జర్నల్ ఆఫ్ జెనరలైజ్డ్ లై థియరీ అండ్ అప్లికేషన్స్

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.

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.

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.

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.

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.

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

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.

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].

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.

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.