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Ph. D. Dissertation Thesis STATE UNIVERSITY OF NEW YORK @ BUFFALO DEPARTMENT OF MATHEMATICS Ph. D. Dissertation Thesis The Representation Theory of Profinite Algebras ∼ Interactions with Category Theory, Algebraic Topology and Compact Groups ∼ Advisor, Samuel D. Schack Miodrag-Cristian Iovanov BUFFALO, APRIL 2009 2 Contents 1 Introduction 1 1.1 The Results . .8 1.2 Categories and (co)Homology . 13 1.3 Algebraic Topology . 17 1.4 Hopf algebras, Compact Groups . 21 2 When Rat splits off in any representation 25 2.1 Splitting Problem . 26 3 When does the rational torsion split off for finitely generated modules 31 3.1 General Considerations . 34 3.1.1 \Almost finite” coalgebras and comodules . 35 3.1.2 The Splitting Property . 37 3.2 Chain Coalgebras . 40 3.3 The co-local case . 47 3.4 Serial coalgebras . 50 4 The Dickson Subcategory Splitting for Pseudocompact Algebras 57 4.1 General results . 60 4.1.1 Some general module facts . 62 4.2 The domain case . 64 4.3 The Main Result . 68 3 CONTENTS CONTENTS 5 Generalized Frobenius Algebras and Hopf Algebras 73 5.1 Quasi-co-Frobenius Coalgebras . 76 5.1.1 Categorical characterization of QcF coalgebras . 84 5.2 co-Frobenius coalgebras and Applications to Hopf Algebras . 85 6 Abstract integrals in algebra 91 6.1 The General results . 94 6.2 Co-Frobenius coalgebras and Hopf algebras . 97 6.3 Examples and Applications . 101 6.4 Locally Compact Groups . 112 7 The Generating Condition for Coalgebras 119 7.1 Loewy series and the Loewy length of modules . 122 7.2 The generating condition . 124 7.3 A general class of examples . 127 4 Chapter 1 Introduction Let K be a field. Throughout this work we shall write ⊗ without a subscript and will understand the tensor product over K. The classical definition of a K-algebra A can be axiomatized in the following way: A is a K-vector space together with two K-linear maps m : A ⊗ A ! A and u : K ! A such that m(m ⊗ IdA) = m(IdA ⊗ m) and m(u ⊗ IdA) = m(IdA ⊗ u) = IdA. Equivalently, this means that the following diagrams are commutative: m⊗IdA A ⊗ A ⊗ A / A ⊗ A IdA⊗m m A ⊗ A m / A and A ⊗ A s9 eKK IdA⊗u s KKu⊗IdA sss KK ss KKK sss K A ⊗ K m K ⊗ A K KKK ss KK sss =∼ KKK ss=∼ KK sss % A ys The first diagram is nothing else than the associativity axiom for the multiplication, while the second one says that 1 · a = a · 1 = a for a 2 A, i.e. 1 = u(1K ) is a unit for A. The advantage of this definition is that it is a categorical one, and it can be dualized as well as generalized and considered in any suitable category. The dual notion is that of a coalgebra: this is a triple (C; ∆;") with C a K-vector space, ∆ : C ! C ⊗ C (called comultiplication) and " : C ! K (called counit) K-linear maps, such that 1 CHAPTER 1. INTRODUCTION (IdC ⊗∆)∆ = (∆⊗IdC )∆ and ("⊗IdC )∆ = IdC = (IdC ⊗")∆. The first equation is called coassociativity and the second one is called the counit property. In categorical terms, this is expressed by the following diagrams which are the above with the arrows reversed, that is, the above two diagrams considered in the dual to the category of vector spaces (for the reader familiar with the categorical language, one says that a coalgebra in a monoidal category C is an algebra in the dual category Cop): ∆⊗IdC C ⊗ C ⊗ CCo ⊗ C O O IdC ⊗∆ ∆ C ⊗ C o ∆ C and C ⊗ C s O KK IdC ⊗" s KK"⊗IdC sss KK ss KKK ysss K% C ⊗ K ∆ K ⊗ C LLL rr LLL rrr =∼ LL rr=∼ LLL rrr % C yr We will use the Sweedler notation (sometimes also called sigma notation), which is explained in the following. Let c 2 C and write X ∆c = ci(1) ⊗ ci(2); (1.1) i with the subscripts (1) and (2) indicating positions in the tensor product. In the Sweedler notation, we omit the index i and write more simply X ∆(c) = c(1) ⊗ c(2): (c) In fact, when no confusion can arise, we frequently suppress the summation sign and the parantheses in the subscripts, writing simply ∆(c) = c1 ⊗ c2. But the reader should always be aware that this is short hand for (1.1) - the summation is implicit and the subscripts 1; 2 indicate only the tensor factors and do not identify specific elements c1; c2 2 C. In this formula, one can apply the comultiplication ∆ to the first position, and the notation 2 CHAPTER 1. INTRODUCTION P for that would then be ∆(c(1)) = c(1)(1) ⊗ c(1)(2); similarly, for the elements in the (c(1)) P second position ∆(c(2)) = c(2)(1) ⊗ c(2)(2). Using Sweedler notation, the axioms of the (c(2)) coassociativity and the counit property become: X X X X c(1)(1) ⊗ c(1)(2) ⊗ c(2) = c(1) ⊗ c(2)(1) ⊗ c(2)(2) (coassociativity) (c) (c(1)) (c) (c(2)) X X c(1)"(c(2)) = c = "(c(1))c(2) (counit property) (c) (c) (2) (2) P Because of this, if we put ∆ = (IdC ⊗∆)∆ = (∆⊗IdC )∆ and we write ∆ (c) = c(1) ⊗ (c) P P P P P c(2) ⊗c(3), then c(1) ⊗c(c2) ⊗c(3) = c(1)(1) ⊗c(1)(2) ⊗c(2) = c(1) ⊗c(2)(1) ⊗c(2)(2). (c) (c) (c1) (c) (c(2)) (n) p n−1−p More generally, we write ∆ = (I ⊗ ∆ ⊗ I ) ◦ ∆n−1, and use the Sweedler notation (n) P ∆ (c) = c(1) ⊗ c(2) ⊗ ::: ⊗ c(n+1). (c) In a way dual to the definition of modules over an algebra A, one can define comodules over a coalgebra C. More specifically, a left module over an algebra A is a K-vector space M together with a K-linear map µ : A ⊗ M ! M satisfying the commutative diagrams: IdA⊗µ A ⊗ A ⊗ M / A ⊗ M A ⊗ M r8 u⊗IdM rrr rrr rrr m⊗IdM µ K ⊗ M µ LL LLL ∼LL = LLL L& A ⊗ M µ / MM By dualizing this definition, we obtain the notion of a right comodule over a coalgebra C: this is a K-vector space M together with a K-linear map called coaction ρ : M ! M ⊗ C such that the following diagrams commute: ρ⊗IdC M ⊗ C ⊗ CMo ⊗ C M ⊗ C O O r O IdM ⊗"rrr rrr xrrr IdM ⊗∆ ρ M ⊗ K ρ fLLL LLL =∼ LL LLL M ⊗ C o ρ M M 3 CHAPTER 1. INTRODUCTION P The Sweedler notation convention for comodules is then ρ(m) = m[0] ⊗ m[1], or more (m) briefly, ρ(m) = m0 ⊗ m1. The 0 position indicates elements in the comodule M, and the other positions indicate the number of times some comultiplication (of the comodule or of the coalgebra) has been applied. The above commutative diagrams are then translated to m00 ⊗ m01 ⊗ m1 = m0 ⊗ m11 ⊗ m12 =: m0 ⊗ m1 ⊗ m2 and m = m0"(m1). In analogy to morphisms of modules over an algebra, one defines a morphism of comodules over a coalgebra f :(M; ρM ) ! (N; ρN ) to be a K-linear map f : M ! N satisfying (f ⊗Id)ρM = ρN f; equivalently, using Sweedler notation one has f(m0) ⊗ m1 = f(m)0 ⊗ f(m)1. This determines a category (right comodules over the coalgebra C), and this is denoted by MC or Comod-C. For now, the above definitions are only motivated by categorical reasons: a coalgebra is just an algebra in the category dual to that of vector spaces. However, there are many places where coalgebras and their corepresentations (comodules) occur naturally, such as representation theory of groups and algebras, algebraic topology, compact groups, quantum groups and category theory, as we shall see in what follows. First, let us just notice that there is a perfect duality between finite dimensional alge- bras and finite dimensional coalgebras: for any finite dimensional algebra (A; m; u), since (A ⊗ A)∗ = A∗ ⊗ A∗, the triple (A∗; m∗; u∗) has exactly the required properties defin- ing a coalgebra, by duality through ()∗. Also, if C is a coalgebra (not necessarily finite dimensional), C∗ becomes an algebra with the so called convolution product defined by f ∗ g = (f ⊗ g)∆, so (f ∗ g)(c) = f(c1)g(c2). Moreover, if M is a right C-comodule, then M ∗ ∗ ∗ P ∗ becomes a left C -module by the action c · m = m0 ⊗ c (m1) = m[0]c (m[1]). This gives (m) a functor MC ! C∗ − Mod from C-comodules to C∗-modules which is a full embedding and is an equivalence of categories when C is finite dimensional. This is easily explained by the isomorphism (M ⊗ C)∗ ' C∗ ⊗ M ∗ for finite dimensional C. Recall that if A is an algebra and M is a left A-module, then the map A ! EndK (M): a 7! a · − is an algebra map and is called a representation of A. Also, conversely, every representation ρ : A ! EndK (V ) induces a module structure on V by av = ρ(a)(v). We call the category of finite dimensional representations Rep(A) and, using the foregoing correspondence, identify it with A − fdmod, the category of finite dimensional left A- modules. For any representation η : A ! End(V ) with V a K-vector space, fix a basis (vi)i=1;:::;n of V and write η(a) = (ηij(a))i;j for the matrix representation of η(a) with 4 CHAPTER 1. INTRODUCTION ∗ respect to this basis. Let R(A) be the subspace of A spanned by all these functions ηij, for all finite dimensional representations η and all choices of bases fvig.
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