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Register a new userOn the inverse limit of finite dimensional lie algebras
حول النهاية المعكوسة لجبور لـي منتهية البعد
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Damascus University ورقة بحثية
Publication date
2006
fields
Mathematics
and research's language is
العربية
Created by
Shamra Editor
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عممت في هذا البحث مبرهنة Malcev – Levi عن جبور لي منتهية البعد، على النهاية المعكوسة لجبور لي منتهية البعد . كما أُثبت أن كل صورة تشاكلية image homomorphic منتهية البعد للجداء الديكارتي لمجموعة من جبور لي العدومة nilpotent و المنتهية البعد هي أيضاً عدومة.
We extend the well Known Levi-Malcev decomposition theorem of finite dimensional Lie algebras to the case of pro-finite dimensional Lie algebras L = limLn (n ∈ N). We also prove that every finite dimensional homomorphic image of the Cartesian product of finite dimensional nilpotent Lie algebras is also nilpotent.
References used
Bourbaki, N. (1989.), "Lie Groups and Lie Algebras", Springer-Verlag
Hochschild, G. (1981), "Basic theory of algebraic groups and Lie algebras". Graduate Texts in Math. 75, Springer-Verlag
Humphreys, J. E. (1972), "Introduction to Lie algebras and representation theory". Second printing, Springer-Verlag
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We prove that the sum A + B of closed subspaces A and B of the inverse
limit of finite dimensional vector spaces, V = limVn (n ∈ N) over an
algebraically closed field of characteristic 0 is closed.
We extend also the basic fact that every ideal of a finite dimensional
semisimple Lie algebra has a unique complement to the case of closed ideals of
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A Lie algebra g over a field F is a vector space together with a bilinear
map [ , ] satisfying [x ,x ] = 0 in addition to Jacobi identity . A Lie subalgebra
B of a Lie algebra g is said to be a Cartan subalgebra if it is a nilpotent and
equals its
normalizer, and it is proved that semi simple Lie algebra g
decomposes into weight spaces for B.
In this scientific paper we present the conception of distinguished
element 0 h in finite dimensional semi simple Lie algebra over a field F has
characteristic 0 and we will prove that the previous decomposition g into
weight spaces for B is the same to decomposition g as a direct sum of h0 ad eigen
spaces. This leads us to construct algorithm to test simple Lie algebras.
We programmed the previous algorithm to test simple linear Lie algebras
over a numeral field by Mathematica 5.0 program where applied this algorithm
on semi simple linear Lie algebra SL(3, ) to prove that it is simple.
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Cooling technique helps prolonging the life of laser systems ,its stability and optimizing its operation. So the total cooling efficient lasers , Scientific and Industrial is essential when designing a laser and helps improving the specifications of lasers , gaseous or solid or semiconductor . In addition to that cooling has a good impact on the age of the laser , and safety of its use .
The designed cooling system , necessary for a certain laser system , depends on the amount of heat generated from the system(in Joule), which must be removed by the cooling system to keep the laser at specific temperature
In this paper, We study the dual representation. We proved that if
p is completely redusibile, decomposable and unitary then p* is
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