Crossed Modules and Quantum Groups in Braided Categories

Let A be a Hopf algebra in a braided category . Crossed modules over A are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group the corresponding braided category of modules is identified with a full subcategory in . The connection with cross products is discussed and a suitable cross product in the class of quantum braided groups is built. Majid–Radford theorem, which gives equivalent conditions for an ordinary Hopf algebra to be such a cross product, is generalized to the braided category. Majid's bosonization theorem is also generalized.     

Positively Convex Spaces

We give a construction of the left adjoint of the comparison functor in one step and we give a characterization of separated (finitely) positively convex spaces.     

Hopf Algebra Dual to a Polynomial Algebra over a Commutative Ring

For a polynomial algebra in several variables over a commutative ring R with a Hopf algebra structure the existence of the dual Hopf algebra is proved.     

THE TENSOR CATEGORY OF LINEAR MAPS AND LEIBNIZ ALGEBRAS

We equip the category of linear maps of vector spaces with a tensor product which makes it suitable for various constructions related to Leibniz algebras. In particular, a Leibniz algebra becomes a Lie object in and the universal enveloping algebra functor UL from Leibniz algebras to associative algebras factors through the category of cocommutative Hopf algebras in . This enables us to prove a Milnor-Moore type theorem for Leibniz algebras.     

Iterating the Bar Construction

For a 1-connected space X Adams's bar construction B(C*(X)) describes H*(X) only as a graded module and gives no information about the multiplicative structure. Thus it is not possible to iterate the bar construction in order to determine the cohomology of iterated loop spaces ⁱ X. In this paper for an n-connected pointed space X a sequence of A()-algebra structures , is constructed, such that for each there exists an isomorphism of graded algebras      

On the dimension of the cartesian product of relations and orders

It is known that for incidence structures and , max , wheref dim stands for Ferrers relation. We shall show that under additional assumptions on and , both bounds can be improved. Especially it will be shown that the square of a three-dimensional ordered set is at least four-dimensional.     

Hom-tensor relations for (quasi-) comodule algebras

We discuss quasi-Hopf algebras as introduced by Drinfeld and generalize the Hom-tensor adjunctions from the Hopf case to the quasi-Hopf setting, making the module category over a quasi-Hopf algebra H into a biclosed monoidal category. However, in this case, the unit and counit of the adjunction are not trivial and should be suitably modified in terms of the reassociator and the quasi-antipode of the quasi-Hopf algebra H. In a more general case, for a comodule algebra $ \mathcal{B} $ over a quasi-Hopf algebra H, the module category over $ \mathcal{B} $ need not to be monoidal. However, there is an action of a monoidal category on it. Using this action, we consider some kind of tensor and Hom-endofunctors of module category over $ \mathcal{B} $ and generalize some Hom-tensor relations from module category on H to this module category.     

K - Theory of the Weil Transfer FunctorDedicated to Daniel Quillen

We consider motivic category of pairs (smooth projective scheme, separable algebra) over a given field. For any separable extension L/Fwe construct the Weil transfer functor . This enables us to compute K-groups of schemes obtained by means of Weil restriction.     

Correction to: ‘The UCT, the Milnor Sequence, and a Canonical Decomposition of the Kasparov Groups’

In this note we correct a mistake in K-Theory 10 (1996), 49–72. In that paper we asserted that under bootstrap hypotheses the short exact sequence

<Emphasis Type="Italic">γ</Emphasis>-Frames,GΓ-Algebras and Measurable Spaces

The category of γ -Frm of γ-frames, which is isomorphic to the category GΓ-Alg of \(\mathbb D\)-algebras satisfying certain identities, and the category γ -Top of γ-topological spaces provide the background for the category γ -Mbl of γ-measurable spaces. As in the category of frames, the functor \(\Omega_\gamma: \gamma {\text{ - }}Top \ni (X,\Omega_X) \mapsto \Omega_\S \in \gamma {\text{ - }}Frm^{op}\) has a right adjoint \(Pt_\gamma: G\Gamma {\text{ - }}Alg^{op} \to \gamma {\text{ - }}Top\).     

Skew Boolean algebras derived from generalized Boolean algebras

Every skew Boolean algebra S has a maximal generalized Boolean algebra image given by S/ where is the Green’s relation defined initially on semigroups. In this paper we study skew Boolean algebras constructed from generalized Boolean algebras B by a twisted product construction for which . In particular we study the congruence lattice of with an eye to viewing as a minimal skew Boolean cover of B. This construction is the object part of a functor from the category GB of generalized Boolean algebras to the category LSB of left-handed skew Boolean algebras. Thus we also look at its left adjoint functor . This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.     

The conductor of a cyclic quartic field using Gauss sums

Let Q denote the field of rational numbers. Let K be a cyclic quartic extension of Q. It is known that there are unique integers A, B, C, D such that where A is squarefree and odd, D=B ²+C ² is squarefree, B $$ " align="middle" border="0"> 0 , C $$ " align="middle" border="0"> 0, GCD(A,D)=1. The conductor f(K) of K is f(K) = 2 ^l |A|D, where A simple proof of this formula for f(K) is given, which uses the basic properties of quartic Gauss sums.     

A Sequence to Compute the Brauer Group of Certain Quasi-Triangular Hopf Algebras

A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra B in a braided monoidal category ${{\mathcal C}}$ , and under certain assumptions on the braiding (fulfilled if ${{\mathcal C}}$ is symmetric), we construct a sequence for the Brauer group ${{\rm{BM}}}({{\mathcal C}};B)$ of B-module algebras, generalizing Beattie’s one. It allows one to prove that ${{\rm{BM}}}({{\mathcal C}};B) \cong {{\rm{Br}}}({{\mathcal C}}) \times {\operatorname{Gal}}({{\mathcal C}};B)$ , where ${{\rm{Br}}}({{\mathcal C}})$ is the Brauer group of ${{\mathcal C}}$ and ${\operatorname{Gal}}({{\mathcal C}};B)$ the group of B-Galois objects. We also show that ${{\rm{BM}}}({{\mathcal C}};B)$ contains a subgroup isomorphic to ${{\rm{Br}}}({{\mathcal C}}) \times {\operatorname{H^2}}({{\mathcal C}};B,I),$ where ${\operatorname{H^2}}({{\mathcal C}};B,I)$ is the second Sweedler cohomology group of B with values in the unit object I of ${{\mathcal C}}$ . These results are applied to the Brauer group ${{\rm{BM}}}(K,B \times H,{{\mathcal R}})$ of a quasi-triangular Hopf algebra that is a Radford biproduct B × H, where H is a usual Hopf algebra over a field K, the Hopf subalgebra generated by the quasi-triangular structure ${{\mathcal R}}$ is contained in H and B is a Hopf algebra in the category ${}_H{{\mathcal M}}$ of left H-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that ${{\rm{BM}}}(K,H,{{\mathcal R}}) \times {\operatorname{H^2}}({}_H{{\mathcal M}};B,K)$ is a subgroup of ${{\rm{BM}}}(K,B \times H,{{\mathcal R}})$ , confirming the suspicion that a certain cohomology group of B × H (second lazy cohomology group was conjectured) embeds into it. New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.     

辫子Hopf代数的积分(英)

本文引进了无限维辫子Hopf代数$H$的忠实拟对偶$H^d$和严格拟对偶$H^{d'}$.证明了每个严格拟对偶$H^{d'}$是一个$H$-Hopf 模. 发现了$H^{d}$的极大有理$H^{d}$-子模$H^{d {\rm rat} }$ 与积分的关系, 即: $H^{d {\rm rat}}\cong \int ^l_{H^d} \otimes H$.给出了在Yetter-Drinfeld范畴$(^B_B{\cal YD},C)$中的辫子Hopf代数的积分的存在性和唯一性.     

Localizations of Transfors

Let be n-dimensional teisi, i.e., higher-dimensional Gray-categorical structures. The following questions can be asked. Does a left q-transfor , i.e., a functor 2 _q , induce a right q-transfor , i.e., a functor More generally, does a functor induce a functor For k-arrows c and whose (k – 1)-sources and targets agree, does a q-transfor induce a q-transfor , for appropriate k-arrows For k-arrows c and whose (k – 1)-sources and targets agree, does a q-transfor induce a (q + k + 1)-transfor , for appropriate k-arrows I give answers to these questions in the cases where n-dimensional teisi and their tensor product have been defined, i.e., for n 3, and for n up to 5 in some cases that do not need all data and axioms of n-dimensional teisi.I apply the above to compositions in teisi, in particular to braidings and syllepses. One of the results is that a braiding on a monoidal 2-category induces a pseudo-natural transformation , where is the reverse of ? –, which is almost, but not quite, equal to – ?. However, in higher dimensions need not be reversible, so a braiding on a higher-dimensional tas can not be seen as a transfor A B B A.     

Geometric constant term functor(s)

We study the Eisenstein series and constant term functors in the framework of geometric theory of automorphic functions. Our main result says that for a parabolic \(P\subset G\) with Levi quotient M, the !-constant term functor

$$\begin{aligned}{\text {CT}}_!:{\text {D-mod}}({\text {Bun}}_G)\rightarrow {\text {D-mod}}({\text {Bun}}_M)\end{aligned}$$

is canonically isomorphic to the *-constant term functor

$$\begin{aligned} {\text {CT}}^-_*:{\text {D-mod}}({\text {Bun}}_G)\rightarrow {\text {D-mod}}({\text {Bun}}_M), \end{aligned}$$

taken with respect to the opposite parabolic \(P^-\).

     

Two-sided Two-cosided Hopf Modules and Yetter–Drinfeld Modules for Quasi-Hopf Algebras

For a quasi-Hopf algebra H, an H-bicomodule algebra and an H-bimodule coalgebra C we will show that the category of two-sided two-cosided Hopf modules is equivalent to the category of right–left generalized Yetter–Drinfeld modules . Using alternative versions of this result we will recover the category isomorphism between the categories of left–left and left–right Yetter–Drinfeld modules over a quasi-Hopf algebra.      

Determinantal Inequalities for Accretive-Dissipative Matrices

A matrix is said to be accretive-dissipative if, in its Hermitian decomposition , both matrices B and C are positive definite. Further, if B= I _n, then A is called a Buckley matrix. The following extension of the classical Fischer inequality for Hermitian positive-definite matrices is proved. Let be an accretive-dissipative matrix, k and l be the orders of A ₁₁ and A ₂₂, respectively, and let m = min{k,l}. Then For Buckley matrices, the stronger bound is obtained. Bibliography: 5 titles.