27), Ezhov, McLaughlin and Schmalz showed how to perform in an effective way Tanaka's prolongation procedure valid generally for filtered structures of constant type when the distribution is equipped with an integrable complex structure, so as to derive the principal curvature invariants and (co)frame(s) associated to strongly pseudoconvex real hypersurfaces M3 in C2, an approach which is alternative and complementary to the `hyperspherical' connection of Elie Cartan, and to the so-called calculi of Fefferman, of Chern-Moser, of Webster. By choosing an initial frame for TM which is explicit in terms of a local graphing function v = f(x, y, u) for M, we provide a Cartan-Tanaka connection all elements of which are completely explicit in terms of f(x, y, u), assuming only C6-smoothness of M. The Gaussian requirement for systematic computational effectiveness then shows - a bit unexpectedly - that the two main curvatures are rational differential expressions in the sixth-order jet of f(x, y, u), the lengths of which are about 1000 pages long on a computer - just for the simplest instance of local embedded CR geometry. Large parts of the memoir aim at formulating general statements that will be useful for further constructions of Cartan-Tanaka connections related to the equivalence problem for (local) embedded CR manifolds whose CR-automorphism group is not semi-simple, cf. e.g. some model lists by Beloshapka.

math.DG↗