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Journal de théorie et applications du mensonge généralisé

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Volume 4, Problème 1 (2010)

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(α, β)-fuzzy Lie algebras over an (α, β)-fuzzy field

P. L. ANTONY a and P. L. LILLY

The concept of ( ; )-fuzzy Lie algebras over an ( ; )-fuzzy eld is introduced. We provide characterizations of an (2;2 _q)-fuzzy Lie algebra over an (2;2 _q)-fuzzy eld.
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Double cross biproduct and bi-cycle bicrossproduct Lie bialgebras

Tao ZHANG

We construct double cross biproduct and bi-cycle bicrossproduct Lie bialgebras from braided Lie bialgebras. The main results generalize Majid's matched pair of Lie algebras and Drinfeld's quantum double and Masuoka's cross product Lie bialgebras.

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Classi cations of some classes of Zinbiel algebras

J.Q.ADASHEV, A.Kh.KHUDOYBERDIYEV, and B.A.OMIROV

In this work, the nul- liform and liform Zinbiel algebras are described up to iso- morphism. Moreover, the classi cation of complex Zinbiel algebras dimensions  3 is extended up to dimension 4.

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Cheban loops

J.D.Phillips a and V.A.Shcherbacov

Left Cheban loops are loops that satisfy the identity x(xy · z) = yx · xz. Right Cheban loops satisfy the mirror identity (z · yx)x = zx · xy. Loops that are both left and right Cheban are called Cheban loops. Cheban loops can also be characterized as those loops that satisfy the identity x(xy · z) = (y · zx)x. These loops were introduced by A. M. Cheban. Here we initiate a study of their structural properties. Left Cheban loops are left conjugacy closed. Cheban loops are weak inverse property, power associative, conjugacy closed loops; they are centrally nilpotent of class at most two.

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Canonical endomorphism eld on a Lie algebra

Jerzy KOCIK

We show that every Lie algebra is equipped with a natural (1; 1)-variant tensor eld, the \canonical endomorphism eld", determined by the Lie structure, and satisfying a certain Nijenhuis bracket condition. This observation may be considered as complementary to the Kirillov-Kostant-Souriau theorem on symplectic geometry of coadjoint orbits. We show its relevance for classical mechanics, in particular for Lax equations. We show that the space of Lax vector elds is closed under Lie bracket and we introduce a new bracket for vector elds on a Lie algebra. This bracket de nes a new Lie structure on the space of vector elds.

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Polynomial k-ary operations, matrices, and k-mappings

G. BELYAVSKAYA

We establish connection between product of two matrices of order k  k over a eld and the product of the k-mappings corresponding to the k-operations, de ned by these matrices. It is proved that, in contrast to the binary case, for arity k  3 the components of the k-permutation inverse to a k-permutation, all components of which are polynomial k-quasigroups, are not necessarily k-quasigroups although are invertible at least in two places. Some transformations with the help of permutations of orthogonal systems of polynomial k-operations over a eld are considered.

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A formula for the number of Gelfand-Zetlin patterns

H. REFAGHAT and M. SHAHRYARI

In this article, we give a formula for the number of Gelfand-Zetlin patterns, using dimensions of the symmetry classes of tensors.

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On jets, extensions and characteristic classes I

Helge MAAKESTAD

In this paper, we give general de nitions of non-commutative jets in the local and global situation using square zero extensions and derivations. We study the functors Exank(A; I), where A is any k-algebra, and I is any left and right A-module and use this to construct ane non-commutative jets. We also study the Kodaira-Spencer class KS(L) and relate it to the Atiyah class.

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On the product of conjugacy classes in unitary group and singular connections

Jafar SHAFFAF

We are attempting to give a new proof to the problem of characterization of the support of the product of conjugacy classes in the compact Lie group SU(n) without any reference to the Mehta-Seshadri theorem in algebraic geometry as it was the case in [1].

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(α, β)-fuzzy Lie algebras over an (α, β)-fuzzy field

P.L.ANTONY and P. L. LILLY

The concept of ( ; )-fuzzy Lie algebras over an ( ; )-fuzzy eld is introduced. We provide characterizations of an (2;2 _q)-fuzzy Lie algebra over an (2;2 _q)-fuzzy eld.

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Poisson structures on C[X1, . . . , Xn] associated with rigid Lie algebras

Nicolas GOZE

We present the classical Poisson-Lichnerowicz cohomology for the Poisson algebra of polynomials C[X1; : : : ;Xn] using exterior calculus. After presenting some non-homogenous Poisson brackets on this algebra, we compute Poisson cohomological spaces when the Poisson structure corresponds to a bracket of a rigid Lie algebra
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Quantization of the q-analog Virasoro-like algebras

Yongsheng CHENG and Yiqian SHI

We use the general method of quantization by Drinfel'd twist element to quantize explicitly the Lie bialgebra structures on the q-analog Virasoro-like algebras studied in Comm. Algebra, 37 (2009), 1264{1274.

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