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Galois Theory for H-Extensions

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Galois Theory for H-Extensions UNIVERSIDADE TÉCNICA DE LISBOA INSTITUTO SUPERIOR TÉCNICO Galois Theory for H-extensions Marcin Wojciech Szamotulski Supervisor: Doctor Roger Francis Picken Co-supervisor: Doctor Christian Edgar Lomp Thesis approved in public session to obtain the PhD Degree in Mathematics Jury final classification: Pass with Merit Jury Chairperson: Chairman of the IST Scientific Board Members of the Committee: arXiv:1304.7643v1 [math.QA] 29 Apr 2013 Doctor Roger Francis Picken Doctor Christian Edgar Lomp Doctor Paulo Jorge da Rocha Pinto Doctor Lars David Kadison 2013 UNIVERSIDADE TÉCNICA DE LISBOA INSTITUTO SUPERIOR TÉCNICO Galois Theory for H-extensions Marcin Wojciech Szamotulski Supervisor: Doctor Roger Francis Picken Co-supervisor: Doctor Christian Edgar Lomp Thesis approved in public session to obtain the PhD Degree in Mathematics Jury final classification: Pass with Merit Jury Chairperson: Chairman of the IST Scientific Board Members of the Committee: Doctor Roger Francis Picken, Professor Associado do Instituto Superior Técnico, da Universidade Técnica de Lisboa; Doctor Christian Edgar Lomp, Professor Auxiliar da Faculdade de Ciências, da Universidade do Porto; Doctor Paulo Jorge da Rocha Pinto, Professor Auxiliar do Instituto Superior Técnico, da Universidade Técnica de Lisboa; Doctor Lars David Kadison, Investigador Auxiliar da Faculdade de Ciências, da Universidade do Porto. Funding Institutions Fundação para a Ciência e a Tecnologia, Portugal 2013 v Teoria de Galois para H-extensões Marcin Szamotulski Doutoramento em Matemática Orientador: Prof. Roger Francis Picken Co-orientador: Prof. Christian Edgar Lomp Resumo Mostramos que existe uma correspondência de Galois entre subálgebras de uma álgebra de H-comódulo A sobre um anel base R e cocientes generaliza- dos de uma álgebra de Hopf H, se ambos A e H são módulos de Mittag-Leffler chatos. Fornecemos ainda critérios novos para subálgebras e cocientes gen- eralizados serem elementos fechados da conexão de Galois construída. Gen- eralizamos a teoria de objetos admissíveis de Schauenburg a este contexto mais geral. Depois consideramos coextensões de coálgebras de H-módulo. Construímos uma conexão de Galois para elas e provamos que coextensões H-Galois são fechadas. Aplicamos os resultados obtidos à própria álgebra de Hopf, dando uma prova simples que existe uma correspondência biun- ívoca entre ideais coideais à direita de H e as suas subálgebras de coideal à esquerda, quando H é de dimensão finita. Formulamos ainda uma con- dição necessária e suficiente para uma correspondência de Galois ser bijetiva quando A=H. Consideramos também álgebras de produto cruzado. Palavras-chave: conexão de Galois, álgebra de Hopf, álgebra de comódulo, coálgebra, biálgebra, reticulado completo, conjunto partialmente ordenado, elementos fechados, produto cruzado, propriedade de Mittag–Leffler. vi Galois theory for H-extensions Marcin Szamotulski PhD Degree in Mathematics Supervisor: Prof. Roger Francis Picken Co-supervisor: Prof. Christian Edgar Lomp Abstract We show that there exists a Galois correspondence between subalgebras of an H-comodule algebra A over a base ring R and generalised quotients of a Hopf algebra H if both A and H are flat Mittag–Leffler modules. We also provide new criteria for subalgebras and generalised quotients to be closed elements of the Galois connection constructed. We generalise the Schauenburg theory of admissible objects to this more general context. Then we consider coex- tensions of H-module coalgebras. We construct a Galois connection for them and we prove that H-Galois coextensions are closed. We apply the results obtained to the Hopf algebra itself and we give a simple proof that there is a bijective correspondence between right ideal coideals of H and its left coideal subalgebras when H is finite dimensional. Furthermore, we formu- late a necessary and sufficient condition for a bijective Galois correspondence for A=H. We also consider crossed product algebras. Key-words: Galois connection, Hopf algebra, comodule algebra, coalgebra, bialgebra, complete lattice, partially ordered set, closed elements, crossed product, Mittag–Leffler property, action by monomorphisms. vii Acknowledgements I am deeply thankful to my advisors Professor Roger Picken and Professor Christian Lomp for their guidance and support throughout my research. I also would like to thank IST (Instituto Superior Técnico) and FCT (Fundação para a Ciência e a Tecnologia, grant SFRH/BD/44616/2008) for the financial support essential to successfully complete this thesis. I would like to thank my friend Dorota Marciniak. Long and deep dis- cussions with her led to the initial development of this subject. She also in- vited me to study Universal Algebra and Category Theory. Understanding the Universal Algebra approach was essential to solve the stated problems. I also would like to thank my dear parents and my brother for their support and a longtime friend of my mother Heike Ortmann. I would like to thank Stavros Papdakis for his friendship. I would also like to thank Prof. Marian Marciniak whose support and guidance over the years was invaluable. Table of Contents Introduction1 1 Preliminaries5 1.1 Posets and Lattices . .6 1.2 Galois connections . 11 1.3 Algebras, coalgebras and Hopf algebras . 14 Algebras . 14 Coalgebras . 14 Comodules . 15 Bialgebras and Hopf algebras . 17 H-comodule algebras . 20 H-module algebras . 24 Finite Galois field extensions . 27 2 Modules with the intersection property 29 3 Lattices of quotients and subobjects 41 3.1 Coalgebras . 43 3.2 Bialgebras and Hopf Algebras . 54 Quotients of k[G] and its dual Hopf algebra . 57 Quotients of U(g) .......................... 61 Quantum enveloping algebras . 68 3.3 Comodule subalgebras . 69 3.4 Summary . 69 4 Galois Theory for Hopf–Galois extensions 71 4.1 Schauenburg’s approach . 73 4.2 Galois correspondence for H-extensions . 77 4.3 Closed Elements . 86 4.4 Correspondence between admissible objects . 89 ix x TABLE OF CONTENTS 4.5 Galois correspondence for Galois coextensions . 96 4.6 Takeuchi correspondence . 98 4.7 Galois theory of crossed products . 102 5 Coring approach 107 5.1 Classical Galois theory and corings . 108 5.2 Lattices of subcorings and quotient corings . 110 5.3 Galois connection for H-module algebras . 115 5.4 Maszczyk’s approach in a noncommutative way . 117 Bibliography 127 Index 131 Introduction Hopf–Galois extensions have roots in the approach of Chase et al.[1965] who generalised the classical Galois Theory for field extensions to rings which are commutative. Chase and Sweedler[1969] extended these ideas to coactions of Hopf algebras on commutative algebras over rings. The general definition of a Hopf–Galois extension was first introduced by Kreimer and Takeuchi [1981]. Under the assumption that H is finite dimensional their definition is equivalent to the following now standard one. Definition An H-extension A=Aco H is called an H-Galois extension if the canonical map of right H-comodules and left A-modules: can : A Aco H A A H; a b ab(0) b(1) (1) 1 co H is an isomorphism , where A := fa 2 A : a(0) a(1) = a 1H g. A breakthrough in Hopf–Galois theory was made by van Oystaeyen and Zhang extending the results of Chase and Sweedler[1969] to a noncommut- ative setting. They construct a Galois correspondence for Hopf–Galois exten- sions. Van Oystaeyen and Zhang introduced a remarkable construction of an associated Hopf algebra to an H-extension A=Aco H , where A as well as H are supposed to be commutative (see [van Oystaeyen and Zhang, 1994, Sec. 3], for a noncommutative generalisation see: Schauenburg[1996, 1998]). We will denote this Hopf algebra by L(A; H). Schauenburg[1998, Prop. 3.2] general- ised the van Oystaeyen and Zhang correspondence (see also [Schauenburg, 1996, Thm 6.4]) to a Galois connection between generalised quotients of the associated Hopf algebra L(A; H) (i.e. quotients by right ideal coideals) and 1We use the Sweedler notation for coactions and comultiplications: here δ : A A H Pn 0 and for a 2 A, δ(a) 2 A H and hence it is a sum of simple tensors k=1 ak hk for some 0 P ak 2 A and hk 2 H. In Sweedler notation this tensor is denoted as a(0) a(1) 2 A H. We prefer to use the sumless Sweedler notation, e.g. we write δ(a) = a(0) a(1), where the suppressed summation symbol is understood. 1 2 TABLE OF CONTENTS H-comodule subalgebras of A. We denote the poset of generalised quotients of a Hopf algebra H by Quotgen (H). In this work we construct the Galois cor- respondence: co Q SubAlg (A=B) Quotgen (H); φ(Q) := A (2) φ where B is the coinvariants subalgebra of A, without the assumption that B is equal to the commutative base ring R and we also drop the Hopf–Galois assumption (see Theorem 4.2.2 on pager 78). We add some module theor- etic assumptions on A and H. Instead of the Hopf theoretic approach of van Oystaeyen, Zhang and Schauenburg we propose to look from the lattice theor- etic perspective. Using an existence theorem for Galois connections we show that if a comodule algebra A and a Hopf algebra H are flat Mittag–Leffler R-modules (Definition 2.1.29 on page 33) then the Galois correspondence (2) exists. The Mittag–Leffler property appears here since it turns out that a flat R-module M has this property if and only if the endofunctor M − of the category of left R-modules preserves arbitrary intersections of submodules (see Definition 2.1.26 and Corollary 2.1.36 on page 35). We consider modules with the intersection property in Chapter2, where we also give examples of flat and faithfully flat modules which fail to have it. For an H-extension co H + A=A over a field we show that (S) = H=KS H where KS is the smallest left coideal subalgebra of H with the property: δA(S) ⊆ A KS, where δA is the comodule structure map of A (see Theorem 4.2.4 on page 81).
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