Normalizers of subsystem subgroups in finite groups of Lie type

Finite groups of Lie type form the greater part of known finite simple groups. An important class of subgroups of finite groups of Lie type are so-called reductive subgroups of maximal rank. These arise naturally as Levi factors of parabolic groups and as centralizers of semisimple elements, and also as subgroups with maximal tori. Moreover, reductive groups of maximal rank play an important part in inductive studies of subgroup structure of finite groups of Lie type. Yet a number of vital questions dealing in the internal structure of such subgroups are still not settled. In particular, we know which quasisimple groups may appear as central multipliers in the semisimple part of any reductive group of maximal rank, but we do not know how normalizers of those quasisimple groups are structured. The present paper is devoted to tackling this problem. Supported by RFBR (grant No. 05-01-00797) and by SB RAS (Young Researchers Support grant No. 29 and Integration project No. 2006.1.2). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 3–30, January–February, 2008.     

Frobenius splitting and geometry of G-Schubert varieties

Let X be an equivariant embedding of a connected reductive group G over an algebraically closed field k of positive characteristic. Let B denote a Borel subgroup of G. A G-Schubert variety in X is a subvariety of the form diag(G)⋅V, where V is a B×B-orbit closure in X. In the case where X is the wonderful compactification of a group of adjoint type, the G-Schubert varieties are the closures of Lusztig's G-stable pieces. We prove that X admits a Frobenius splitting which is compatible with all G-Schubert varieties. Moreover, when X is smooth, projective and toroidal, then any G-Schubert variety in X admits a stable Frobenius splitting along an ample divisors. Although this indicates that G-Schubert varieties have nice singularities we present an example of a nonnormal G-Schubert variety in the wonderful compactification of a group of type G₂. Finally we also extend the Frobenius splitting results to the more general class of R-Schubert varieties.     

Mod p classification of Shimura F ‐crystals

Let k be an algebraically closed field of positive characteristic p. We first classify the D ‐truncations mod p of Shimura F ‐crystals over k and then we study stratifications defined by inner isomorphism classes of these D ‐truncations. This generalizes previous works of Kraft, Ekedahl, Oort, Moonen, and Wedhorn. As a main tool we introduce and study Bruhat F ‐decompositions; they generalize the combined form of Steinberg theorem and of classical Bruhat decompositions for reductive groups over k (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Subdirectly irreducible medial quandles

We describe all subdirectly irreducible medial quandles. We show that they fall within one of four disjoint classes. In particular, in the finite case they are either connected (and therefore Alexander quandles) or reductive. Moreover, we provide a representation of all non-connected subdirectly irreducible medial quandles.     

Selective Cross-Coupling of Unsaturated Substrates on AlI

The Al^I compound NacNacAl ( 1 , NacNac = [ArNC(Me)CHC(Me)NAr]⁻, Ar = 2,6-iPr₂C₆H₃) serves as a template for the chemoselective coupling between carbonyls (benzophenone, fenchone, isophorone, p-tolyl benzoate, N,N-dimethylbenzamide, (1-phenylethylidene)aniline) and pyridine. With the CH-acidic ketone (1R)-(+) camphor, the reaction affords a hydrido alkoxide compound of Al, formed as the result of enolization, whereas an enolizable imine, (1-phenylethylidene)aniline, and the bulky ketone isophorone, still chemoselectively couple with pyridine. In contrast, reaction with the ester p-tolyl benzoate results in cleavage of the ester bond together with replacement of the alkoxy group by a hydrogen atom of the pyridine moiety. This study demonstrates that for carbonyl substrates featuring phenyl substituents, the reaction proceeds via intermediate formation of η²(C,X)-coordinated (X = O, N) carbonyl adducts, whereas the reaction of 1 with (R)-(−)-fenchone in the absence of pyridine leads to CH activation in the pendant isopropyl group of the Ar substituent of the NacNac ligand.     

A Catalyst and Solvent Free Route for the Synthesis of N-Substituted Pyrrolidones from Levulinic Acid

An efficient, metal-free, catalyst-free and solvent-free methodology for the reductive amination of levulinic acid with different anilines has been developed using HBpin as the reducing reagent. This protocol offers an excellent method to avoid solvents and added catalysts on the synthesis of different kinds of N-substituted pyrrolidones under metal free conditions. It is also the first report for the synthesis of different pyrrolidones by solvent-free as well as catalyst-free methods. The proposed mechanism for the formation of pyrrolidone has been supported by DFT calculations and control experiments.     

Interaction of secondary amines with aromatic aldehydes-efficient method for synthesis of the functionalized heterocyclic amines

A method is proposed for the benzylation of secondary heterocyclic amines with functionalized derivatives of benzaldehyde in the presence of formic acid under conditions close to amination according to the Leuckart-Wallach reaction. __________ Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 12, pp. 1820–1823, December, 2007.     

四(4－N-吡啶基)卟啉衍生物的合成及表征

阳离子卟啉配体;配合物;还原电位;四(4－N-吡啶基)卟啉衍生物的合成及表征