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

This quantity provides the lectures given through the moment French-Uzbek Colloquium on Algebra and Operator thought which happened in Tashkent in 1997, on the Mathematical Institute of the Uzbekistan Academy of Sciences. one of the algebraic subject matters 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 genuine *K*-theory. a few contributions have a geometric point, equivalent to supermanifolds. The papers on operator idea care for the examine of specific sorts of operator algebras. This quantity additionally encompasses a precise creation to the speculation of quantum teams. *Audience:* This ebook 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 resources for Algebra and Operator Theory: Proceedings of the Colloquium in Tashkent, 1997**

**Example text**

Theorem 3 Let 9 be a non-decomposable faithful Lie algebra with a filiform nilradical n. Then 9 = {C. ~;;2l {3ihi , ({32,"" {3n-l} E Cn- 2 - {O} where hi are the nilpotent derivations defined by hi (Xo ) = 0 , hi (X j ) = X i+j , j = 1, ... ,n-i. (ii) n = L n , Is = d l + dz and In is defined by In (Xo) = Xl and In (Xi) = 0 for i =f. O. ::~3 {3i h2i+l , ({3t, ... , {3m-d E cm - l _ {O}, where hi is the nilpotent derivation ofn defined by h2i+l (Zo) = 0, hzi+l (Zj) = Z2i+j+l, j = 1, ... ,n - 2i - 2.

We deduce the following description of non-decomposable faithful Lie algebras with a filiform nilradical. Theorem 3 Let 9 be a non-decomposable faithful Lie algebra with a filiform nilradical n. Then 9 = {C. ~;;2l {3ihi , ({32,"" {3n-l} E Cn- 2 - {O} where hi are the nilpotent derivations defined by hi (Xo ) = 0 , hi (X j ) = X i+j , j = 1, ... ,n-i. (ii) n = L n , Is = d l + dz and In is defined by In (Xo) = Xl and In (Xi) = 0 for i =f. O. ::~3 {3i h2i+l , ({3t, ... , {3m-d E cm - l _ {O}, where hi is the nilpotent derivation ofn defined by h2i+l (Zo) = 0, hzi+l (Zj) = Z2i+j+l, j = 1, ...

2 Filiform Lie algebras of rank 1. Let g be a filiform Lie algebra of dimension n + 1 and of rank 1 . There is a basis (Yo, Y I , . ยท . , Y n ) of g such that g is one of the following Lie algebras: (i) g=~+l(al, ... =-LI) Yi+j+r, 1 ~ i < j ~ n, i+ j +r g=B~+l(al, ... ,Cl't), n=2m+l, 1~r~n-4,t=[n-;-2], [Yo,Yi] = Yi+l' 1~i~n-2, ~ n. KHAKIMDJANOV 52 [Yi, Yn-iJ = (-l)i yn , 1::; i::; m, [Yi,1jJ = (E~=i ak (_l)k-i ~;~LI) Yi+j+r, 1 ::; i < j ::; n - 1, i + j + r ::; n - 1. (iii) 9 = Cn+l (al, ... , at), n = 2m + 1, t = m - 1, [Yo, Yi] = Yi+1, 1::; i ::; n - 2, [Yi, Yn-iJ = (_l)i Y n , 1::; i ::; m, [Yi, Y n -i-2kJ = (_l)i akYn, 1::; k ::; m - 1, 1::; i ::; n - 2k - 1, where cg are the binomial coefficients (we suppose that cg = 0 if q < 0 or q < s), (al, ...