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This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras A by first determining the graded Hopf algebra grA associated to the coradical filtration of A. The A0-coinvariants elements form a braided Hopf algebra R in the category of Yetter–Drinfeld modules over the coradical… (More)
The energy, E(G), of a simple graph G is defined to be the sum of the absolute values of the eigen values of G. If G is a k-regular graph on n vertices,then E(G) k+√k(n− 1)(n− k)= B2 and this bound is sharp. It is shown that for each > 0, there exist infinitely many n for each of which there exists a k-regular graph G of order n with k < n− 1 and B2 < . Two… (More)
We classify finite-dimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements G(A) is abelian such that all prime divisors of the order of G(A) are > 7. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum… (More)
In this paper, we obtain a theorem on the distribution of eigenvalues for Schur complements of H-matrices. Further, we give some properties of diagonal-Schur complements on diagonally dominant matrices and their distribution of eigenvalues. © 2004 Elsevier Inc. All rights reserved. AMS classification: 15A45; 15A48
We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra grA. Then grA is a graded Hopf algebra, since the coradical A0 of A is a Hopf subalgebra. In addition, there is a… (More)
– We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p > 17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius–Lusztig kernels of semisimple Lie… (More)
is an isomorphism, which can be interpreted as the correct algebraic formulation of the condition that the G-action of X should be free, and transitive on the fibers of the map X → Y . In many applications surjectivity of the Galois map β, which, in the commutative case, means freeness of the action of G, is obvious, or at least easy to prove (it is… (More)
We introduce the concept of homotopy equivalence for Hopf Galois extensions and make a systematic study of it. As an application we determine all H-Galois extensions up to homotopy equivalence in the case when H is a Drinfeld-Jimbo quantum group.
These notes contain the material presented in a series of five lectures at the University of Córdoba in September 1994. The intent of this brief course was to give a quick introduction to Hopf algebras and to prove as directly as possible (to me) some recent results on finitedimensional Hopf algebras conjectured by Kaplansky in 1975. In particular, in the… (More)
The operator trigonometry of symmetric positive definite (SPD) matrices is extended to arbitrary invertible matrices A and to arbitrary invertible bounded operators A on a Hilbert space. Some background and motivation for these results is provided. © 2000 Elsevier Science Inc. All rights reserved.