A b is a linear map which is both an algebra and a coalgebra homomorphism. Hopf algebras, quantum groups and yangbaxter equations. Notes on strict bicommutative hopf algebras contents. This last relation is a general property of hopf algebras with an involution. Computations with the adams spectral sequence require the extensive use of homological algebra in the category of amodules. Each comes with an appropriate notion of a module and thus with an associated theory of representations. Full crossed products by hopf calgebras internet archive. Say that acts finitely on a left module algebra if the image of is finite dimensional.
If a is commutative, the nakayama involution is the identity. In the cocommutative case, bialgebras and hopf algebras can be viewed as monoids and groups in the symmetric monoidal category of cocommutative coalgebras. Hopf algebras and hopf algebroids university of rochester. This starts with a different concept of bialgebroid, which is discussed here on the nlab. August 1998, ihesm9860 abstract we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations. If a is commutative or cocommutative s2 ida and therefore s is bijective. In the case of lie groups or lie algebras, star operations are used to define real. All the maps in the diagrams are algebra homomorphisms given our hy. Kernels and cokernels in the category of augmented. Operads, algebras, modules, and motives 3 introduction there are many di erent types of algebra. The rst examples are given by the quantised enveloping algebra of a semisimple lie algebra, and the second are given by an involutive hopf algebra.
Westudy thestrucure of arbitrary splitinvolutive regular homliealgebras. Introduction to hopf algebras and representations aalto math. We show that the connesmoscovici negative cyclic cohomology of a hopf algebra equipped with a character has a lie bracket of. The present notes were prepared for the lectures of my course introduction to hopf algebras and representations, given in the university of helsinki in the spring of 2011. The order of the antipode of a hopf algebra has been an important subject of study since the beginning of the theory in the 60s. To get some idea what it looks like, group itself is a hopf algebra, considered over a field with one element. Moreover any semisimple hopf algebra over a field of characteristic zero is involutive.
Hopf algebras of dimension p2 siuhung ng mathematics department, towson university, baltimore, md 21252 abstract let p be a prime number. Hopf algebras, algebraic, f ormal, and quan tum groups. If s 2 id h, then the hopf algebra is said to be involutive and the underlying algebra with involution is a algebra. Hopf algebras, quantum groups and topological field theory. Representations of hopf algebras 477 here are the main results of the article. On the structure of split involutive regular homlie algebras. Aof algebras which is universal with respect to maps of algebras f. In particular, no vector calculus, abstract algebra or topology is needed. Poissonhopf algebra deformations of liehamilton systems. In the cocommutative case, bialgebras and hopf algebras can be viewed as.
Hopf stars, twisted hopf stars and scalar products on quantum spaces. A class of examples of such hopf algebroids internally in a monoidal category of cocomplete cofiltered vector spaces is in. Schmitt department of mathematical sciences memphis state university memphis, tennessee 38152 abstract. A hopf algebra is said to be almost involutive ai if it admits a companion automorphism that can be viewed as a special additional symmetry. The crux will be to consider a poissonhopf algebra structure that replaces the lh algebra of the nondeformed lh system, thus allowing for an explicit construction of tindependent. We exhibit an internal coproduct on the hopf algebra of finite topologies recently defined by the second author, c. Batalinvilkovisky algebras and cyclic cohomology of hopf algebras luc menichi abstract. An isomorphism between the hopf algebras a b of jacobi. H such that b is a braided hopf algebra in the category of. The hopf algebra of finite topologies and tpartitions. In the rst main result we show that a coalgebra is qcf if and only if cis weakly isomorphic to ratc c as left cmodules, in the sense that some coproduct. Informally, this framework amounts among other tasks to represent the abstract involutive algebras of operators stemming from the above mentioned coordinate algebras on well chosen involutive algebras of. Hopf algebras of combinatorial structures 1 william r. On twisting of finitedimensional hopf algebras math.
Free braided nonassociative hopf algebras and sabinin. We present examples and study some of the basic properties and constructions of aihopf algebras centering the attention in the finite dimensional case. Hopf algebras and representation theory of hopf algebras faubox. Any groupoid algebra is an involutive weak hopf algebra. This invariant is known to distinguish lens spaces of the same homotopy type which. Hopf algebras, renormalization and noncommutative geometry alain connes ihes dirk kreimer mainz univ. An isomorphism between the hopf algebras a and b of jacobi diagrams in the theory of knot invariants j anis lazovskis december 14, 2012 abstract we construct a graded hopf algebra b from the symmetric algebra of a metrized lie algebra, and. Hopf algebras of dimension 2 department of mathematics. Bialgebras and hopf algebras university of chicago. Above we showed that the antipode is an involution if the hopf algebra is commutative or. In mathematics, a hopf algebra, named after heinz hopf, is a structure that is simultaneously an. It is not a paper describing basic properties and applications of hopf algebras. Moreover, we prove that the free braided algebra k v has a natural structure of a braided nonassociative hopf algebra such that every element of the space of generators v is primitive. So, we get the following reconstruction theorem for.
The notion of hopf algebra emerged slowly from the work of the topologists in the 1940s dealing with the cohomology of compact lie groups and their homogeneous spaces. Hopf algebras were constructed as vector spaces on certain diagrams, such as rooted trees, which can be combined accounting for a product. Rather, it is a paper providing the necessary structures in order to construct a. A hopf algebra structure on hall algebras christopher d. In mathematics, a hopf algebra, named after heinz hopf, is a structure that is simultaneously an unital associative algebra and a counital coassociative coalgebra, with these structures compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. Involutive representations of coordinate algebras and. Schmitt memphis state university, memphis, tn 38152 1. We prove several properties of kernels and cokernels in the category of augmented involutive stereotype algebras. Coinvolutive hopfvon neumann algebras springerlink.
In particular, in the last part of the course i gave a complete proof from scratch of zhous theorem 1994. In the connected case, the result specializes to give the following. Pdf almost involutive hopfore extensions of low dimension. Patras, dual to the composition of quasiormoulds, which are. We say that a weak hopf algebra is involutive if its antipode is an involution. All algebras are associative and have identity, we use the standard hopf algebra notation and terminology and we always assume that the antipode of a hopf algebra is bijective. Victor reiner, hopf algebras in combinatorics math user home. In chapter4, familiarity with representation theory of nite groups over c is assumed, along.
In tak 1971 a hopf algebra is given where s is not bijective. If h is finitedimensional semisimple over a field of characteristic zero, commutative, or cocommutative, then it is involutive. I thank the participants of the course for making it a very enjoyable experience. One can thus give an axiomatic characterization of the category of representations of a hopf algebra, and this is one of the most fruitful ways to deal with quantum groups. Moscovici hopf algebra of differential operators in the onedimensional case. If a is generated as an algebra by a set b, to prove that a is a bialgebra we only need to check the diagrams in 1. A basic introduction to hopf algebras becca ebert april 18, 2014 abstract this paper o ers an introduction to hopf algebras. A hopf algebra is called involutory if its antipode is an involution. In taft 1971 fd examples tn, where n 1, are given where s2 has.
Our approach uses the algebra of quaternions and illustrates some of the algebraic and geometric properties of the hopf bration. On the structure of split involutive regular homlie algebras jian zhang,chiping zhang andyan cao abstract. Walker department of mathematics, university of california riverside, ca 92521 usa october 16, 2010 abstract one problematic feature of hall algebras is the fact that the standard multiplication and comultiplication maps do not satisfy the bialgebra com. Incorporating antipodes sometimes called conjugations, we obtain the notion of a hopf algebra. Hopf algebras in combinatorics 3 the main prerequisite for reading these notes is a good understanding of graduate algebra3, in partic ular multilinear algebra tensor products, symmetric powers and exterior powers4 and basic categorical language5. Hopf algebra is nice object full of structure a bialgebra with an antipode. After the pioneer work of connes and kreimer1, hopf algebras have be come an established tool in perturbative quantum. Throughout, h denotes a finitedimensional hopf algebra over a field iof characteristic p g 0, and s is the antipode of h. If s2 idh, then the hopf algebra is said to be involutive and the underlying algebra with involution is a. Hopf algebras and hopf algebroids commutative, noncocommutative hopf algebras, such as the dual of the steenrod algebra a3. In the matrix model approach of ncft, one regards the noncommutative coordinate algebras as abstract involutive algebras of. A hopf algebra is said to be \emphalmost involutive ai if it admits a companion automorphism that can be viewed as a special additional symmetry. This chapter is devoted to the structure of coinvolutive hopfvon neumann algebras, which has been introduced by ernest 44.
Introduction in this paper we introduce a hopf algebraic framework for studying invariants of graphs, matroids, and other combinatorial structures. We note that many authors use the term hopf algebra for any bialgebra with an antipode. Selfdual hopf algebras the homology hopf algebras h bu. Hopf algebras, renormalization and noncommutative geometry.
Hopf algebras in combinatorics contents introduction 2 1. Throughout we work with involutive algebras over the. A hopf algebra is said to be almost involutive ai if it admits a companion automorphism. Eine hopfalgebra benannt nach dem mathematiker heinz hopf h \ displaystyle h h.
The universal enveloping algebra of the lie algebra g is the factor algebra of the tensor algebra tg by the ideal. Skoda, lie algebra type noncommutative phase spaces are hopf algebroids, arxiv1409. It is known that any nonsemisimple hopf algebra of dimension p2 over an algebraically closed. The graph algebra is a commutative, cocommutative, graded, connected incidence hopf algebra, whose basis elements correspond to nite graphs and whose hopf product and coproduct admit simple com. Generalized frobenius algebras and hopf algebras 3 techniques of i can be extended and applied to obtain a symmetric characterization of these coalgebras.