By Sh. A. Ayupov, B. A. Omirov (auth.), Yusupdjan Khakimdjanov, Michel Goze, Shavkat A. Ayupov (eds.)

ISBN-10: 9401061300

ISBN-13: 9789401061308

ISBN-10: 9401150729

ISBN-13: 9789401150729

This quantity provides the lectures given through the moment French-Uzbek Colloquium on Algebra and Operator idea which happened in Tashkent in 1997, on the Mathematical Institute of the Uzbekistan Academy of Sciences. one of the algebraic issues mentioned listed here are deformation of Lie algebras, cohomology concept, the algebraic number of the legislation of Lie algebras, Euler equations on Lie algebras, Leibniz algebras, and actual *K*-theory. a few contributions have a geometric point, corresponding to supermanifolds. The papers on operator thought care for the research of particular types of operator algebras. This quantity additionally encompasses a certain creation to the speculation of quantum teams. *Audience:* This booklet is meant for graduate scholars specialising in algebra, differential geometry, operator thought, and theoretical physics, and for researchers in arithmetic and theoretical physics.

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**Additional info for Algebra and Operator Theory: Proceedings of the Colloquium in Tashkent, 1997**

**Example text**

On peut remarquer que l'algebre de Weyl AI(C) est 2-commutative ([5]). Supposons dans la suite que (A, F) est r-commutative. ) Considerons maintenant [A, A], filtree par (Ann[A, A])nEN, puis la filtration quotient sur HHo(A) = A/[A, A] definie par (Ani An n lA, A])nEN ; par ailleurs, on peut considerer Ht ois (8) = 8/ {S, 8} pour la structure de Poisson definie par (23). Afin de comparer gr HHo(A) et Ht ois (8), nous aurons besoin de la propriete suivante exprimant une compatibilite graduee entre la filtration F sur A et la structure d'algebre de Lie induite sur A.

29, Tashkent, 700143, Uzbekistan. Abstract. We examine the problem of describing filiform Lie algebras over the field C admitting a semisimple derivation. As applications of obtained results we give a description of Lie algebras with a given filiform nilradical. Introduction The class of nilpotent Lie algebras is the union of two importent subclasses: subclass of characteristically nilpotent Lie algebras and subclass of nilpotent Lie algebras admitting a semisimple derivation. These subclasses have been studied in the papers [5], [6], [7], [8], [9].

Hn-l, adXo, adXl, ... , adXn _ d) form a basis of the derivations algebra DerLn and a maximal toros of derivations of L n is spanned by d l and dz . Caseg=Qn, n=2k+1. Let (Zo, Zl, ... ,Zn) be a basis of Qn introduced in Example 2. We consider the endomorphisms d l , dz, t, h3, hs, ... ,hZk-1 defined by = Zo , dl (Zi) = (i - l)Zi, 1 ~ i ~ n - 1, d l (Zn) = (n - 2)Zn. d2(ZO) = 0, d2(Z;) = Zi, 1 ~ i ~ n - 1, d2(Zn) = 2Zn t (Zo) = Zn , t (Xi) = 0 , 1 ~ i ~ n. h2j +l (Zo) = 0 , h2j+l (Zi) = Zi+2j+l, 1 ~ j ~ k - 1, 1 ~ i ~ n - j.

### Algebra and Operator Theory: Proceedings of the Colloquium in Tashkent, 1997 by Sh. A. Ayupov, B. A. Omirov (auth.), Yusupdjan Khakimdjanov, Michel Goze, Shavkat A. Ayupov (eds.)

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