Title Upotpunjeni Hopfovi algebroidi Title (english) Completed Hopf algebroids Author Martina Stojić Mentor Zoran Škoda (mentor)Committee member Dražen Adamović (predsjednik povjerenstva)Committee member Zoran Škoda (član povjerenstva)Committee member Tomislav Šikić (član povjerenstva)Granter University of Zagreb Defense date and country 2017-10-20, Croatia Scientific / art field, NATURAL SCIENCES Universal decimal classificationUDC ) 51 - Mathematics Abstract U ovoj disertaciji uvodimo prirodno poopćenje pojma Hopfovog algebroida unutar simetrične monoidalne kategorije s koujednačiteljima koji komutiraju s monoidalnim produktom. U radu također konstruiramo simetričnu monoidalnu kategoriju (indproVect,\(\tilde{\otimes} ,k)\) filtrirano-kofiltriranih vektorskih prostora, čiji morfizmi su linearna preslikavanja koja u slabom smislu poštuju filtracije i kofiltracije, a monoidalni produkt je obični tenzorski produkt vektorskih prostora formalno upotpunjen i s odgovarajućom filtracijom kofiltracija. Za tu kategoriju u radu dokazujemo da zadovoljava gore navedene uvjete za postojanje unutarnjeg Hopfovog algebroida. Ona kao potkategorije sadrži kategoriju (indVect,\(\otimes ,k)\) filtriranih vektorskih prostora i njoj dualnu kategoriju (proVect,\(\hat{\otimes} ,k)\) kofiltriranih vektorskih prostora. Njen monoidalni produkt objedinjuje obični tenzorski produkt i upotpunjeni tenzorski produkt. Jednu od važnijih klasa običnih Hopfovih algebroida nad nekomutativnom bazom čine poludirektni produkti \(H \sharp A\) Hopfove algebre \(H\) i pleteničasto-komutativne algebre \(A\) u kategoriji Yetter-Drinfeldovih modula nad \(H\). Tu klasu Hopfovih algebroida zovemo skalarnim proširenjima. U ovom radu dokazujemo da poludirektni produkti u kojima su Hopfova algebra \(H\) i Yetter-Drinfeldova modulna algebra \(A\) zamijenjene svojim unutarnjim analogonima u kategoriji indproVect imaju strukturu Hopfovih algebroida u toj monoidalnoj kategoriji. Time među ostalim postavljamo temelj za proučavanje poopćenja Heisenbergovih udvojenja \(A^\ast \sharp A\), ona u kojima je \(A\) beskonačno-dimenzionalna Hopfova algebra umjesto konačno-dimenzionalna, te postojanje strukture poopćenog Hopfovog algebroida na njima. U disertaciji su zatim proučavana Hopfova sparivanja filtriranih Hopfovih algebri \(A\) i kofiltriranih Hopfovih algebri \(H\) koja su nedegenerirana u \(H\), te su nađeni dovoljni uvjeti uz koje \(A\) postaje pleteničasto-komutativna Yetter-Drinfeldova modulna algebra nad \(H\) u kategoriji indproVect. Proučavana je također manja klasa primjera, u kojima je \(A\) Hopfova algebra filtrirana konačno-dimenzionalnim komponentama. Tu su nađeni nužni i dovoljni uvjeti na Hopfove algebre \(A\) i \(A^\ast\), odnosno \(A\) i \(H\), u vidu konačne dimenzionalnosti adjungiranih orbita od \(A\) te postojanja određenih kanonskih elemenata unutar \(H \sharp A\). Time je dakle inducirana konstrukcija nekih filtrirano-kofiltriranih Hopfovih algebroida tipa skalarnog proširenja. Važni primjeri takvih skalarnih proširenja su oni u kojima je \(A\) univerzalna omotačka algebra \(U(\mathfrak{g})\) konacno-dimenzionalne Liejeve algebre \(\mathfrak{g}\). Ako je \(H\) njen algebarski dual \(U(\mathfrak{g})^\ast\) s induciranom kofiltracijom, pripadno skalarno proširenje, koje je poopćeno Heisenbergovo udvojenje od \(U(\mathfrak{g})\), može se identificirati s algebrom diferencijalnih operatora na formalnoj okolini jedinice pripadne Liejeve grupe što sugerira neke od primjena u geometriji i matematičkoj fizici.
Abstract (english) In this thesis, a natural generalization of the definition of a Hopf algebroid is introduced, internal to any symmetric monoidal category with coequalizers that commute with the monoidal product. In this thesis we also construct a symmetric monoidal category (indproVect,\(\tilde{\otimes} ,k)\) of filtered-cofiltered vector spaces, whose morphisms are linear maps which in a weak sense respect the filtrations and cofiltrations, and whose monoidal product is the usual tensor product of vector spaces formally completed and with a corresponding filtration of cofiltrations. We prove that this category satisfies the above conditions for the existence of internal Hopf algebroids. It contains two dual subcategories, the category (indVect,\(\otimes ,k)\) of filtered vector spaces and the category (proVect,\(\hat{\otimes} ,k)\) of cofiltered vector spaces. The monoidal product in it combines the ordinary tensor product and a completed tensor product. An important class of Hopf algebroids over a noncommutative base is comprised of smash products of a Hopf algebra \(H\) and a braided-commutative algebra \(A\) in the category of YetterDrinfeld modules over \(H\). Such Hopf algebroids are called scalar extensions. In this thesis, we prove that the smash products in which \(H\) and \(A\) are replaced by their analogues in the monoidal category of filtered-cofiltered vector spaces have the structure of Hopf algebroids in that monoidal category. By doing this, we set the basis for studying the Heisenberg doubles \(A^\ast \sharp A\) in which \(A\) is an infinite-dimensional Hopf algebra instead of a finite-dimensional one, among other examples, and the existence of the Hopf algebroid structure on them internal to the category indproVect. We then study Hopf pairings of a filtered Hopf algebra \(A\) and a cofiltered Hopf algebra \(H\) which are non-degenerate in the variable in \(H\), and find sufficient conditions for \(A\) to be a braided-commutative Yetter-Drinfeld module algebra over \(H\) in the indproVect category. A smaller class of examples is also studied, for which \(A\) is a Hopf algebra countably filtered by finite-dimensional vector spaces. Here we find necessary and sufficient conditions on Hopf algebras \(A\) and \(A^\ast\), or \(A\) and \(H\), in the form of finite dimensionality of the adjoint orbits of \(A\) and the existence of certain canonical elements in \(H \sharp A\). Thus a construction of some filtered-cofiltered Hopf algebroids of scalar extension type is obtained. Important examples of such scalar extensions are the ones with \(A\) the universal enveloping algebra \(U(\mathfrak{g})\) of a finite-dimensional Lie algebra \(\mathfrak{g}\). When \(H\) is equal to its algebraic dual \(U(\mathfrak{g})^\ast\) with induced cofiltration, the corresponding scalar extension, that is the Heisenberg double of \(U(\mathfrak{g})\), can be identified as an algebra with the algebra of differential operators on the formal neighborhood of the unit of a Lie group integrating \(\mathfrak{g}\), suggesting applications in geometry and mathematical physics.
Keywords
Hopfov algebroid
formalno upotpunjenje
filtrirani vektorski prostor
kofiltrirani vektorski prostor
filtrirano-kofiltrirani vektorski prostor
strogi ind-objekt
strogi pro-objekt
strogi ind-pro-objekt
formalna suma
formalna baza
univerzalna omotačka algebra
upotpunjeni tenzorski produkt
unutarnji bialgebroid
dualne Hopfove algebre
Yetter-Drinfeldov modul
skalarno proširenje
Heisenbergovo udvojenje
Keywords (english)
Hopf algebroid
formal completion
filtered vector space
cofiltered vector space
filtered-cofiltered vector space
strict ind-object
strict pro-object
strict ind-pro-object
formal sum
formal basis
universal enveloping algebra
completed tensor product
internal bialgebroid
dual Hopf algebra
Yetter-Drinfeld module
scalar extension
Heisenberg double
Language croatian URN:NBN urn:nbn:hr:217:875774 Study programme Title: Mathematics Type of resource Text Extent ix, 306 str. File origin Born digital Access conditions Open access Terms of use Created on 2019-03-21 13:30:18
