相关论文: Crystals, quiver varieties and coboundary categori…
Commuting pairs of algebraic structures on a set have been studied by several authors and may be described equivalently as algebras for the tensor product of Lawvere theories, or more basically as certain bifunctors that here we call bifold…
We construct commutative algebra spectra that represent the operator $K$-theory of $C^*$-algebras, which are algebras over the commutative ring spectra that represent topological $K$-theory. The spectral multiplicative structure introduces…
Over-extended Kac-Moody algebras contain so-called gradient structures - a gl(d)-covariant level decomposition of the algebra contains strings of modules at different levels that can be interpreted as spatial gradients. We present an…
A Littelmann path model is constructed for crystals pertaining to a not necessarily symmetrizable Borcherds-Cartan matrix. Here one must overcome several combinatorial problems coming from the imaginary simple roots. The main results are an…
Given a braided pivotal category $\mathcal C$ and a pivotal module tensor category $\mathcal M$, we define a functor $\mathrm{Tr}_{\mathcal C}:\mathcal M \to \mathcal C$, called the associated categorified trace. By a result of…
We prove for $n\geq c-1$ that the functor taking an animated ring $R$ to its mod $(p^c,v_1^{p^n})$ syntomic cohomology factors through the functor $R \mapsto R/p^{c(n+2)}$, a phenomenon we term crystallinity for mod $(p^c,v_1^{p^n})$…
A duality is discussed for Lie group bundles vs. certain tensor categories with non-simple identity, in the setting of Nistor-Troitsky gauge-equivariant K-theory. As an application, we study C*-algebra bundles with fibre a fixed-point…
Fix a semisimple Lie algebra g. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for g-representations. These algebras depend on a parameter which is a point in the Deligne-Mumford moduli space of marked…
Multiplicative Unitaries are described in terms of a pair of commuting shifts of relative depth two. They can be generated from ambidextrous Hilbert spaces in a tensor C*-category. The algebraic analogue of the Takesaki-Tatsuuma Duality…
We consider the space of $n$-tuples of pairwise commuting elements in the Lie algebra of $U(m)$. We relate its one-point compactification to the subquotients of certain rank filtrations of connective complex $K$-theory. We also describe the…
We define commutants mod normed ideals associated with compact smooth manifolds with boundary. The results about the K-theory of these operator algebras include an exact sequence for the connected sum of manifolds, derived from the…
We introduce a discrete cobordism category for nested manifolds and nested cobordisms between them. A variation of stratified Morse theory applies in this case, and yields generators for a general nested cobordism category. Restricting to a…
Cartesian differential categories come equipped with a differential combinator that formalizes the derivative from multi-variable differential calculus, and also provide the categorical semantics of the differential $\lambda$-calculus. An…
We give a new interpretation and proof of the dilogarithm identities, associated to the affine Kac-Moody algebra sl(2)^, using the path description of the corresponding crystal basis. We also discuss connections with algebraic K-theory.
A representation of $\mathfrak{gl}(V)=V \otimes V^*$ is a linear map $\mu \colon \mathfrak{gl}(V) \otimes M \to M$ satisfying a certain identity. By currying, giving a linear map $\mu$ is equivalent to giving a linear map $a \colon V…
We provide a complete classification of the class of unital graph $C^*$-algebras - prominently containing the full family of Cuntz-Krieger algebras - showing that Morita equivalence in this case is determined by ordered, filtered…
We give a combinatorial model structure to the category of, not necessarily conilpotent, differential graded (dg) cocommutative coalgebras and an $\infty$-category structure to the category of curved Lie algebras over an algebraically…
A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant $\mathcal{C}'$ of a fully faithful representation $\mathcal{C}\to\operatorname{Bim}(R)$ of a…
For a commutative ring $A$, we have the category of (bounded-below) chain complexes of $A$-modules $Ch_{+}(A\mymod)$, a closed symmetric monoidal category with a compatible stable Quillen model structure. The associated homotopy category is…
We produce braided commutative algebras in braided monoidal categories by generalizing Davydov's full center construction of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative…