About this book These lecture notes provide a self-contained introduction to a wide range of generalizations of Hopf algebras. Since Sweedler's work in the s, Hopf algebras have earned a noble place in the garden of mathematical structures. Their use is well accepted in fundamental areas such as algebraic geometry, representation theory, algebraic topology, and combinatorics.
Now, similar to having moved from groups to groupoids, it is becoming clear that generalizations of Hopf algebras must also be considered.
This book offers a unified description of Hopf algebras and their generalizations from a category theoretical point of view. The author applies the theory of liftings to Eilenberg—Moore categories to translate the axioms of each considered variant of a bialgebra or Hopf algebra to a bimonad or Hopf monad structure on a suitable functor.
Covered structures include bialgebroids over arbitrary algebras, in particular weak bialgebras, and bimonoids in duoidal categories, such as bialgebras over commutative rings, semi-Hopf group algebras, small categories, and categories enriched in coalgebras.
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Notes. 1. If A and B are two k-algebras then so is A⊗B if we define (a⊗b)(c⊗d) = ac ⊗ db. . A Hopf algebra is a bialgebra with an antipode. Morphisms of Hopf. The present notes were prepared for the lectures of my course Introduction to Hopf algebras and representations, given in the University of Helsinki in the spring.
A, An Introduction to noncommutative differential geometry on quantum groups. The lectures are self-contained but assume a basic knowledge of general relativity. Plan of the lecture: 1.
Event horizons 2. Equilibrium states 3.
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Conservations laws 4. Quasi-equilibrium states Lecture notes: notes pdf Advised references: P.
Wald, "The Thermodynamics of black holes", Living Rev. Leclercq, S. Cnockaert, N. Bouatta, C. Introduction to the BRST antifield formalism 2.
Applications of the antifield formalism 3. Henneaux, C. Some reviews: -J. Gomis, J.
Paris and S. Fuster, M. Henneaux, A. For part 4.