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Topological quantum D-branes and wild embeddings from exotic smooth R^4  [PDF]
T. Asselmeyer-Maluga,J. Krol
Mathematics , 2011, DOI: 10.1142/S0217751X11053857
Abstract: This is the next step of uncovering the relation between string theory and exotic smooth R^4. Exotic smoothness of R^4 is correlated with D6 brane charges in IIA string theory. We construct wild embeddings of spheres and relate them to a class of topological quantum Dp-branes as well to KK theory. These branes emerge when there are non-trivial NS-NS H-fluxes where the topological classes are determined by wild embeddings S^2 -> S^3. Then wild embeddings of higher dimensional $p$-complexes into S^n correspond to Dp-branes. These wild embeddings as constructed by using gropes are basic objects to understand exotic smoothness as well Casson handles. Next we build C*-algebras corresponding to the embeddings. Finally we consider topological quantum D-branes as those which emerge from wild embeddings in question. We construct an action for these quantum D-branes and show that the classical limit agrees with the Born-Infeld action such that flat branes = usual embeddings.
Classical derived functors as fully faithful embeddings  [PDF]
Pedro Nicolas,Manuel Saorin
Mathematics , 2014,
Abstract: Given associative unital algebras $A$ and $B$ and a complex $T^\bullet$ of $B-A-$bi\-modules, we give necessary and sufficient conditions for the total derived functors, $\Rh_A(T^\bullet,?):\D(A)\longrightarrow\D(B)$ and $?\Lt_BT^\bullet:\D(B)\longrightarrow\D(A)$, to be fully faithful. We also give criteria for these functors to be one of the fully faithful functors appearing in a recollement of derived categories. In the case when $T^\bullet$ is just a $B-A-$bimodule, we connect the results with (infinite dimensional) tilting theory and show that some open question on the fully faithfulness of $\Rh_A(T,?)$ is related to the classical Wakamatsu tilting problem.
Quantum Geometry and Wild embeddings as quantum states  [PDF]
T. Asselmeyer-Maluga,J. Krol
Physics , 2012, DOI: 10.1142/S0219887813500552
Abstract: In this paper we discuss wild embeddings like Alexanders horned ball and relate them to fractal spaces. We build a $C^{\star}$-algebra corresponding to a wild embedding. We argue that a wild embedding is the result of a quantization process applied to a tame embedding. Therefore quantum states are directly the wild embeddings. Then we give an example of a wild embedding in the 4-dimensional spacetime. We discuss the consequences for cosmology.
Several adjoint constructions for biset functors via Mackey-functorial interpretation  [PDF]
Hiroyuki Nakaoka
Mathematics , 2014,
Abstract: We consider analogs of Jacobson's $F$-Burnside construction and Boltje's $(-)_+$-construction for biset functors, using Mackey-functor theoretic interpretation of biset functors.
Poincaré inequalities, embeddings, and wild groups  [PDF]
Assaf Naor,Lior Silberman
Mathematics , 2010, DOI: 10.1112/S0010437X11005343
Abstract: We present geometric conditions on a metric space $(Y,d_Y)$ ensuring that almost surely, any isometric action on $Y$ by Gromov's expander-based random group has a common fixed point. These geometric conditions involve uniform convexity and the validity of nonlinear Poincar\'e inequalities, and they are stable under natural operations such as scaling, Gromov-Hausdorff limits, and Cartesian products. We use methods from metric embedding theory to establish the validity of these conditions for a variety of classes of metric spaces, thus establishing new fixed point results for actions of Gromov's "wild groups".
Submodule Categories of Wild Representation Type  [PDF]
Claus Michael Ringel,Markus Schmidmeier
Mathematics , 2004,
Abstract: Let $\Lambda$ be a commutative local uniserial ring of length at least seven with radical factor ring $k$. We consider the category $S(\Lambda)$ of all possible embeddings of submodules of finitely generated $\Lambda$-modules and show that $S(\Lambda)$ is controlled $k$-wild with a single control object $I\in S(\Lambda)$. In particular, it follows that each finite dimensional $k$-algebra can be realized as a quotient $\End(X)/\End(X)_I$ of the endomorphism ring of some object $X\in S(\Lambda)$ modulo the ideal $\End(X)_I$ of all maps which factor through a finite direct sum of copies of $I$.
Embeddings of representations  [PDF]
Kathrin Kerkmann,Markus Reineke
Mathematics , 2014,
Abstract: We derive "numerical" criteria for the existence of embeddings of representations of finite dimensional algebras.
Categorification of a linear algebra identity and factorization of Serre functors  [PDF]
Sefi Ladkani
Mathematics , 2009,
Abstract: We provide a categorical interpretation of a well-known identity from linear algebra as an isomorphism of certain functors between triangulated categories arising from finite dimensional algebras. As a consequence, we deduce that the Serre functor of a finite dimensional triangular algebra A has always a lift, up to shift, to a product of suitably defined reflection functors in the category of perfect complexes over the trivial extension algebra of A.
A Mackey-functor theoretic interpretation of biset functors  [PDF]
Hiroyuki Nakaoka
Mathematics , 2013,
Abstract: In this article, we consider a formulation of biset functors using the 2-category of finite sets with variable finite group actions. We introduce a 2-category $\mathbb{S}$, on which a biset functor can be regarded as a special kind of Mackey functors. This gives an analog of Dress' definition of a Mackey functor, in the context of biset functors.
Product preserving functors of infinite dimensional manifolds  [PDF]
Andreas Kriegl,Peter W. Michor
Mathematics , 1996,
Abstract: The theory of product preserving functors and Weil functors is partly extended to infinite dimensional manifolds, using the theory of $C^\infty$-algebras.
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