A. M. Alexeev,1 R. R. Hartmann,2 and M. E. Portnoi1,3,* 1School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom 2De La Salle University-Manila, 2401 Taft Avenue, 1004 Manila, Philippines 3International Institute of Physics, Universidade Federal do Rio Grande do Norte, Natal, RN, Brazil (Received 28 April 2015; revised manuscript received 31 October 2015; published...

We give the classification of (co-)path Hopf algebras and semi-path Hopf algebras with pointed module structures. This leads to the classification of multiple crown algebras and multiple Taft algebras as well as pointed Yetter-Drinfeld kG-modules and their corresponding Nichols algebras. Moreover, we characterize quantum enveloping algebras in terms of semi-path Hopf algebras.

We give the classification of (co-)path Hopf algebras and semi-path Hopf algebras with pointed module structures. This leads to the classification of multiple crown algebras and multiple Taft algebras as well as pointed Yetter-Drinfeld kG-modules and their corresponding Nichols algebras. Moreover, we characterize quantum enveloping algebras in terms of semi-path Hopf algebras.

In the paper, we describe Drinfel'd double structure of $n$-rank Taft algebra and all its simple modules, then endow $R$-matrices with some application to knot invariants. The invariants get is a generalization Jones polynomial, in particular, it recovers polynomial rank $1$ case, while $2$ one-parameter specialization two-parameter unframed Dubrovnik higher case composite ($n$-power) polynomial.

As President of the United States from 1909 to 1913, William Howard Taft's minimum body mass index was 42 kg/m(2). This article presents evidence that he suffered from obstructive sleep apnea, manifested by excessive daytime somnolence, snoring, systemic hypertension and, perhaps, cognitive and psychosocial impairment. As president, Taft's hypersomnolence was severe and obvious, but never promp...

Over an algebraically closed field k of characteristic zero, the Drinfeld double Dn Taft algebra that is defined using a primitive nth root unity q?k for n?2 quasitriangular Hopf algebra. Kauffman and Radford have shown has ribbon element if only n odd, unique; however there been no explicit description this element. In work, we determine explicitly. For any n?2, use R-matrix to construct actio...

We quantize explicitly the Witt algebra in characteristic 0 equipped with its Lie bialgebra structures discovered by Taft. Then, we study the reduction modulo p of our formulas. This gives p − 1 families of polynomial noncocommutative deformations of a restricted enveloping algebra of a simple Lie algebra in charactristic p (of Cartan type). In particular, this yields new families of noncommuta...

Patients with active cirrhosis of the liver may show evidence of multiple system involvement (Read, Sherlock, and Harrison, 1963). The suggestion has also been made that in such patients autoimmune phenomena play an important part in the pathogenesis of the disease (Mackay, Taft, and Cowling, 1956). These findings prompted us to undertake an investigation into the frequency with which antinucle...