# Hopf Algebras in Noncommutative Geometry and Physics by Stefaan Caenepeel

By Stefaan Caenepeel

This entire reference summarizes the court cases and keynote shows from a up to date convention held in Brussels, Belgium. delivering 1155 demonstrate equations, this quantity comprises unique learn and survey papers in addition to contributions from world-renowned algebraists. It makes a speciality of new leads to classical Hopf algebras in addition to the class concept of finite dimensional Hopf algebras, express features of Hopf algebras, and up to date advances within the conception of corings and quasi-Hopf algebras. It presents examples and easy homes of corings and their comodules when it comes to ring and Hopf algebra thought and analyzes entwining buildings and Morita idea for corings.

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Aly, F. Van Oystaeyen, Hopf Filtrations and Larson-Type Orders in Hopf Algebras, J. Algebra 267 (2003), 756-772. [2] R. G. Larson, Hopf Algebra Orders Determined by Group Valuations, J. Algebra 38 (1976), 414-452. [3] R. G. Larson, Orders in Hopf Algebras, J. Algebra 22 (1972), 201-210. [4] H. S. Li, F. Van Oystaeyen, “Zariskian Filtrations”, K-Monographs Math. 3, Kluwer Aca demic Publishers, Dordrecht, 1996. [5] S. Montgomery, “Hopf algebras and their actions on rings”, American Mathematical Society, Providence, 1993.

In [18] J. H. Lu introduced the notion of Hopf algebroid as a triple consisting ofa left bialgebroid A l = {A^L^SL^ti^^i^TTL) such thatA and ¿ are algebras over a commutative ring k. It is equipped with an antipode S : A A. The S is required to be an anti-automorphism of the fc-algebra A satisfying S o t i = Sl rriA O(S 0L id/l) O7L = OTTLo 5 (8) rriA o (id/i slottl (9) S )o^ o^ L = (7) where rriA is the multiplication in A and ^ s a section of the canonical projec tion pl : a iS>k a A l (S>l A that is ^ is a map A l ^ l A ^ A® k A satisfying Following the result of [31, 17] - proving that an irreducible finite index depth 2 extension of von Neumann factors can be realized as a crossed product with a finite dimensional C*-Hopf algebra - big effort has been made in order to make connection between more and more general kinds of extensions and of ‘quantum symmetries’ [25, 8 , 9, 35, 24].

Indeed, if both S\ and 38 G. BOHM S2 make Si) and {A^y A r , S2) Hopf algebroids then 52 (a) = 5H0 7rH(a(i))52(a(2^) = 5i(a^i)(i))a^i)(2)52(a^2^) = *S'l(^(l))a(2)^^^52(a(2)^^^) = 5i(a(i))5L o 7TL(a(2)) = 5i(a). There are some examples however in which only the left bialgebroid structure is naturally given and we have some ambiguity in the choice of the right bialgebroid structure and the corresponding antipode. (See for example the Hopf algebroid symmetry of a depth 2 Probenius extension of rings in Section 3 of [1] or at the end of Section 6 below.