Trivial examples of semisimple hopf algebras are the group algebra kg and its dual hopf algebra kg. Let h be an arbitrary pointed hopf algebra of rank one and g the group of group like elements of h. Hopf 2cocycles also are related with hopfgalois theory, since there exists a biyective correspondence between hopf 2cocycle and cleft galois object see hopf bigalois extensions, and for finite dimensional hopf algebras, the classification of hopf 2. For the love of physics walter lewin may 16, 2011 duration. Taking direct sums of the representations works just like before. It is clear that if gis a pivotal element, then the coset gzgh is the set of all pivotal elements of hwhere zgh is the group of central grouplike elements of h. According to the properties of the initial hopf algebra and the geometric requirements of the reader, this gives then a. Universal algebra of a homlie algebra and grouplike elements. Ravenel uses his version to show that a particular hopf algebra is a group algebra, so he found it useful. A primer of hopf algebras 3 basis, and the multiplication in gis extended to kgby linearity. Antipodes and grouplikes in finite hopf algebras sciencedirect.
Quasitriangular hopf algebras whose grouplike elements form an abelian group sara westreich communicated by lance w. We will mainly report on recent progress in the classi. These results were proved independently by cartier and gabriel and by kostant in the early 1960s. Over fields we know the grouplikes form a group of at most rz elements, so if g is a grouplike of a the class of g n is trivial modulo pa. In this paper we prove some properties of the set of group like elements of a, ga, for a pointed minimal quasitriangular hopf algebra a over a eld k of characteristic 0, and for a pointed. Ribbon element on cofrobenius quasitriangular hopf algebras. The group hopf algebra of g over k, denoted kg or kg, is as a set and vector space the free vector space on g over k. Request pdf universal algebra of a homlie algebra and grouplike elements we construct the universal enveloping algebra of a homlie algebra and endow it with a homhopf algebra structure. Subalgebra of a group algebra planned maintenance scheduled april 23, 2019 at 00. Let h be an arbitrary pointed hopf algebra of rank one and g the group of grouplike elements of h. Beginning of definition of hopf algebra by string diagrams. This hopf algebra is generated by skewprimitive and grouplike elements, and depends on a complex parameter. The group hopf algebra of g over k, denoted kg or kg, is as a set and a vector space the free vector space on g over k.
In this paper, we find a necessary and sufficient condition for h, r to have a ribbon element. In the cocommutative case, bialgebras and hopf algebras can be viewed as monoids. Quasitriangular hopf algebras whose group like elements form an abelian group sara westreich communicated by lance w. Here fp gg g2g is the dual basis of the basis given by group elements. Qan is called the module of aindecomposable elements of n and is. Cartier ca describing hopf algebras in topology and group. Group elements of some semisimple finitedimensional hopf. In this paper we prove some properties of the set of grouplike elements of a, ga, for a pointed minimal quasitriangular hopf algebra a over a eld k of characteristic 0, and for a pointed. Morphisms of hopf algebras are just bialgebra maps \preserving the antipode exercise 3.
Introduction to hopf algebras and representations aalto university. Like hopf algebras, weak hopf algebras form a selfdual class of algebras. Bialgebras and hopf algebras university of chicago. The counit map is given by, xn 1, if n 0 0, if n 6 0 as another example consider cg, the group algebra of a group g. Show that kg has no nonzero primitive elements, that is ph 0. Recall that the coalgebra structure on kg is given by dgg g and sg g 1. Thus l is recovered from the coalgebra structure of ul in this case. It is shown in this paper that for odd prime n the set of grouplike elements of these algebras is a cyclic group of order 2n. An easy example of a hopf algebra is the complex group algebra of a nite group. The grouplike elements form a group with inverse given by the antipode. Homework assignment 4 bialgebras and hopf algebras. Abstractwe study the group of group like elements of a weak hopf algebra and derive an analogue of radfords formula for the fourth power of the antipode s, which implies that the antipode has a finite order modulo, a trivial automorphism. Further, the algebra and co algebra structures must be compatible. The structure of pointed hopf algebras a with nonabelian group ga is largely unknown.
Representations of hopf algebras 477 here are the main results of the article. Request pdf universal algebra of a homlie algebra and group like elements we construct the universal enveloping algebra of a homlie algebra and endow it with a hom hopf algebra structure. This paper is concerned with versions of these results. Now np n is the nilradical of r, and since a is free we have npa na. This space is also both an algebra and a coalgebra. Pointed hopf algebras from enveloping algebras to quantum. Decomposable elements, grouplike elements, primitive elements. The sum of a finite group of weights of a hopf algebra. In particular, an application of this result has enabled us to nd a deformation of such a smash product algebra which. Bialgebras and hopf algebras i let abe an augmented algebra. A hop f order o over an integral domain r with field of fractions k is an order in a hopf alg ebra h over k which is closed unde r the alg ebra and coalgebra operations.
In the case of pointed hopf algebras certainly a big glass of hopf algebras where the group likes are the coradical, the strucure is basically dictated by the group likes and by a nichols algebra the bosonization of both actually gives the graded algebra associated to the coradical filtration of your hopf algebra. The studied hopf algebra has only one nonone dimensional irreducible representation. Subalgebra of a group algebra planned maintenance scheduled. The corepresentation ring of a pointed hopf algebra of. Let gbe an algebraic group gl n, sl n, sp 2n, so n. The condition gives a connection with the order of g h and the order of s2. Deformations of group hopf algebras are foundational in the. Such grouplike element is called a pivotal element and his called a pivotal hopf algebra. Chubarov proved a criterion under which an element of some semisimple finitedimensional hopf algebra is group like. If r has no nilpotents, or contains q, the bound is the same as it would be over fields. Throughout c denotes a coalgebra and h denotes a hopf algebra. The corepresentation ring of a pointed hopf algebra of rank. We then discuss results for products of skewprimitive elements.
On hopf 2algebras international mathematics research. Denote by kgthe group algebra which has gas a basis for its vector space. Show that the grouplike elements of a group ring of g form a group that can be. The literature cited for each talk overlaps and sometimes only parts of the cited chapters will enter the talks. Such group like element is called a pivotal element and his called a pivotal hopf algebra. Note that qais wellde ned even if ais not associative. Let h, r be a cofrobenius quasitriangular hopf algebra with antipode s. Examples include groups, quantum groups over lie algebras, the small quantum groups of lusztig, and their variations. Hopf algebras of dimension 2 iowa state university. Motivated by the orthogonality relations for irreducible characters of a finite group, we evaluate the sum of a finite group of linear characters of a hopf algebra, at all grouplike and skewprimitive elements.
Hopf algebras says that a cocommutative hopf algebra is a group algebra if and only if it has a basis of group like elements. It is clear that if gis a pivotal element, then the coset gzgh is the set of all pivotal elements of hwhere zgh is the group of central group like elements of h. In this paper we prove some properties of the set of group like elements of a, ga, for a pointed minimal quasitriangular hopf algebra a over a field k of characteristic 0, and for a pointed quasitriangular. It is shown in this paper that for odd prime n the set of group like elements of these algebras is a cyclic group of order 2n. As an algebra, its product is defined by linear extension of the group composition in g, with multiplicative unit the identity in g. In this paper we prove some properties of the set of grouplike elements of a, ga, for a pointed minimal quasitriangular hopf algebra a over a field k of characteristic 0, and for a pointed quasitriangular. In this paper, we give the decomposition of a tensor product of finite dimensional indecomposable right hcomodules into a direct sum of indecomposables. Then the group finite hopf algebras 295 like elements of a have finite order. Hopf algebra is nice object full of structure a bialgebra with an antipode. Ve dim v and for two group elements that are conjugates, g2 hg1h. Exercise 5 primitive elements aif gis a group then the group algebra kg is a hopf algebra with all g2gbeing group like elements.
As another example consider cg, the group algebra of a group g. Exercise 39 show that the grouplike elements of a hopf algebra form a group. Show that this group can also be defined as the group of isomorphism classes. Throughout, h denotes a finitedimensional hopf algebra over a field iof characteristic p g 0, and s is the antipode of h. If an dimensional hopf algebra contains distinct grouplike elements, then it follows that, the set of grouplike elements in, is linearly independent and contains at least elements. A more general result for semisimple hopf algebras of dimension pq, where p, q are odd prime, is obtained by eg98. It is therefore tempting to define the object integrating the homlie algebra g as being the set of grouplike elements in the homhopf algebra u g. For the dual algebra one has dp g auv p u p v and sp g p g 1. Walker department of mathematics, university of california. For any hopf algebra h over k an element e 2 h which satis. G given in 7 is failed to be a universal rmatrix for the hopf algebra d. Exercise 5 primitive elements aif gis a group then the group algebra kg is a hopf algebra with all g2gbeing grouplike elements. This hopf algebra is generated by skewprimitive and group like elements, and depends on a complex parameter.
To get some idea what it looks like, group itself is a hopf algebra, considered over a field with one element. A hopf algebra structure on hall algebras christopher d. In particular, grouplike elements of any hopf algebra h form a group, denoted g h. Noncommutative, noncocommutative semisimple hopf algebras. Cartier ca describing hopf algebras in topology and group theory and applications in combinatorics. Elementary super groups a university like no other. A hopf algebra is an associative algebra h with unity 1 which has a coalgebra structure, such that are algebra maps, and a certain algebra antiendomorphism s. Abstractwe study the group of grouplike elements of a weak hopf algebra and derive an analogue of radfords formula for the fourth power of the antipode s, which implies that the antipode has a finite order modulo, a trivial automorphism.
Such a ring is not commutative if g is not abelian. Namely, having a coproduct means that you can define the tensor product of representations and having an antipode means that you can define the dual of a. Let g be any group and let kg be the group algebra. Homology of the primitive part prima of a cocommutative graded hopf algebra aover a eld of characteristic zero. Chubarov proved a criterion under which an element of some semisimple finitedimensional hopf algebra is grouplike. The multiplication of kgis the obvious one, induced by the multiplication of gand extended linearly. This enables us to describe the corepresentation ring of h in terms of generators and relations.