We establish results about the second cohomology with coefficients in trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over commutative associative without unit. These provide simple unified approach to number questions treated earlier completely separated ways: periodization semisimple algebras (Anna Larsson), derivation algebras, prescribed part, ni...

Let R be a commutative ring with identity. By a Bres̃ar generalized derivation of R we mean an additive map g : R→ R such that g (xy) = g (x) y + xd (y) for all x, y ∈ R, where d is a derivation of R. And an additive mapping f : R → R is called a generalized derivation in the sense of Nakajima if it satisfies f(xy) = f(x)y + xf(y) − xf(1)y for all x, y ∈ R. In this paper we extend some results o...

We define the Poisson quasi-Nijenhuis structures with background on Lie algebroids and we prove that to any generalized complex structure on a Courant algebroid which is the double of a Lie algebroid is associated such a structure. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis structure with background constitutes, with its dual, a quasi-Lie bialgebroid. We also prove that any ...

We give a constructive account of the fundamental ingredients of Poisson Lie theory as the basis for a description of the classical double group D. The double of a group G has a pointwise decomposition D ∼ G × G * , where G and G * are Lie subgroups generated by dual Lie algebras which form a Lie bialgebra. The double is an example of a factorisable Poisson Lie group, in the sense of Reshetikhi...

The 5'-AMP (adenosine monophosphate)-activated protein kinase (AMPK) coordinates metabolic function with energy availability by responding to changes in intracellular ATP (adenosine triphosphate) and AMP concentrations. Here, we report crystal structures at 2.9 and 2.6 A resolution for ATP- and AMP-bound forms of a core alphabetagamma adenylate-binding domain from the fission yeast AMPK homolog...

Consider a Lie color algebra, denoted by L. Our aim in this paper is to study the triple derivations TDer(L) and generalized GTDer(L) of algebras. We discuss centroids, quasi centroids central algebras, where we show relationship derivation with algebras A classification algebra all perfect given, prove that for centerless TDer(L)=Der(L) TDer(Der(L))=Inn(Der(L)).

Theorem ([11, Theorem 2, page 282]). Let R be a prime ring, L a noncommutative Lie ideal of R and d 6= 0 a derivation of R. If [d(x), x] ∈ Z(R), for all x ∈ L, then either R is commutative, or char(R) = 2 and R satisfies s4, the standard identity in 4 variables. Here we will examine what happens in case [d(x), x]n ∈ Z(R), for any x ∈ L, a noncommutative Lie ideal of R and n ≥ 1 a fixed integer....