Some examples of nil Lie algebras

Generalizing Petrogradsky`s construction, we give examples of infinite-dimensional nil Lie algebras of finite Gelfand-Kirillov dimension over any field of positive characteristic.

Nil graded self-similar algebras

In [19], [24] we introduced a family of self-similar nil Lie algebras L over fields of prime characteristic p > 0 whose properties resemble those of Grigorchuk and Gupta-Sidki groups. The Lie algebra L is generated by two derivations v(1) = partial derivative(1) + t(0)(p-1) (partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...))))), v(2) = partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...)))) of the truncated polynomial ring K[t(i), i is an element of N vertical bar t(j)(p) =0, i is an element of N] in countably many variables. The associative algebra A generated by v(1), v(2) is equipped with a natural Z circle plus Z-gradation. In this paper we show that for p, which is not representable as p = m(2) + m + 1, m is an element of Z, the algebra A is graded nil and can be represented as a sum of two locally nilpotent subalgebras. L. Bartholdi [3] andYa. S. Krylyuk [15] proved that for p = m(2) + m + 1 the algebra A is not graded nil. However, we show that the second family of self-similar Lie algebras introduced in [24] and their associative hulls are always Z(p)-graded, graded nil, and are sums of two locally nilpotent subalgebras.

Granulocyte Colony-stimulating Factor Reduces Mortality by Suppressing Ventricular Arrhythmias in Acute Phase of Myocardial Infarction in Rats

Our aim was to evaluate the effects of granulocyte colony-stimulating factor (G-CSF) on early cardiac arrhythmias after myocardial infarction (MI) and the impact on survival. Male Wistar rats received repeated doses of 50 mu g/kg G-CSF (MI-GCSF group) or vehicle (MI group) at 7, 3, and 1 days before surgery. MI was induced by permanent occlusion of left corollary artery. The electrocardiogram was obtained before occlusion and then for 30 minutes after surgery. Events and duration of ventricular arrhythmias were analyzed. The levels of connexin43 (Cx43) were measured by Western blot immediately before MI production. Survival was significantly increased in MI-GCSF pretreated group (74% versus 52.0% MI. P < 0.05). G-CSF pretreatment also significantly reduced the ventricular premature beats when compared with the untreated-MI group (201 +/- 47 versus 679 +/- 117, P < 0.05). The number and the duration of ventricular tachycardia were smaller in the MI-G-CSF group, as well as the number of ventricular fibrillation episodes (10% versus 69% in NIL P < 0.05). Cx43 levels were significantly increased by G-CSF treatment (1.27 +/- 0.13 versus 0.86 +/- 0.11; P < 0.05). The MI size 24 hours after occlusion was reduced by G-CSF pretreatment (36 +/- 3% versus 44 +/- 2% of left ventricle in MI group; P < 0.05). The increase of Cx43 expression in the heart may explain the reduced incidence in ventricular arrhythmias in the early phases after coronary artery occlusion in rats, thus increasing survival after MI.

Synthesis and Characterization of Homoleptic and Heteroleptic Cobalt, Nickel, Copper, Zinc and Cadmium Compounds with the 2-(Tosylamino)-N-[2-(tosylamino)benzylidene]aniline Ligand

The electrochemical oxidation of anodic metal (cobalt, nickel, copper, zinc and cadmium) in an acetonitrile solution of the Schiff-base ligand 2-(tosylamino)-N-[2-(tosylamino)-benzylidene] aniline (H(2)L) afforded the homoleptic compounds [ML]. The addition of 1,1-diphenylphosphanylmethane (dppm), 2,2`-bipyridine (bipy) or 1,10-phenanthroline (phen) to the electrolytic phase gave the heteroleptic complexes [NiL(dppm)], [ML(bipy)] and [ML(phen)]. The crystal structures of H(2)L (1), [NiL] (2), [CuL] (3), [NiL(dppm)] (4), [CoL(phen)] (5), [CuL(bipy)] (6) and [Zn(Lphen)] (7) were determined by X-ray diffraction. The homoleptic compounds [NiL] and [CuL] are mononuclear with a distorted square planar [MN(3)O] geometry with the Schiff base acting as a dianionic (N(amide)N(amide)N(imine)O(tosyl)) tetradentate ligand. Both compounds exhibit an unusual pi-pi stacking interaction be-tween a six-membered chelate ring containing the metal and a phenylic ring of the ligand. In the heteroleptic complex [NiL(dppm)], the nickel atom is in a distorted tetrahedral [NiN(3)P] environment defined by the imine, two amide nitrogen atoms of the L(2-) dianionic tridentate ligand and one of the phosphorus atoms of the dppm molecule. In the other heteroleptic complexes, [CoL(phen)], [CuL(bipy)] and [ZnL(phen)], the metal atom is in a five-coordinate environment defined by the imine, two amide nitrogen atoms of the dianionic tridentate ligand and the two bipyridine or phenanthroline nitrogen atoms. The compounds were characterized by microanalysis, IR and UV/Vis (Co, Ni and Cu complexes) spectroscopy, FAB mass spectrometry and (1)H NMR ([NiL] and Zn and Cd complexes) and EPR spectroscopy (Cu complexes).

Examples of Self-Iterating Lie Algebras, 2

We study properties of self-iterating Lie algebras in positive characteristic. Let R = K[t(i)vertical bar i is an element of N]/(t(i)(p)vertical bar i is an element of N) be the truncated polynomial ring. Let partial derivative(i) = partial derivative/partial derivative t(i), i is an element of N, denote the respective derivations. Consider the operators v(1) = partial derivative(1) + t(0)(partial derivative(2) + t(1)(partial derivative(3) + t(2)(partial derivative(4) + t(3)(partial derivative(5) + t(4)(partial derivative(6) + ...))))); v(2) = partial derivative(2) + t(1)(partial derivative(3) + t(2)(partial derivative(4) + t(3)(partial derivative(5) + t(4)(partial derivative(6) + ...)))). Let L = Lie(p)(v(1), v(2)) subset of Der R be the restricted Lie algebra generated by these derivations. We establish the following properties of this algebra in case p = 2, 3. a) L has a polynomial growth with Gelfand-Kirillov dimension lnp/ln((1+root 5)/2). b) the associative envelope A = Alg(v(1), v(2)) of L has Gelfand-Kirillov dimension 2 lnp/ln((1+root 5)/2). c) L has a nil-p-mapping. d) L, A and the augmentation ideal of the restricted enveloping algebra u = u(0)(L) are direct sums of two locally nilpotent subalgebras. The question whether u is a nil-algebra remains open. e) the restricted enveloping algebra u(L) is of intermediate growth. These properties resemble those of Grigorchuk and Gupta-Sidki groups.

The bar-radical of a class of almost alternative baric algebras

In this article we prove that, if (U, ) is a finite dimensional baric algebra of (gamma, delta) type over a field F of characteristic not equal 2,3,5 such that gamma(2) - delta(2) + delta = 1 and 0,1, then rad(U) = R(U)boolean AND(bar(U))(2), where R(U) is the nilradical (maximal nil ideal) of U.