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Register a new userGlobalization for geometric partial comodules
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We discuss globalization for geometric partial comodules in a monoidal category with pushouts and we provide a concrete procedure to construct it, whenever it exists. The mild assumptions required by our approach make it possible to apply it in a number of contexts of interests, recovering and extending numerous ad hoc globalization constructions from the literature in some cases and providing obstruction for globalization in some other cases.
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The aim of this paper is to prove the statement in the title. As a by-product, we obtain new globalization results in cases never considered before, such as partial corepresentations of Hopf algebras. Moreover, we show that for partial representations of groups and Hopf algebras, our globalization coincides with those described earlier in literature. Finally, we introduce Hopf partial comodules over a bialgebra as geometric partial comodules in the monoidal category of (global) modules. By applying our globalization theorem we obtain an analogue of the fundamental theorem for Hopf modules in this partial setting.
In partial action theory, a pertinent question is whenever given a partial (co)action of a Hopf algebra A on an algebra R, it is possible to construct an enveloping (co)action. The authors Alves and Batista, in [2],have shown that this is always possible if R has unit. We are interested in investigating the situation where both algebras A and R are nonunitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [11], which is called multiplier Hopf algebra. Therefore, we will consider partial (co)actions of multipliers Hopf algebras on algebras not necessarily unitary and we will present globalization theorems for these structures. Moreover, Dockuchaev, Del Rio and Simon, in [5], have shown when group partial actions on nonunitary algebras are globalizable. Based in [5], we will establish a bijection between group partial actions on an algebra R not necessarily unitary and partial actions of a multiplier Hopf algebra on the algebra R.
We study the globalization of partial actions on sets and topological spaces and of partial coactions on algebras by applying the general theory of globalization for geometric partial comodules, as previously developed by the authors. We show that this approach does not only allow to recover all known results in these settings, but it allows to treat new cases of interest, too.
Let $A$ and $B$ be two point sets in the plane of sizes $r$ and $n$ respectively (assume $r leq n$), and let $k$ be a parameter. A matching between $A$ and $B$ is a family of pairs in $A times B$ so that any point of $A cup B$ appears in at most one pair. Given two positive integers $p$ and $q$, we define the cost of matching $M$ to be $c(M) = sum_{(a, b) in M}|{a-b}|_p^q$ where $|{cdot}|_p$ is the $L_p$-norm. The geometric partial matching problem asks to find the minimum-cost size-$k$ matching between $A$ and $B$. We present efficient algorithms for geometric partial matching problem that work for any powers of $L_p$-norm matching objective: An exact algorithm that runs in $O((n + k^2) {mathop{mathrm{polylog}}} n)$ time, and a $(1 + varepsilon)$-approximation algorithm that runs in $O((n + ksqrt{k}) {mathop{mathrm{polylog}}} n cdot logvarepsilon^{-1})$ time. Both algorithms are based on the primal-dual flow augmentation scheme; the main improvements involve using dynamic data structures to achieve efficient flow augmentations. With similar techniques, we give an exact algorithm for the planar transportation problem running in $O(min{n^2, rn^{3/2}} {mathop{mathrm{polylog}}} n)$ time.
Let $det_2(A)$ be the block-wise determinant (partial determinant). We consider the condition for completing the determinant $det(det_2(A)) = det(A),$ and characterize the case for an arbitrary Kronecker product $A$ of matrices over an arbitrary field. Further insisting that $det_2(AB)=det_2(A)det_2(B)$, for Kronecker products $A$ and $B$, yields a multiplicative monoid of matrices. This leads to a determinant-root operation $text{Det}$ which satisfies $text{Det}(text{Det}_2(A)) = text{Det}(A)$ when $A$ is a Kronecker product of matrices for which $text{Det}$ is defined.
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