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Topic: Hopf algebra

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	Hopf algebra Encyclopedia (Site not responding. Last check: 2007-10-16)
		Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, and in numerous other places, making them probably the most familiar type of bialgebra.
		Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other.
		Graded Hopf algebras are often used in algebraic topology: they are the natural algebraic structure of the totality of all homology or cohomology groups of a space.
	www.hallencyclopedia.com /topic/Hopf_algebra.html (968 words)

	PlanetMath: Hopf algebra
		The prime spectrum of a commutative Hopf algebra is an affine group scheme of multiplicative units.
		Further, a commutative Hopf algebra is a cogroup object in the category of commutative algebras.
		This is version 9 of Hopf algebra, born on 2002-10-18, modified 2005-07-12.
	planetmath.org /encyclopedia/HopfAlgebra.html (269 words)

	Non-commutative Hopf algebra of formal diffeomorphisms (Site not responding. Last check: 2007-10-16)
		This paper deals with two Hopf algebras which are the non-commutative analogues of two different groups of formal power series.
		Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients.
		Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double tensor algebra on a bi-algebra.
	www.mat.univie.ac.at /~kratt/artikel/NCHopf.html (295 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-16)
		is a Hopf algebra, and the Hopf algebra dual to it is the homology algebra
		A graded Hopf algebra is a graded bi-algebra with antipode
		Hopf algebras, under the name quantum groups, and related objects have also become important in physics; in particular in connection with the quantum inverse-scattering method [a3], [a4].
	eom.springer.de /h/h047970.htm (577 words)

	[No title]
		The Hopf algebra structure of Lemma 1.4 passes to a Hopf algebra structure in the graded case, if we use the grading conventions of Remark 1.2 It is convenient to have a name for these Hopf algebras.
		Hopf algebras withs such lif* ts of the Frobenius are fairly rare; for example the primitively generated Hopf* al* gebra Zp[x] cannot support a lift of the Frobenius as there is no primitive which red *uces to xp modulo p.
		Consider the Hopf algebra HBU ~=Zp[a1; a2; a3; : :]: 16 PAUL G. with the degree of ai equal to 2i and X ak = ai aj: i+j=k This represents the functor on graded Zp algebras* (A) = (1 + tA[[t]])0 where A[[t]] is the graded power series on A with deg(t) = -2.
	hopf.math.purdue.edu /Goerss/hopfring.txt (14186 words)

	On the Braiding on a Hopf Algebra in a Braided Category (ResearchIndex)
		By definition, a bialgebra H in a braided monoidal category (C; ø) is an algebra and coalgebra whose multiplication and comultiplication (and unit and counit) are compatible; the compatibility condition involves the braiding ø.
		The present paper is based upon the following simple observation: If H is a Hopf algebra, that is, if an antipode exists, then the compatibility condition of a bialgebra can be solved for the braiding.
		Algebraic Non-Integrability of the Cohen Map - Rychlik, Torgerson (1998)
	citeseer.ist.psu.edu /14261.html (316 words)

	Dear Stefan:
		This Hopf algebra is one of particular importance among several ``combinatorial Hopf algebras'' that have arisen recently in different contexts, like Gessel's Hopf algebra of quasi-symmetric functions and Loday-Ronco's Hopf algebra of planar binary trees.
		The Hopf algebras we consider are all abelian extensions; as a special case, they include the Drinfeld double of a group algebra.
		In fact this Hopf algebra is the degree zero part of a larger one which is based on all the planar trees.
	condor.depaul.edu /~scatoiu/seminar/february2002/abstracts.html (1659 words)

	Álgebras de Hopf
		The quantum coordinate algebra Fq (G) is a deformation of the algebra F(G) of ``polynomial functions'' on a complex semisimple group G. More precisely, the dual of the quantum enveloping algebra Uq(g) (the ``simply connected'' version) corresponding to the complex semisimple Lie algebra g, has a natural algebra structure.
		The algebra Fq(G) is defined as the subalgebra of the dual of Uq(g) generated by the matrix coefficients corresponding to finite dimensional representations of Uq(g).
		If the given algebra is finitely generated then every differential left ideal is generated by constants, a non-commutative Taylor series decomposition formula is valid, and the category of locally nilpotent modules over the operator algebra is semisimple with the only simple object that is isomorphic to the optimal algebra as a module.
	www.mate.uncor.edu /vaq2001/pf/abstracts_alg-hopf.html (1197 words)

	[No title]
		The Andre-Quillen homology of a simplicial commutative algebra is a total d* erived functor of abelianization in the category of commutative algebras, and, as such , relies on the existence of free commutative algebras*.
		To compute the Andre-Quillen homol* ogy of a bicommutative Hopf* algebra, therefore, one may be forced to leave the categor* y - the free commutative algebra* on a coalgebra is not necessarily a Hopf algebra.
		Hopf Algebras, Dieudonne modules, and derived functors of indecompos- ables The category of abelian Hopf algebras over Fp has a set of generators and, * *as such, is equivalent to a category of modules.
	hopf.math.purdue.edu /Goerss-Turner/cartan.txt (8890 words)

	Hopf algebra - Definition, explanation
		In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map
		The most exciting Hopf algebras however are certain "deformations" or "quantizationss" of those from example 3 and 4 which are neither commutative nor co-commutative.
		These Hopf algebras are often called quantum groups, a term that is only loosely defined.
	www.calsky.com /lexikon/en/txt/h/ho/hopf_algebra.php (544 words)

	Seminar: Hopf Algebras
		We refine the concept of Hopf Algebra by adding the notion of antipode, patterned after the inverse map in a group, and that of bialgebra.
		The Hopf Algebra of rooted trees II We continue exploring Connes and Kreimer's Hopf Algebra generated by rooted trees, in particular the constructions of the coproduct and the antipode.
		We aim in particular at describing a notable Hopf algebra structure related to the one of rooted trees and to interesting Lie algebras.
	www.math.fsu.edu /~ealdrov/seminar/spring2004/hopf.html (654 words)

	[No title]
		It is a Hopf algebra generated by its subgroup ${\cal K}$ of primitive elements.
		P.G. Abstract: Hopf algebras over the prime field with $p$ elements is an abelian category which is equivalent, by work of Schoeller, to a category of graded modules, known as Dieudonn\'e modules.
		Graded ring objects in Hopf algebras are called Hopf rings, and they arise in the study of unstable cohomology operations for extraordinary cohomology theories.
	www.lehigh.edu /dmd1/public/www-data/h120 (837 words)

	Introduction to "Taming Wild Extensions" of Lindsay N. Childs
		Galois module theory is the branch of algebraic number theory which studies rings of integers of Galois extensions of number fields as modules over the integral group ring of the Galois group.
		Their classification transforms the problem of classifying Hopf Galois structures on a Galois extension into a purely group-theoretic problem involving the Galois group G, which, however, is often unmanageable.
		Chapter 12 then presents the Kummer theory of formal groups, from {CM94}, {Mo94}, {Mo96}, which shows that if a Hopf algebra H is constructed as in Chapter 11, then the principal homogeneous spaces for H, or equivalently, the Galois extensions for H^*, are easily and explicitly described.
	math.albany.edu:8000 /~lc802/mono.html (1829 words)

	Representation of a Hopf algebra - Definition, explanation
		Let's say we have an unital associative algebra H. We know what a representation of H is. It's just a module.
		This is merely one justification of a Hopf algebra.
		If V is a rep of a Hopf algebra H, then X in V is said to be invariant under H if for all A in H, A[X]=ε(A)X. Note that the subset of all invariant elements of V forms a subrep of V. This article describes a vector space rep of a Hopf algebra.
	www.calsky.com /lexikon/en/txt/r/re/representation_of_a_hopf_algebra.php (745 words)

	Overview Research Activities 2005 - PNA0
		The Hopf algebra, NSymm, of noncommutative symmetric functions and the Hopf algebra, QSymm, of quasisymmetric functions are two important generalizations of the very rich (in theory as well as applications) and important Hopf algebra of symmetric functions.
		The current situation is that there is a strict rigidity theorem: the only Hopf algebra automorphisms of MPR is the identity and that as a PtwsH (positive twisted selfadjoint Hopf) algebra is unique up to and including level 5.
		Rigidity for MPR, the Malvenuto-Poirier-Reutenauer Hopf algebra of permutations.
	www.cwi.nl /publications/annual-reports/2005/ORA/HTML/PNA0.shtml (1015 words)

	Discrete Maths at UBC
		Peak functions and Eulerian enumeration: In 1995 the complete duality of Q -- the Hopf algebra of quasisymmetric functions -- and NC -- the Hopf algebra of nonommutative symmetric functions -- was realised by Malvenuto and Reutenauer.
		Combinatorial Hopf Algebras: A Combinatorial Hopf algebra [CH-algebra] is a pair (H, z) where H is a graded connected Hopf algebra over a field F, and z: H -> F is a multiplicative functional.
		Now, in general, we can fix a partition \lambda of n, and ask for algebraic conditions on the coefficients of F so that the roots follow the pattern dictated by the parts in \lambda.
	www.math.ubc.ca /~steph/spring2003.html (922 words)

	CJM - The Representation Ring and the Centre of a Hopf Algebra
		When $H$ is a finite dimensional, semisimple, almost cocommutative Hopf algebra, we examine a table of characters which extends the notion of the character table for a finite group.
		We give a basis of the centre of $H$ which generalizes the conjugacy class sums of a finite group, and express the class equation of $H$ in terms of this basis.
		We show that the representation ring and the centre of $H$ are dual character algebras (or signed hypergroups).
	www.journals.cms.math.ca /cgi-bin/vault/view/witherspoon0939 (142 words)

	Entropic Hopf algebras and models of non-commutative logic (Site not responding. Last check: 2007-10-16)
		It has recently been demonstrated that Hopf algebras provide an excellent framework for modeling a number of variants of multiplicative linear logic, such as commutative, braided and cyclic.
		We extend these ideas to the entropic setting by developing a new type of Hopf algebra, which we call entropic Hopf algebras.
		We show that the category of modules over an entropic Hopf algebra is an entropic category (possibly after application of the Chu construction).
	www.tac.mta.ca /tac/volumes/10/17/10-17abs.html (182 words)

	Hopf Algebra Structure
		The Hopf algebra structure that is used by default is the one described in Section Representations of U_q(L).
		Given a quantized universal enveloping algebra U and (anti-) automorphisms f and g of U where g is the inverse of f (this is not checked by Magma) set U to use the corresponding twisted Hopf algebra structure.
		This command has to be given before using the Hopf algebra structure, otherwise the default structure will be used.
	www.math.lsu.edu /magma/text1093.htm (358 words)

	Serban Raianu
		Hopf Algebras: an Introduction, Marcel Dekker Monographs and Textbooks, vol.
		Induction functors for the Doi-Koppinen unified Hopf modules, in Abelian groups and modules, Proceedings of the Padova Conference, Padova, Italy, June 23-July 1, 1994, Kluwer, 1995, eds.
		Co-Frobenius Hopf algebras with a separable Hopf-Galois extension are finite dimensional, (with M.
	www.csudh.edu /math/sraianu/cv.htm (758 words)

	Richard G. Larson -- Publications
		Characters of Hopf algebras, J. Algebra 17(1971), 352-368.
		Cosemisimple Hopf algebras with small simple components are involutory, Comm.
		(with D.E. Radford), Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple, J. Algebra 117 (1988), 267-289.
	tigger.uic.edu /~rgl/allpubl.html (737 words)

	A generalization of Connes-Kreimer Hopf algebra
		“Bonsai” Hopf algebras, introduced here, are generalizations of Connes-Kreimer Hopf algebras, which are motivated by Feynman diagrams and renormalization.
		We introduce a new differential on these bonsai Hopf algebras, which is inspired by the tree differential.
		The cohomologies of these are computed here, and the relationship of this differential with the appending operation * of Connes-Kreimer Hopf algebras is investigated.
	repository.upenn.edu /dissertations/AAI3165650 (111 words)

	1
		Westreich, “Quasitriangular Hopf Algebras whose Group of Grouplike Elements Form an Abelian Group”, Proc.
		Gelaki and S. Westreich, “Hopf algebras of type Uq(sln)’ and Oq(SLn)’ which give rise to certain invariants of knots, links and 3-manifolds”, Trans.
		Westreich, “A Galois-type correspondence theory for actions of finite dimensional pointed Hopf algebras on prime algebras”, J. of Algebra, 219 (1999) (2), 606-624.
	www.biu.ac.il /faculty/westreich/publ.html (289 words)

	Algebraic & Geometric Topology, Volume 5 (2005)
		ring spectrum) R naturally has the structure of a commutative R–algebra in the strict sense, and of a Hopf algebra over R in the homotopy category.
		We show, under a flatness assumption, that this makes the Bökstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence.
		This is part of a program to make systematic computations of the algebraic K–theory of S–algebras, by means of the cyclotomic trace map to topological cyclic homology.
	www.msp.warwick.ac.uk /agt/2005/05/p049.xhtml (148 words)

	M. Susan Montgomery
		Hopf Algebras and Their Actions on Rings, CBMS Lecture Notes vol 82, American Math Society, Providence, RI, 1993 (238 pages).
		Skew-derivations of finite-dimensional algebras and actions of the double of the Taft Hopf algebra (with H.-J. Schneider), Tsukuba J. of Math 25 (2001), 337-358.
		Algebra properties invariant under twisting, Hopf algebras in Non-commutative Geometry and Physics, S. Caenepeel and F. Van Oystaeyen, editors, Lecture Notes in Pure and Applied Math, vol 239, Marcel Dekker, 2005, 61-75.
	www-rcf.usc.edu /~smontgom (725 words)

	Transactions of the American Mathematical Society
		Abstract: We give a different proof for a structure theorem of Hausser and Nill on Hopf modules over quasi-Hopf algebras.
		We extend the structure theorem to a classification of two-sided two-cosided Hopf modules by Yetter-Drinfeld modules, which can be defined in two rather different manners for the quasi-Hopf case.
		The category equivalence between Hopf modules and Yetter-Drinfeld modules leads to a new construction of the Drinfeld double of a quasi-Hopf algebra, as proposed by Majid and constructed by Hausser and Nill.
	www.ams.org /tran/2002-354-08/S0002-9947-02-02980-X/home.html (172 words)

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