HOM-TENSOR CATEGORIES. Paul T. Schrader. A Dissertation

Transcription

1 HOM-TENSOR CATEGORIES Paul T. Schrader A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 2018 Committee: Mihai Staic, Advisor Hong Peter Lu, Graduate Faculty Representative Rieuwert Blok Xiangdong Xie

2 Copyright c May 2018 Paul T. Schrader All rights reserved

3 ABSTRACT iii Mihai Staic, Advisor Braided monoidal categories and Hopf algebras have applications for invariants in knot theory and 3-dimensional manifolds. The classical results involving the relationship between k-bialgebras (quasi-triangular k-bialgebras and monoidal categories (braided monoidal categories have been known for some time. Motivated by problems in the deformation of Witt algebras Jonas T. Hartwig, Daniel Larsson, and Sergei D. Silvestrov introduced hom-lie algebras in Over the last decade, many hom-associative algebraic structures and their properties were established. This dissertation addresses the categorical settings of hom-associative algebras analogous to the aforementioned classical results. To facilitate this objective we first introduce a new type of category called a hom-tensor category (4.1.1 and show that it provides the appropriate categorical framework for modules over a hom-bialgebra ( Next we introduce the notion of a hom-braided category (4.2.3 and show that this is the right categorical setting for modules over quasitriangular hom-bialgebras ( We also prove that, under certain conditions, one can obtain a pre-tensor category (respectively a quasi-braided category from a hom-tensor category (respectively a hom-braided category and explain how the hom-yang-baxter equation fits into the framework of hom-braided categories. Finally, we show how the category of Yetter-Drinfeld modules over a hom-bialgebra with a bijective structural map can be organized as a hom-braided category and discuss some open questions.

4 iv I dedicate this dissertation to my mother June ( , my father Richard ( , and my late wife Pandora (

5 ACKNOWLEDGMENTS v First, I like to thank my all my family (both living and passed, new and old who encouraged, supported, and strengthened me through my many years of academia. A special thanks goes to Rick, Jane, and Amanda whose undying devotion are forever giving me the courage to endure. I like to thank all my instructors and peers of Bowling Green State University and Cleveland State University. They provided the knowledge, challenge, and inspiration which contributed to my growth as a mathematician. A sincere thanks goes to Florin Panaite for his collaborative insights and expertise. I would also like to thank my committee members Riewert Blok, Hong Peter Lu, and Xiangdong Xie for their thoughtful guidance, suggestions, and feedback in my dissertation process. Lastly, special thanks and gratitude goes to my advisor Mihai Staic. His unwavering patience, careful instruction, formidable knowledge, and passions for mathematics led me through the realities of mathematical research making this dissertation possible.

6 vi TABLE OF CONTENTS CHAPTER 1 INTRODUCTION Page CHAPTER 2 PRELIMINARIES Algebras and Coalgebras Modules and Comodules Hom-Structures for (CoAlgebras and (CoModules CHAPTER 3 MONOIDAL CATEGORIES AND BRAIDED MONOIDAL CATEGORIES Category Theory Fundamentals Monoidal Categories Braided Monoidal Categories The Yang-Baxter Equation in a Braided Monoidal Category Relating Algebras to Pre-Tensor Categories and Quasi-Braided Categories CHAPTER 4 HOM-TENSOR CATEGORIES AND HOM-BRAIDED CATEGORIES Hom-Tensor Categories Hom-Braided Categories CHAPTER 5 PROPERTIES OF HOM-TENSOR AND HOM-BRAIDED CATEGORIES Hom-Tensor Categories versus Pre-Tensor Categories Hom-Braided Categories versus Quasi-Braided Categories The Hom-Yang-Baxter Equation Yetter-Drinfeld Modules CHAPTER 6 OPEN PROBLEMS AND FUTURE RESEARCH

7 BIBLIOGRAPHY vii

8 viii LIST OF FIGURES Figure Page 1.1 The Pentagon Axiom for the associativity constraint a The commutative diagrams of the maps m A and u A in Definition The commutative diagrams for a k-algebra morphism f The commutative diagram of Definition The commutative diagrams of the maps C and ɛ C in Definition The commutative diagrams for a morphism of k-coalgebras g The commutative diagram of Definition The compatibility relations of a k-bialgebra H The commutative diagrams of the left action µ M in Definition The commutative diagram of a left A-module morphism in Definition The commutative diagrams of the coaction δ U in Definition The commutative diagram of a C-comodule morphism in Definition The commutative diagram of the compatibility relation in Definition The Commutative Diagram of a Natural Transformation η : F G The Commutative Diagram of a Natural Transformation det : GL n ( The Triangle axiom of a monoidal category The Pentagon axiom for the associativity constraint a The naturality of a X,Y,Z : (X Y Z X (Y Z The naturality of l X : I X X (r X : X I X The naturality of d V,W : V W W V (Hex1 of the Hexagon Axiom

9 ix 3.9 (Hex2 of the Hexagon Axiom The naturality of c V,W : V W W V The categorical version of the Yang-Baxter equation Proof for the categorical version of the Yang-Baxter equation The naturality of the quasi-braiding c The research scheme for hom-tensor categories and hom-bialgebras The Pentagon axiom for the hom-associativity constraint a The research scheme for hom-braided categories and quasitriangular hom-bialgebras The naturality of d V,W : V W G (W G (V The (H1 property The (H2 property The definition of b U,V,W : (U V W U (V W The diagram for 2# Hom-associativity for a implies associativity for b The diagram for 4# The diagram for 5# Definition of c U,V : U V V U The diagram for 2# The (H1 property implies the first Hexagon axiom The diagram for 3# The diagram for 5# The diagram for 6# Θ G(V Φ V = Φ F (V Θ V The (H 2 property Naturality of a 1 and F (Φ G(W = Φ F G(W The hom-yang-baxter property

10 x 5.16 The commutative bold portion of Figure The commutative dashed portion of Figure Proof of the hom-yang-baxter property Naturality for d and G(d U,V = d G(U,G(V The (wh1 property The (wh2 property The commutative diagram for the (wh 2 property The weak hom-yang-baxter property The commutative bold portion of Figure The commutative dashed portion of Figure Proof of the weak hom-yang-baxter property Final step in the proof of Proposition The (H1 property implies the (wh1 property

11 1 CHAPTER 1 INTRODUCTION Most of us are familiar with additive and multiplicative properties such as associativity and identity from the integers (denoted Z or from polynomials of a single variable with real coefficients (denoted R[x]. These structures are formally known as rings. Besides being an abelian group, a ring has a multiplication which must satisfy an associativity condition (i.e., (abc = a(bc for any elements a, b, and c of the ring together with a compatibility condition between addition and multiplication called distributivity (i.e., (a + bc = ac + bc and a(b + c = ab + ac. It may also have a multiplicative identity for a unique identity element 1 in the ring (i.e., 1a = a = a1. A k-algebra is a ring together with a ring homomorphism from a field k (e.g., the real numbers R into the center of the ring. For example, n n matrices with real entries together with matrix addition, matrix multiplication, and the identity matrix I n form the R-algebra M(n, R. Equivalently, a k-algebra is a k-vector space together with two distinct k-linear maps (called the multiplication and unit maps of the algebra that satisfy associative and unital conditions. The dual of a k-algebra is a k-coalgebra. Furthermore, a k-bialgebra is both a k-algebra and a k-coalgebra in a compatible way. Finally, a Hopf algebra H is a k-bialgebra together with a k-linear map S : H H that acts as the inverse of the identity under the convolution product. We will detail these definitions in Chapter 2. For other more complete references see Milnor, John W. and Moore, John C. (1965, Sweedler, Moss E. (1969, Abe, Eiichi (1980, Montgomery, Susan (1993, Turaev, Vladimir (1994, Kassel, Christian (1995, Majid, Shahn (1995, and Dăscălescu, Sorin, Năstăsescu, Constantin and Raianu, Şerban (2001, et al. There are also algebras with generalized associativity properties or even non-associative multiplication. A well known example of a non-associative algebra is a Lie algebra. A Lie algebra is a k-algebra where the multiplication is given by a k-bilinear map called the bracket which is antisymmetric and satisfies the Jacobi identity. Ongoing research into deformations of classical differential calculus led to interest in further generalizations for Lie algebras and other algebraic structures with key differential properties.

12 2 In 2006 (see Hartwig, Jonas T., Larsson, Daniel and Silvestrov, Sergei D. (2006 J. Hartwig, D.Larsson, and S. Silvestrov introduced a generalization called a hom-lie algebra. A hom-lie algebra is an analogue to a Lie algebra whose non-associative behavior is further deformed by a linear map. Hom-Lie algebras became the prototype for many hom-structures (e.g., hom-algebras, hom-coalgebras, hom-bialgebras, etc.. These were developed over the next several years (e.g., see Makhlouf, Abdenacer and Silvestrov, Sergei (2010a, Yau, Donald (2008. In particular, these developments showed a hom-associative k-algebra as a unitless k-algebra A together with a k- linear map α A : A A where the multiplicative hom-associativity is expressed as (abα A (c = α A (a(bc for all elements a, b, c A. It is hom-structures of these types that are at the center of my research interests. A tool commonly used for investigating the properties of the aforementioned mathematical objects is Category Theory. Category theory gives an algebraic framework for several mathematical structures (e.g., sets, groups, rings, modules, algebras, topological spaces, differentiable manifolds, etc.. Roughly speaking, a category is a collection of objects and mappings between them that obey identity, composition, and associative properties. For example, one can consider the category of groups and group homomorphisms or the category of differentiable manifolds and smooth maps between them. A monoidal category is a category with some additional structure (called a tensor product which behaves just like the tensor product of k-vector spaces. This means that a tensor product in a monoidal category has an associative behavior (i.e., (X Y Z = X (Y Z for all objects X, Y, Z in the tensor category. Formally, we say that the monoidal category has a natural isomorphism a X,Y,Z : (X Y Z X (Y Z called an associativity constraint which satisfies the commutative diagram as seen in Figure 1.1 called the Pentagon Axiom. Monoidal categories were first introduced in the early 1960 s by J. Bénabou and S. MacLane (see Bénabou, Jean (1963, Mac Lane, Saunders (1963. They were studied intensely in Joyal, André and Street, Ross (1991a, Joyal, André and Street, Ross (1991b, Joyal, André and Street, Ross (1993, Kassel, Christian (1995 and Turaev, Vladimir (1994.

13 3 (X Y (Z T a X Y,Z,T a X,Y,Z T ((X Y Z T X (Y (Z T a X,Y,Z id T id X a Y,Z,T a X,Y Z,T (X (Y Z T X ((Y Z T Figure 1.1: The Pentagon Axiom for the associativity constraint a Braided monoidal categories are monoidal categories with some additional structure that corresponds to the commutative behavior of a tensor product (i.e., X Y = Y X for all objects X, Y in a braided category. They were introduced in the 1990 s by A. Joyal and R. Street (see Joyal, André and Street, Ross (1991a, Joyal, André and Street, Ross (1993. Braided monoidal categories have applications in knot theory and 3-dimensional manifolds. There are classical results involving the relationship between k-bialgebras (quasi-triangular bialgebras and monoidal categories (braided monoidal categories which can be found in Montgomery, Susan (1993 and Kassel, Christian (1995. Concisely, we have Proposition (folklore, see Kassel, Christian (1995 and Montgomery, Susan (1993. Let H = (H, m H, u H be a k-algebra and let both H : H H H and ɛ H : H k be morphisms of k-algebras.. Then the following statements are equivalent. (A (H, m H, H, u H, ɛ H is a k-bialgebra. (B The category H = (H-mod,, I, a, l, r is a monoidal category (see for details. and Proposition (folklore, see Kassel, Christian (1995 and Montgomery, Susan (1993. Let H = (H, m H, u H, H, ɛ H be a k-bialgebra, let H be the monoidal category described in

14 Proposition 3.5.1, and let R = i s i t i be an invertible element of H H. Then following statements are equivalent. 4 (A (H, m H, u H, H, ɛ H, R is a quasi-triangular k-bialgebra. (B The category H = ( H-mod,, I, a, l, r, c R is a braided monoidal category (see for details. The known research involving the categorical frameworks for hom-associative algebras analogous to these classical results contain certain restrictions. For example, the k-linear map α of the hom-associative structure needs to be invertible. (see Caenepeel, Stefaan and Goyvaerts, Isar (2011 and Graziani, Giacomo (2013. This dissertation addresses the categorical settings of hom-structures without the aforementioned restrictions. We first define a new type of categorical structure called a hom-tensor category (4.1.1 and show that it provides the appropriate framework for modules over a hom-bialgebra ( Next we describe the notion of the hom-braided category (4.2.3 and show that this is the right setting for modules over quasitriangular hom-bialgebras ( We also prove that, under certain conditions, one can obtain a pre-tensor category (respectively a quasi-braided category from a hom-tensor category (respectively a hom-braided category. Finally, we show how the hom- Yang-Baxter equation fits into the framework of hom-braided categories and how the category of Yetter-Drinfeld modules over a hom-bialgebra with bijective structure map can be organized as a hom-braided category. The content in this thesis is as self-contained as possible. A brief summary is as follows. In Chapter 2 we begin by recalling some necessary algebraic theory including (coalgebras and (comodules while introducing notation used throughout this thesis. The chapter concludes with a section on known hom-structure theory applicable to the results in the later chapters. In Chapter 3 we first review some fundamentals of category theory. Next we recall definitions and results of monoidal and pre-tensor categories. This is followed by a detailed statement of the classical relationship between k-bialgebras and monoidal categories (3.5.1 together with its analo-

15 5 gous statement for pre-tensor categories. Then we review some definitions and results of monoidal braided and quasi-braided categories, state the classical relationship between k-bialgebras and monoidal categories (3.5.3, and describes the analogous result for quasi-braided categories. Up to this point all the results presented are classical or known. Chapter 4 thoroughly details several results found in Panaite, Florin, Schrader, Paul and Staic, Mihai (2017. In particular, we define hom-tensor categories, state and prove the relationship between hom-tensor categories and hom-bialgebras (4.1.16, define hom-braided categories, and state and prove the relationship between hom-braided categories and quasi-triangular hom-bialgebras ( We use Chapter 5 to present some applications and properties from Panaite, Florin, Schrader, Paul and Staic, Mihai (2017 for the results in Chapter 4. We begin by showing that under certain conditions one can associate a pre-tensor category to a hom-tensor category. This is followed by a comparison (under certain conditions of hom-braided category to quasi-braided categories, a categorical description of the Hom-Yang-Baxter equation, and finally how Yetter-Drinfeld modules over a hom-bialgebra are related to hom-braided categories (under certain conditions. Chapter 6 is dedicated to presenting a few open problems which naturally follow from the results presented in this dissertation.

16 6 CHAPTER 2 PRELIMINARIES This chapter consists of known definitions, examples, and results for algebraic structures and hom-algebraic structures. These concepts form the foundation and motivate the theory presented in later chapters. We also establish notational conventions utilized throughout this thesis. 2.1 Algebras and Coalgebras In this section we recall some necessary algebra theory. The following definitions, examples, and known results are taken from Montgomery, Susan (1993, Kassel, Christian (1995 and Dăscălescu, Sorin, Năstăsescu, Constantin and Raianu, Şerban (2001. One can find the origins and detailed studies of the algebraic theory discussed here in Milnor, John W. and Moore, John C. (1965, Sweedler, Moss E. (1969, Abe, Eiichi (1980, Turaev, Vladimir (1994, Majid, Shahn (1995, et al. In this thesis we denote a field by k and any unlabeled tensor product is a tensor product over k. That is, = k. We start by recalling some classical definitions, results, and examples. Definition An algebra over a field k (or a k-algebra is a triple (A, m A, u A where A is a k-vector space, m A : A A A is a k-linear map, and u A : k A is a k-linear map satisfying the commutative diagrams as seen in Figure 2.1. A A A id A m A A A u A id A id A u A k A A A A k m A id A m A = m A = A A m A A A. Figure 2.1: The commutative diagrams of the maps m A and u A in Definition Remark If A = (A, m A, u A is a k-algebra then: We will call the k-linear maps m A and u A the multiplication and unit of A respectively.

17 We will denote the multiplication on A by the juxtaposition of elements in A. That is, 7 m A (a b = ab for all a, b A. Observe that the leftmost commutative diagram in Figure 2.1 is equivalent to stating that (ab c = a (bc for all a, b, c A. We will call this property the associativity of the multiplication in A. In the rightmost commutative diagram in Figure 2.1 the two isomorphisms k A = A and A k = A are given by scalar multiplication. So this diagram implies that u A (λa = λa = au A (λ λ k and a A. In particular, we have 1 A = u A (1 k where 1 A is the identity element in A and 1 k is the identity element in k. Example Let n be a positive integer and let k be a field. Then the vector space M (n, k of all n n matrices over k is an n 2 -dimensional k-algebra. The multiplication in M (n, k is the usual matrix multiplication and the unit map sends 1 k to the identity matrix I n. Example Let End(V be the k-linear space of k-linear endomorphisms for a vector space V. Then End(V is a k-algebra whose product is given by the composition and whose unit is given by the identity map id V : V V. Example Given a k-algebra A one can construct the k-algebra A[x] of all polynomials of the form n i=0 a ix i such that n is any non-negative integer or the k-algebra A[x, x 1 ] of all Laurent polynomials of the form n i=m a ix i where m, n Z. One can also construct additional examples of k-algebras from a given k-algebra using the following definition.

18 Definition Let (A, m A, u A be a k-algebra and τ : A A A A be a k-linear map defined by τ (a b = b a for all a b A A. Then the opposite k-algebra denoted A op is the triple (A, m op A, u A where m op A = m A τ. 8 Obviously if the k-algebra A is commutative then A op = A. Example Suppose that (A, m A, u A and (B, m B, u B are k-algebras. Then one can show that (A B, (m A m A (id A τ id A, u A u A is also a k-algebra. In this case we say that A B has the tensor product algebra structure. Next we consider maps between k-algebras. Definition (Dăscălescu, Sorin, Năstăsescu, Constantin and Raianu, Şerban (2001. Let (A, m A, u A and (B, m B, u B be two k-algebras. A morphism of k-algebras (or a k-algebra morphism f : (A, m A, u A (B, m B, u B is a k-linear map f : A B such that f m A = m B (f f and f u A = u B. Remark The properties f m A = m B (f f and f u A = u B of a k-algebra morphism f as described in Definition can be respectively interpreted as commutative diagrams seen in Figure A A f f B B A f B m A m B u A u B A f B k Figure 2.2: The commutative diagrams for a k-algebra morphism f We also have a diagrammatic notion of what a commutative k-algebra is.

19 9 Definition We say that a given a k-algebra (A, m A, u A is commutative if ab = ba for all a, b A. That is, the diagram as seen in Figure 2.3 commutes where τ is the k-linear map described in Definition A A τ A A m A A m A Figure 2.3: The commutative diagram of Definition Dual to the notion of a k-algebra is a k-coalgebra. Definition (Kassel, Christian (1995. A coalgebra over a field k (or a k-coalgebra is a triple (C, C, ɛ C where C is a k-vector space and both C : C C C and ɛ C : C k are k-linear maps satisfying the commutative diagrams in Figure 2.4. C C C C ɛ C id C id C ɛ C k C C C C k C id C C = C = C C C id C C C C C Figure 2.4: The commutative diagrams of the maps C and ɛ C in Definition Remark If C = (C, C, ɛ C is a k-coalgebra then: We will call C the comultiplication of C, ɛ C the counit of C, and the leftmost commutative diagram in Figure 2.4 the coassociativity of C. A useful notation for writing long compositions of comultiplication in a compressed way is the so called Sweedler notation. In the context of this notation co-multiplication in C is written as C (c = c (1 c (2. (2.1.13

20 10 for some c C. In the usual summation notation for the formal sum denoting an element in C C, we would have wrote i=1,n c i(1 c i(2. Sweedler notation suppresses the index i and helps to emphasize the form of C (c. The expression c (1 c (2 is utilized throughout this paper and will be referred to as Sweedler s notation when applied to any computation involving comultiplication. For example, we can express the coassociativity of C (id C C C = ( C id C C ( in the following way. Let c C. For the left hand side of ( we have ( ((id C C C (c = (id C C c(1 c (2 = ( c (1 C c(2 = c (1 ( c(2 ( c (1 (2 (2 = c (1 ( c (2 ( c (1 (2 (2 and for the right hand side of ( we have ( ( C id C C (c = ( C id C c(1 c (2 = ( C c(1 c(2 = ( c(1 ( c (1 (1 (2 c (2 = ( c(1 ( c (1 (1 (2 c (2. Thus, the coassociativity of C expressed in Sweedler s notation is c(1 ( c (2 (1 ( c (2 (2 = ( c(1 (1 ( c (1 (2 c (2 for any c C. Also, using Sweedler s notation, the counit behavior of the k-coalgebra C can be expressed

21 as 11 c = (ɛ id C (c ( = (ɛ id C c(1 c (2 = ɛ ( c (1 c(2 = ɛ ( c (1 c(2 for any c C. Here are some examples of k-coalgebras. Example Let S be a set and let k [S] = x S kx be the vector space with basis S. Then we can give k [S] a k-coalgebra structure by defining k[s] (x = x x and ɛ k[s] (x = 1 for any x S (and then extend by linearity. Example Let M c (n, k be a k-vector space of dimension n 2 where n is a positive integer. Let (e ij 1 i,j n be a basis for M c (n, k. One can define a comultiplication M and a counit ɛ M by M (e ij = 1 k n e ik e kj and ɛ M (e ij = δ ij respectively for any i, j where δ ij indicates the Kronecker delta. In this way (M c (n, k, M, ɛ M becomes a k-coalgebra sometimes called the matrix coalgebra. We can also construct additional examples of k-coalgebras from a given k-coalgebra using the following definition. Definition Given a k-coalgebra (C, C, ɛ C, let τ : C C C C be a linear map defined by τ (c d = d c for all c d C C. An opposite k-coalgebra C cop is the triple (C, cop C, ɛ C where cop C = τ C. One can show that C cop is itself a k-coalgebra.

22 Example Let (C, C, ɛ C and (D, D, ɛ D be k-coalgebras. Then 12 (C D, (id C τ id D ( C D, ɛ C ɛ D is a k-coalgebra. We say that C D has the tensor product k-coalgebra structure. We also want to consider maps between k-coalgebras. Definition (Dăscălescu, Sorin, Năstăsescu, Constantin and Raianu, Şerban (2001. Let (C, C, ɛ C and (D, D, ɛ D be two k-coalgebras. A morphism of k-coalgebras or k-coalgebra morphism is a k-linear map g : C D such that (g g C = D g and ɛ C = ɛ D g. Remark For any given k-coalgebra morphism g : (C, C, ɛ C (D, D, ɛ D : The properties (g g C = D g and ɛ C = ɛ D g of a k-coalgebra morphism g as described in Definition can be respectively interpreted as commutative diagrams seen in Figure 2.5. C g D C g D C D ɛ C ɛ D C C g g D D k Figure 2.5: The commutative diagrams for a morphism of k-coalgebras g Given a morphism of k-coalgebras g : (C, C, ɛ C (D, D, ɛ D one can express the condition (g g C = D g of Definition using Sweedler notation as g(c(1 g(c (2 = (g(c (1 (g(c (2 and ɛ D (g(c = ɛ C (c for all c C.

23 We also consider a type of commutativity of k-coalgebras. 13 Definition We say that a given k-coalgebra (C, C, ɛ C is cocommutative if the diagram in Figure 2.6 commutes where τ is the map from Definition C C C C C τ C C Figure 2.6: The commutative diagram of Definition The following propositions whose proofs are found on page 3 of Montgomery, Susan (1993 show the intimate connection of k-algebras and k-coalgebras as dual structures. Proposition (Montgomery, Susan (1993. Let (C, C, ɛ C be a k-coalgebra. Then (C, ( C, (ɛ C has an k-algebra structure where C is the dual k-vector space of C and the multiplication ( C : C C C is defined by ( C (f g(c = (f g( C (c for all f, g C and for all c C. Proposition (Montgomery, Susan (1993. Let (A, m A, u A be a finite dimensional k- algebra. Then (A, (m A, (u A has a k-coalgebra structure where A is the dual k-vector space of A and the comultiplication (m A : A A A is defined by m A (f(a b = f(ab for all f A and for all a, b A. In some cases a k-vector space may simultaneously have compatible k-algebra and k-coalgebra structures. Definition (Kassel, Christian (1995. A bialgebra over a field k or k-bialgebra is a quintuple (H, m H, u H, H, ɛ H where H is a k-vector space, (H, m H, u H a k-algebra and (H, H, ɛ H a k-coalgebra such that the following compatibility conditions are satisfied h, h H: (1 H (hh = H (h H (h,

24 (2 ɛ H (hh = ɛ H (h ɛ H (h, 14 (3 H (1 = 1 1, (4 ɛ H (1 = 1. A morphism of bialgebras is a morphism for both the underlying k-algebra and k-coalgebra structures of (H, m H, u H, H, ɛ H. Remark For any given k-bialgebra (H, m H, u H, H, ɛ H : Property (1 of Definition can be seen as the leftmost commutative diagram in Figure 2.7 where τ is the map from Definition Properties (2 and (3 of Definition can be seen as the center and rightmost commutative diagrams in Figure 2.7 respectively. m H H H H H H H H u H H u H H ɛ H H k k H m H m H m H ɛ H ɛ H u H H H H H id H τ id H H H H H H H H H. Figure 2.7: The compatibility relations of a k-bialgebra H We can express the condition (hh = (h (h of the bialgebra H using Sweedler notation as follows h, h H: (hh (1 (hh (2 = h (1 h (1 h (2 h (2. ( Here are some examples of k-bialgebras. Example Recall that in Example we associated a k-coalgebra (k [S], k[s], ɛ k[s] to a set S. Now assume that S is a unital monoid M. That is, M is a set equipped with an associative binary map φ : M M M along with a left and right unit e. Then the map φ of M

25 and the unit e induces a k-algebra structure (k [M], m k[m], u k[m] where m k[m] = φ extended by linearity and u k[m] = e. Thus, 15 k[m] (xy = xy xy = (x x (y y = k[m] (x k[m] (y and (k [M], m k[m], u k[m], k[m], ɛ k[m] is a k-bialgebra. Moreover, one can show that this k- bialgebra is cocommutative. Example Given a k-bialgebra H = (H, m H, H then H op (H, m H, cop H and Hopcop = (H, m op H, cop H are k-bialgebras. = (H, m op H, H, H cop = Example Let (H, m H, u H, H, ɛ H be a finite dimensional k-bialgebra. Then, by Propositions and , (H, (m H, (u H, ( H, (ɛ H has a natural k-bialgebra structure. Next we recall from Kassel, Christian (1995 the definition of a quasitriangular k-bialgebra. Sometimes a quasitriangular k-bialgebra is called a braided k-bialgebra. Note that both relations (ii and (iii of the following definition will require some additional explanation which will immediately follow. Definition (Kassel, Christian (1995. Let (H, m H, u H, H, ɛ H be a k-bialgebra and R be an invertible element of H H. Then (H, m H, u H, H, ɛ H, R is quasitriangular (or braided if R satisfies the following three relations: (i cop H R = R H. (ii ( H id H (R = R 13 R 23. (iii (id H H (R = R 13 R 12. Remark We employ the following notational conventions and reformulations for a quasitriangular k-bialgebra (H, m H, u H, H, ɛ H, R in the remainder of the thesis.

26 16 Let R be expressed as a formal sum or R = i s i t i for some s i, t i H. Then R kl will be an element of H 3 where s i is in the k th position of tensor product, t i is in the l th position of the tensor product and 1 is in the remaining position. For example, R 31 = i t i 1 s i and R 12 = i s i t i 1. If we set R = i s i t i for some s i, t i H and let h H then relations (i, (ii and (iii in Definition can be reformulated using Sweedler notation as (i i h (2s i h (1 t i = i s ih (1 t i h (2, (ii i (s i (1 (s i (2 t i = i,j s i s j t i t j, (iii i s i (t i (1 (t i (2 = i,j s is j t j t i. Here are some examples of quasitriangular k-bialgebras. Example A cocommutative k-bialgebra is quasitriangular if we set R = 1 1. Example (Montgomery, Susan (1993. Let H be the quotient of the free algebra k {x, y} by the two-sided ideal generated by x 2 1, y 2, yx + xy. One can show that H has a basis {1, x, y, xy}. Then H is a k-bialgebra where H (x = x x, H (y = 1 y +y x, ɛ H (x = 1 and ɛ H (y = 0 x, y H. If we set R α = 1 2 ( x + x 1 x x + α 2 (y y + y xy + xy xy xy y for any scalar α, then R α satisfies the conditions in the definition of a quasitriangular k-bialgebra. 2.2 Modules and Comodules The results presented in this dissertation also require us to consider the module structures on the described types of k-algebras in section 2.1. The following definitions, examples and known results of this section are again taken from Montgomery, Susan (1993, Kassel, Christian (1995 and Dăscălescu, Sorin, Năstăsescu, Constantin and Raianu, Şerban (2001. We begin with the definition of a module over a k-algebra.

27 17 Definition (Kassel, Christian (1995. Let M be a k-vector space and (A, m A, u A be a k- algebra. A left A-module structure on M consists of a k-linear map µ M : A M M (called the left action on M that satisfies the commutative diagrams seen in Figure 2.8. A A M id A µ M A M k M u A id M A M m A id M µ M = µ M A M µ M M M Figure 2.8: The commutative diagrams of the left action µ M in Definition Remark For any given left A-module (M, µ M we will denote the left action of µ M with notation µ M (a m = a m for all a A and for all m M. Thus the leftmost commutative diagram in Figure 2.8 can be written as (ab m = a (b m a, b A, m M and the rightmost commutative diagram as u A (λ m = λm λ k. Similar to how we define a left A-module one can define a right A-module structure using a k-linear map from V A to V where V is a k-vector space. This, however, is nothing else than a left module over the opposite algebra A op. Thus we only need consider left A-modules. We also consider morphisms between left A-modules. Definition Let (M, µ M and (N, µ N be two left A-modules. A morphism of left A-modules or left A-module morphism is a k-linear map f : (M, µ M (N, µ N satisfying the commutative diagram seen in Figure 2.9.

28 18 A M id A f A N µ M µ N M f N Figure 2.9: The commutative diagram of a left A-module morphism in Definition Utilizing the notation from Remark we can express the commutative diagram in Figure 2.9 of any left A-module morphism f as f (a m = a f (m for all a A and for all m M. Dual to the notion of modules is the notion of comodules. Definition (Kassel, Christian (1995. Let (C, C, ɛ C be a k-coalgebra and U a k-vector space. A left C-comodule structure on U consists of a k-linear map δ U : U C U (called the left coaction of U that satisfies the commutative diagrams seen in Figure U δ U C U k U ɛ C id U C U δ U id C δ U = δu C U C id U C C U U Figure 2.10: The commutative diagrams of the coaction δ U in Definition Remark In most instances of computation with left C-comodules we will employ Sweedler type notation. Let C = (C, C, ɛ C be a k-coalgebra and M be a left C-comodule. By convention

29 for the left coaction δ U in Definition we shall write 19 δ U (x = x 1 x 0 for any x U. Thus the leftmost commutative diagram in Figure 2.10 can be expressed as ( x 1 (1 ( x 1 (2 x 0 = x 1 ( x 0 1 ( x 0 0 One can similarly define a right C-comodule for a k-vector space Q using a map β Q : Q Q C subject to satisfying a condition similar to the one in Definition 2.2.4(that is, (id Q C β Q = (β Q id C β Q. However, a right C-comodule is the same as a left comodule over the opposite coassociative k-coalgebra C cop. Thus we only need consider left C-comodules. The following are some examples of C-comodules. Example Any k-coalgebra (C, C, ɛ C can be viewed as a left C-comodule (U, δ U by letting C = U and C = δ U. Example Let (C, C, ɛ C be a k-coalgebra and V be a k-vector space. Then C V is a left C-comodule where the coaction δ V : C V C C V is given by δ V = C id V. That is, δ V (c v = c (1 c (2 v for all c C and for all v V. Example Let H = (H, m H, u H, H, ɛ H be a k-bialgebra and (M, δ M and (N, δ N be left H-comodules. If we define δ M N = (m id M N (id H τ M,H id N (δ M δ N then M N is endowed with a left H-comodule structure. This is called the tensor product of left H-comodules. Example Let C = (M c (n, k, M, ɛ M be the matrix coalgebra described in Example and V be a k-vector space whose basis is given by (v i 1 i n. Then (V, δ V is a right C- comodule whose right coaction δ V : V V C is defined by δ V (v i = v k e ki. 1 k n

30 x 1 g ( x 0 = (g (x 1 (g (x 0 We also consider morphisms between left C-comodules. Definition Let U and V be left C-comodules, with structures δ U 20 : U C U and δ V : V C V, a morphism of left C-comodules or left C-comodule morphism g : U V is a k-linear map satisfying the commutative diagram seen in Figure U g V δ U δ V C U id C g C V Figure 2.11: The commutative diagram of a C-comodule morphism in Definition One can also express the commutative diagram in Figure 2.11 for a C-comodule morphism g using the Sweedler type notation of Remark as for all x U. In a similar way one can define a morphism of right C-comodules. Yetter-Drinfeld modules were first introduced by D. Yetter in Yetter, David N. (1990 as crossed bi-modules. The following definitions and observations were taken from Radford, David E. and Towber, Jacob (1993 where we first see these modules under their modern nomenclature. Definition (Radford, David E. and Towber, Jacob (1993. Let H = (H, m H, u H, H, ɛ H be a k-bialgebra and V a k-vector space which is a left H-module with action H V V where h v h v and a left H-comodule with coaction V H V where v v 1 v 0. Then V is called a (left-leftyetter-drinfeld module over H if the diagram as seen in 2.12 commutes. Remark We will denote the category of (left-leftyetter-drinfeld modules over H as H HYD. Using the Sweedler notation for the comultiplication and left C-comodule structure of the

31 21 H V cop H id V H H V H δ V id H µ V H H H V H V id H τ H,H id V id H δ V H H V H H V m H µ V τ H,H id V H V m H id V H H V Figure 2.12: The commutative diagram of the compatibility relation in Definition (left-left Yetter-Drinfeld module V in Definition , we can rewrite the commutative diagram seen in Fig 2.12 into the following compatibility condition for V which holds for all h H and for all v V : h(1 v 1 ( h (2 v 0 = ( h(1 v 1 h (2 ( h (1 v 0. ( In a similar manner one can define other types of Yetter-Drinfeld modules over a k-bialgebra H. That is, (left-rightyetter-drinfeld modules over H, (right-leftyetter-drinfeld modules over H and (right-rightyetter-drinfeld modules over H. However, it was shown in Radford, David E. and Towber, Jacob (1993 how these can all be viewed as equivalent notions. So it is sufficient for our purposes to just consider (left-left Yetter-Drinfel d modules over H. 2.3 Hom-Structures for (CoAlgebras and (CoModules The origins of hom-structures can be found in the physics literature of the late 1980s to the early 1990s. The motivation behind their development concerned quantum deformations of algebras of vector fields, especially Witt and Virasoro algebras (e.g., see Aizawa, Naruhiko and Sato, Harutada (1991, Chaichian, Masud, Kulish, Petr and Lukierski, Jerzy (1990, Curtright, Thomas L. and Zachos, Cosmas K. (1990, Daskaloyannis, Costas (1992 and Kassel, Christian (1992. These

32 22 classes of examples led to the development first of hom-lie algebras (Hartwig, Jonas T., Larsson, Daniel and Silvestrov, Sergei D. (2006, Larsson, Daniel and Silvestrov, Sergei D. (2005, which are analogues of Lie algebras where the Jacobi identity is twisted by a linear map. This was followed by the development of hom-structures for associative algebras, coalgebras, bialgebras, Hopf algebras, etc. (e.g., see Benayadi, Saïd and Makhlouf, Abdenacer (2014, Caenepeel, Stefaan and Goyvaerts, Isar (2011, Chen, Yuanyuan, Wang, Zhongwei and Zhang, Liangyun (2013, Zheng, Shanghua and Guo, Li (2016, Hassanzadeh, Mohammad, Shapiro, Ilya and Sütlü, Serkan (2015, Liu, Ling and Shen, Bingliang (2014, Makhlouf, Abdenacer and Panaite, Florin (2014, Makhlouf, Abdenacer and Silvestrov, Sergei (2008, Makhlouf, Abdenacer and Silvestrov, Sergei (2009, Makhlouf, Abdenacer and Silvestrov, Sergei (2010a, Sheng, Yunhe (2012, Yau, Donald (2008, Yau, Donald (2009a, Yau, Donald (2010. The reader can find a concise history on hom-structures in the introduction of Makhlouf, Abdenacer and Panaite, Florin (2014. This section provides the necessary known definitions and results for hom-structures taken from Makhlouf, Abdenacer and Silvestrov, Sergei (2008, Makhlouf, Abdenacer and Silvestrov, Sergei (2009, Makhlouf, Abdenacer and Silvestrov, Sergei (2010a, Yau, Donald (2008, Yau, Donald (2010 and Makhlouf, Abdenacer and Panaite, Florin (2014. In particular, we consider the concept of a generalized hom-bialgebra (see Makhlouf, Abdenacer and Silvestrov, Sergei (2008, Makhlouf, Abdenacer and Silvestrov, Sergei (2009. We begin by defining the hom-structure for a k-algebra. Definition A hom-associative k-algebra is a triple (A, m A, α A where A is a k-vector space, m A : A A A is a k-linear map denoted by m A (a b = ab, for all a, b A, and α A : A A is a k-linear map satisfying the following conditions, for all a, b, c A: α A (ab = α A (a α A (b, (2.3.2 α A (a (bc = (ab α A (c. (2.3.3 Let (A, m A, α A and (B, m B, α B be two hom-associative k-algebras. A morphism of hom-

33 associative algebras f : (A, m A, α A (B, m B, α B is a k-linear map f : A B such that α B f = f α A and f m A = m B (f f. 23 Remark As in Makhlouf, Abdenacer and Panaite, Florin (2014, Definition corresponds to the definition found in Yau, Donald (2008 for a hom-associative algebra. This differs from the definition of a hom-associative algebra found in Makhlouf, Abdenacer and Silvestrov, Sergei (2008 and Makhlouf, Abdenacer and Silvestrov, Sergei (2010b (where the extra multiplicative condition (2.3.2 is not made on the map α. Unless indicated otherwise for the remainder of the dissertation, we will assume that any given hom-associative k-algebra is as in Definition Here are some examples of hom-associative k-algebras. Example (a non-multiplicative hom-associative k-algebra. Let {v 1, v 2, v 3 } be a basis of a three dimensional k-vector space V and let α V : V V be a k-linear map. Define the multiplication m V : V V V and the map α V in the following way: m V (v 1 v 1 = av 1, m V (v 1 v 2 = m V (v 2 v 1 = av 2, m V (v 2 v 2 = av 2, m V (v 1 v 3 = m V (v 3 v 1 = bv 3, m V (v 2 v 3 = bv 3, m V (v 3 v 2 = m V (v 3 v 3 = 0, α V (v 1 = av 1, α V (v 2 = av 2, and α V (v 3 = bv 1 for a, b k. Then (V, m V, α V is a nonmultiplicative hom-associative k-algebra (i.e., it only satisfies condition (2.3.3 of Definition as in Makhlouf, Abdenacer and Silvestrov, Sergei (2008 and Makhlouf, Abdenacer and Silvestrov, Sergei (2010b. For example, notice that α V (m V (v 3 v 3 m V (α V (v 3 α V (v 3 when a 0 and b 0. Furthermore, observe that (V, m V, α V is not associative (when a b and b 0 since m V (m V (v 1 v 1 v 3 m V (v 1 (v 1 v 3 = (a bbv 3. The following example gives a construction which converts any k-algebra (not necessarily with a unit into a hom-associative k-algebra using a tensor product.

34 24 Example Let (A, m A be a k-algebra (not necessarily with a unit and α A : A A an k-algebra endomorphism. Define a new multiplication m α : A A A by m αa = α A m A = m A (α A α A. Then A αa = (A, m α, α A is a hom-associative k-algebra called the Yau twist of A. One can also produce additional examples of hom-associative k-algebras from a given homassociative k-algebras. Example Let (A, m A, α A and (B, m B, α B be hom-associative k-algebras, then (A B, m A B, α A B is also a hom-associative k-algebra, where m A B ((a b (a b = aa bb and α A B = α A α B for all a A and for all b B. In this case we say that A B has the tensor product hom-associative algebra structure. Dual to the notion of a hom-associative k-algebra is a hom-coassociative k-coalgebra. Definition A hom-coassociative k-coalgebra is a triple (C, C, ψ C where C is a k-vector space, C : C C C and ψ C : C C are k-linear maps satisfying the following conditions: (ψ C ψ C C = C ψ C, (2.3.9 ( C ψ C C = (ψ C C C. ( Let (C, C, ψ C, (D, D, ψ D be two hom-coassociative k-coalgebras. A morphism of homcoassociative k-coalgebras g : (C, C, ψ C (D, D, ψ D is a k-linear map g : C D such that ψ D g = g ψ C and (g g C = D g. Remark As in Makhlouf, Abdenacer and Panaite, Florin (2014, Definition corresponds to the definition found in Yau, Donald (2008 for a hom-coassociative coalgebra. This differs from the definition of a hom-coassociative coalgebra found in Makhlouf, Abdenacer and Silvestrov, Sergei (2008 and Makhlouf, Abdenacer and Silvestrov, Sergei (2010b (where the

35 25 comultiplicative condition (2.3.2 is not made on the map ψ. Unless indicated otherwise for the remainder of the dissertation, we will assume that any given hom-coassociative k-coalgebra is as in Definition We can reformulate equations and respectively using Sweedler notation in the following way for all c C. ψc (c (1 ψ C (c (2 = ψ C ( c(1 ψc ( c(2, ( c(1(1 c (1 ψ ( ( (2 C c(2 = ψc c(1 c(2(1 c (2. ( (2 Here are some examples of hom-coassociative k-coalgebras. Example Let C = (C, C be a k-coalgebra (not necessarily with a counit and α C : C C a k-coalgebra endomorphism. Define a new comultiplication αc : C C C by αc = C α C. Then C αc = (C, αc, α C is a hom-coassociative k-coalgebra called the Yau twist of C. We can also construct additional examples of hom-coassociative k-coalgebras from a given hom-coassociative k-coalgebras in the following way. Example Let (C, C, ψ C and (D, D, ψ D be hom-coassociative k-coalgebras. Then (C D, (id C τ id D ( C D, ψ C D is also a hom-associative k-algebra, where ψ C D = ψ C ψ D for all c C and for all d D and τ is the map described in Definition In this case we say that C D has the tensor product hom-coassociative coalgebra structure. Next we define a hom-analogous structure for k-bialgebras which is similar to the generalized hom-bialgebra of Makhlouf, Abdenacer and Silvestrov, Sergei (2008 and Makhlouf, Abdenacer and Silvestrov, Sergei (2009. Definition (Makhlouf, Abdenacer and Silvestrov, Sergei (2008. A hom-bialgebra is a 5- tuple (B, m B, B, α B, ψ B, where (B, m B, α B is a hom-associative k-algebra, (B, B, ψ B is a

36 26 hom-coassociative k-coalgebra, B is a morphism of hom-associative k-algebras, α B is a morphism of hom-coassociative k-coalgebras, and ψ B is a morphism of hom-associative k-algebras (in particular we have α B ψ B = ψ B α B. Remark The following statement is equivalent to Definition A hom-bialgebra is a homassociative k-algebra (B, m B, α B together with two k-linear maps B : B B B and ψ B : B B such that α B ψ B = ψ B α B and the following conditions are satisfied for all b, b B: b(1(1 b (1 ψ ( ( (2 B b(2 = ψb b(1 b(2(1 b (2, ( (2 (bb (1 (bb (2 = b (1 b (1 b (2 b (2, ( αb (b (1 α B (b (2 = α B ( b(1 αb ( b(2, ( ψb (b (1 ψ B (b (2 = ψ B ( b(1 ψb ( b(2, ( ψ B (bb = ψ B (bψ B (b. ( Observe that we do not require the existence of a unit or counit within hom-associative, homcoassociative, and hom-bialgebras. Moreover, we do not require the existence of a (counit in the (comodules over these hom-structured (coalgebras. This is due to the origins of hom-structures from Lie algebras which lack unital structure. The following example gives a construction which converts any k-bialgebra (not necessarily with a unit and counit into a hom-bialgebra. Example Let H = (H, m H, H be an k-bialgebra and α H : H H a k-bialgebra endomorphism. Then H αh = (H, m αh, αh is a hom-bialgebra called the Yau twist of H.

37 Remark In the literature, most of the results about hom-bialgebras use the extra assumption that ψ B = α B or ψ B = α 1 B 27 (see Caenepeel, Stefaan and Goyvaerts, Isar (2011, Makhlouf, Abdenacer and Panaite, Florin (2014, Yau, Donald (2012. We treat the general situation, to cover both cases of interest. Next we recall the definition of quasitriangular hom-bialgebras (see Yau, Donald (2009a, Yau, Donald (2012 for the case α = ψ. Definition Let (H, m H, H, α H, ψ H be a hom-bialgebra and let R H H be given as R = i s i t i. We call (H, m H, H, α H, ψ H, R a quasitriangular hom-bialgebra if the following conditions are satisfied: R (h = cop (h R, for all h H, ( ( α (R = i,j (α (R = i,j ψ (s i ψ (s j t i t j, ( s i s j ψ (t j ψ (t i, ( where the opposite comultiplication is denoted as cop (h = (τ with τ being the map described in Definition Remark Let H = (H, m H, H, α H, ψ H, R be a quasitriangular hom-bialgebra and h H. We can reformulate conditions (2.3.26, (2.3.27, and ( respectively in Definition where cop (h = h (2 h (1, for any h H using Sweedler notation as follows: s i h (1 t i h (2 = h (2 s i h (1 t i, ( i i (s i (1 (s i (2 α (t i = ψ (s i ψ (s j t i t j, ( i i,j α (s i (t i (1 (t i (2 = s i s j ψ (t j ψ (t i. ( i i,j Remark Notice that if (ψ ψ(r = R then conditions ( and ( are equivalent

38 to 28 ( (α ψ (R = i,j ((α ψ (R = i,j s i s j t i t j, ( s i s j t j t i. ( The following example gives a construction which converts any quasitriangular k-bialgebra (not necessarily with a unit and counit into a quasitriangular hom-bialgebra. Example Let (H, m H, H, R be a quasitriangular k-bialgebra and α H : H H be a k-bialgebra morphism. Then H αh = (H, m αh, αh, α H, R is a quasitriangular hom-bialgebra in which m αh = α H m H and αh = H α H. It is worth noting that this example holds even for R non-invertible. See pages 9-10 of Yau, Donald (2012. We also consider module structures over hom-analogues of these type of algebras. Definition (Makhlouf, Abdenacer and Panaite, Florin (2014. Let M be a k-vector space, A = (A, m A, α A be a hom-associative k-algebra and α M : M M be a k-linear map. A left A-module structure on (M, α M consists of a k-linear map µ M : A M M, with notation µ M (a m = a m, such that the following conditions are satisfied for all a, b A and m M: α M (a m = α A (a α M (m, ( α A (a (b m = (ab α M (m. ( Let (M, α M and (N, α N be two left A-modules. A morphism of left A-modules is a k-linear map f : M N satisfying the conditions α N f = f α M and f (a m = a f (m for all a A, m M.