Differential calculus on quantized simple lie groups

Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU _q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q are also discussed.     

A new quantum group associated with a ‘nonstandard’ braid group representation

A new quantum group is derived from a nonstandard braid group representation by employing the Faddeev-Reshetikhin-Takhtajan constructive method. The classical limit is not a Lie superalgebra, despite relations like x ²–y ²=0. We classify all finite-dimensional irreducible representations of the new Hopf algebra and find only one- and two-dimensional ones.     

Millettia ferruginea extract attenuates cisplatin-induced alterations in kidney functioning,DNA damage,oxidative stress,and renal tissue morphology

This study aimed to investigate the beneficial role of Millettia ferruginea extract (MF) in preventing cisplatin (Cisp) induced nephrotoxicity in rats. A total of 55 metabolites were identified using LC-MS analysis. The in vivo results indicated that MF pretreatment for 4 weeks (20 mg/kg b.w.) remarkably attenuated the altered renal biomarkers by decreasing the levels of plasma creatinine, urea, and uric acid when compared to the Cisp-group. The nephroprotective capacity of MF was further strengthened by histopathological observations, where Cisp + MF treated rats showed lower number of inflammatory cells and tubular degenerative changes than the Cisp-group. The harmful effects of cisplatin on renal oxidative stress indicators (MDA, SOD, CAT, and GPx), were restored by the treatment of MF. In addition, the reduction of inflammatory markers (IL-6 and TNF-α), associated with alleviating DNA fragmentation, highlighted the preventive effect of MF in kidney tissue. Additionally, MF components presented lower binding energies when docked into the active site of TNF-α and IL-6. The present findings concluded that M. ferruginea extract exhibited nephroprotective potential, which may be attributed to its antioxidant and anti-inflammatory properties. Further work is recommended to confirm the current results, explore the involved mechanism of action, and determine the therapeutic doses and time.     

Extraction Equilibria between Benzene and Water of Uni- and Bivalent Metal Picrates with Lariat 16-Crown-5: Effect of the Side Arms on the Extraction Ability and Selectivity

The overall extraction constants (K_ex) of uni- andbivalent metal picrates with 15-(2,5-dioxahexyl)-15-methyl-16-crown-5(L16C5) were determined between benzene and water at 25°C. TheK_ex values were analyzed into the constituent equilibriumconstants, i.e., the extraction constant of picric acid, the distributionconstant of the crown ether, the stability constant of the metalion–crown ether complex in water, and the ion-pair extraction constantof the complex cation with the picrate anion. The K_ex valuedecreases in the orders Ag⁺ > Na⁺ >Tl⁺ > K⁺ > Li⁺ andPb²⁺ > Ba²⁺ > Sr²⁺ for theuni- and bivalent metals, respectively, which are the same as those observedfor 16C5. The extraction selectivity was found to be governed by theselectivity of the ion-pair extraction of the L16C5–metal picratecomplex rather than by that of the complex formation in water. Theextraction ability of L16C5 is smaller for all the metals than that of 16C5,which is mostly attributed to the higher lipophilicity of L16C5. Differencesin the extraction selectivity between L16C5 and 16C5 were observed for thebivalent metals but little for the univalent metals. The side-arm effect onthe extraction selectivity was interpreted on the basis of the negativecorrelation between the effect on the complex stability constant in waterand that on the ion-pair extraction constant.     

On the homogeneity of additive mappings

LetK be a ring with an identity 1 0 andM, L two unitaryK-modules. Then, for any additive mappingf:M L, the setH _f :={ K f(x)=f(x) for allx M} forms a subring ofK, the homogeneity ring off. It is shown that, forM {0},L {0} and any subringS ofK for whichM is a freeS-module, there exists an additive mappingf:ML such thatH _f =S. This result is applied to the four Cauchy functional equations, and it leads also to an answer to the question as to whether it is possible to introduce onM a multiplication ·:M × M M makingM into a ring but not into aK-algebra.     

Lifting via cocycle deformation

We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such lifting.     

Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by Lp-modulus of smoothness

We prove a sharp estimate for the k-modulus of smoothness, modelled upon a $L^p$-Lebesgue space, of a function f in $W^k{L}^{\frac{pn}{n+kp},p}(\mathrm{\Omega })$, where Ω is a domain with minimally smooth boundary and finite Lebesgue measure, $k,n\in \mathbb{N}$, $k<n$ and $\frac{n}{n?k}<p<+\infty$. This sharp estimate is used to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces into generalized Hölder spaces defined by means of the k-modulus of smoothness. General results are illustrated with examples. In particular, we obtain a generalization of the classical Jawerth embeddings.     

PRME AND k-PRIME IDEALS IN Ω-GROUPS

Abstract

In this paper we define two concepts of prime ideals for Ω-groups. The first generalizes the definitions of prime ideal in rings, nearrings, Γ-rings, associative algebras and Lie algebras. The second generalizes a concept defined for groups by ??ukin ([21]). We show that both lead to radicals in the sense of Hoehnke ([10]). Furthermore in the case of rings, Γ-rings, abelian zero-symmetric nearrings and cubic rings these two definitions coincide, thus obtaining a new characterization for the prime ideal. Zero-symmetric Ω-groups are defined analogously to the nearring case and a new characterization in term of ideals is given.