On Generators and Congenerators

The question of the existence of generators and cogenerators i n a category is of i n t e r e s t i n view of the special adjoint functor theorem. ISBELL has given an example (unpublished) which shows t h a t the existence of a cogenerator i s a necessary part of the hypothesis of the special adjoint functor theorem. This example also shows t h a t the category of groups has no cogenerator. (Clearly the f r e e group on one element i s a generator i n the category of groups.) It is well known t h a t there e x i s t generators and cogenerators i n the categories of commutative groups, Comrnutative Lie algebras (over a f i e l d ) and commutative r e s t r i c t e d Lie algebras, because a l l of these categories are module categories. By ISBELL1s r e s u l t when one drops the condition of cornmutativity for the category of commut a t i v e groups there i s no longer a cogenerator. We have Proved similar r e s u l t s for the categories of commutative Lie algebras and commutative r e s t r i c t e d Lie algebras. The r e s u l t s are summarized i n the l i s t below where we have included some r e l a t e d categories

The Heyneman-Radford Theorem for Monoidal Categories

We prove Heyneman-Radford Theorem in the framework of Monoidal Categories

Lightning Draft

Lightning Draft is a web application for drafting Magic: the Gathering cards. Users can visit www.lightningdraft.online to build a deck from randomly generated booster packs. This app was inspired by digital card games such as Hearthstone. Lightning Draft is a quick, fun, and simple alternative to drafting with physical cards

Additive Deformations of Hopf Algebras

Additive deformations of bialgebras in the sense of Wirth are deformations of the multiplication map of the bialgebra fulfilling a compatibility condition with the coalgebra structure and a continuity condition. Two problems concerning additive deformations are considered. With a deformation theory a cohomology theory should be developed. Here a variant of the Hochschild cohomology is used. The main result in the first part of this paper is the characterization of the trivial deformations, i.e. deformations generated by a coboundary. When one starts with a Hopf algebra, one would expect the deformed multiplications to have some analogue to the antipode, which we call deformed antipodes. We prove, that deformed antipodes always exist, explore their properties, give a formula to calculate them given the deformation and the antipode of the original Hopf algebra and show in the cocommutative case, that each deformation splits into a trivial part and into a part with constant antipodes.Comment: 18 page

Generating loop graphs via Hopf algebra in quantum field theory

We use the Hopf algebra structure of the time-ordered algebra of field operators to generate all connected weighted Feynman graphs in a recursive and efficient manner. The algebraic representation of the graphs is such that they can be evaluated directly as contributions to the connected n-point functions. The recursion proceeds by loop order and vertex number.Comment: 22 pages, LaTeX + AMS + eepic; new section with alternative recursion formula added, further minor changes and correction

Structure of semisimple Hopf algebras of dimension p2q2p^2q^2

Let $p,q$ be prime numbers with $p^4<q$, and $k$ an algebraically closed field of characteristic 0. We show that semisimple Hopf algebras of dimension $p^2q^2$ can be constructed either from group algebras and their duals by means of extensions, or from Radford biproduct R#kG, where $kG$ is the group algebra of group $G$ of order $p^2$, $R$ is a semisimple Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^2$. As an application, the special case that the structure of semisimple Hopf algebras of dimension $4q^2$ is given.Comment: 11pages, to appear in Communications in Algebr

Loops on surfaces, Feynman diagrams, and trees

We relate the author's Lie cobracket in the module additively generated by loops on a surface with the Connes-Kreimer Lie bracket in the module additively generated by trees. To this end we introduce a pre-Lie coalgebra and a (commutative) Hopf algebra of pointed loops on a surface. In the last version I added sections on Wilson loops and knot diagrams.Comment: 13 pages, no figures. Added sections on Hopf algebras, Wilson loops on surfaces and knot diagram

The sum of a finite group of weights of a Hopf algebra

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 skew-primitive elements. We then discuss results for products of skew-primitive elements. Examples include groups, (quantum groups over) Lie algebras, the small quantum groups of Lusztig, and their variations (by Andruskiewitsch and Schneider).Comment: 28 pages, LaTeX, final form (modulo table of contents and typesetting

Simplicial presheaves of coalgebras

The category of simplicial R-coalgebras over a presheaf of commutative unital rings on a small Grothendieck site is endowed with a left proper, simplicial, cofibrantly generated model category structure where the weak equivalences are the local weak equivalences of the underlying simplicial presheaves. This model category is naturally linked to the R-local homotopy theory of simplicial presheaves and the homotopy theory of simplicial R-modules by Quillen adjunctions. We study the comparison with the R-local homotopy category of simplicial presheaves in the special case where R is a presheaf of algebraically closed (or perfect) fields. If R is a presheaf of algebraically closed fields, we show that the R-local homotopy category of simplicial presheaves embeds fully faithfully in the homotopy category of simplicial R-coalgebras.Comment: 24 page

Algebra Structures on Hom(C,L)

We consider the space of linear maps from a coassociative coalgebra C into a Lie algebra L. Unless C has a cocommutative coproduct, the usual symmetry properties of the induced bracket on Hom(C,L) fail to hold. We define the concept of twisted domain (TD) algebras in order to recover the symmetries and also construct a modified Chevalley-Eilenberg complex in order to define the cohomology of such algebras