Generic $\mathfrak{gl}_{2}$-foams, web and arc algebras Title: Generic $\mathfrak{gl}_{2}$-foams, web and arc algebras Authors: Michael Ehrig, Catharina Stroppel and Daniel Tubbenhauer Status: Preprint. Last update: Sun, 7 May 2017 17:12:35 GMT ArXiv link: http://arxiv.org/abs/1601.08010 LaTex Beamer presentation: Slides1, Slides2 Abstract We define parameter dependent $\mathfrak{gl}_{2}$-foams and their associated web and arc algebras and verify that they specialize to several known $\mathfrak{sl}_{2}$ or $\mathfrak{gl}_{2}$ constructions related to higher link and tangle invariants. Moreover, we show that all these specializations are equivalent, and we deduce several applications, e.g. for the associated link and tangle invariants, and their functoriality.   A few extra words Let $\mathtt{Q}$ be a (certain) set of generic parameters. In this paper we introduce a $\mathtt{Q}$-version of singular topological quantum field theories (TQFTs) which we use to define a certain $\mathtt{Q}$-parameter foam $2$-category $\mathfrak{F}[\mathtt{Q}]$, that is a certain $2$-category of topological origin. We obtain from $\mathfrak{F}[\mathtt{Q}]$ several specializations. Among the specializations of this parameter version one can find the main foam $2$-categories studied in the context of higher link and tangle invariants: Khovanov/Bar-Natan's cobordisms Caprau's “foams” Clark-Morrison-Walker's disoriented cobordisms Blanchet's foams A relation that holds for these parameter foams is the following: We also study the web algebra $\mathfrak{W}[\mathtt{Q}]$ corresponding to $\mathfrak{F}[\mathtt{Q}]$, i.e. an algebra which has an associated $2$-category of certain bimodules giving a (fully) faithful $2$-representation of $\mathfrak{F}[\mathtt{Q}]$. Similarly, we study also an algebraic model $\mathfrak{A}[\mathtt{Q}]$ called arc algebra. Our main result is, surprisingly, that any two specializations (under some conditions) are isomorphic/equivalent (our results are even stronger since everything is explicit). Similarly for the associated $2$-categories of bimodules and foams. This includes the four examples from above. We discuss several applications of our explicit isomorphisms/equivalences. For example, we can show the higher tangle invariants constructed from the various $2$-categories are the same (they get identified by the above equivalence) and not just the associated link homologies. NEWS I am still a fool. My paper got accepted. The arXiv version of this paper was updated. My paper got accepted. The arXiv version of this paper was updated. "There are two ways to do mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else - but persistent." - based on a quotation from Raoul Bott. Upcoming event where you can meet me: Visit Faro Click
 Last update: 20.01.2018 or later · email