Mathematics

We find a substantial class of pairs of ∗ -homomorphisms between graph C*-algebras of the form C ∗ (E)↪ C ∗ (G)↞ C ∗ (F) whose pullback C*-algebra is an AF graph C*-algebra. Our result can be interpreted as a recipe for determining the quantum space obtained by shrinking a quantum subspace. There is a variety of examples from noncommutative topology, such as quantum complex projective spaces (including the standard Podleś quantum sphere) or quantum teardrops, that instantiate the result. Furthermore, to go beyond AF graph C*-algebras, we consider extensions of graphs over sinks and prove an analogous theorem for the thus obtained graph C*-algebras.

In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder Σ , which gives rise to the quasi Poisson bracket of G.Massuyeau and V.Turaev on the group algebra k π 1 (Σ,p) of the fundamental group of a surface based at p∈∂Σ . This bracket also induces a noncommutative Goldman Poisson bracket on the cyclic space C ♮ , which is a k -linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative r -matrix formalism. This gives a more conceptual proof of the result of N. Ovenhouse that traces of powers of Lax matrix form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on C ♮ .

In this paper, we introduce the notion of a noncommutative Poisson bialgebra, and establish the equivalence between matched pairs, Manin triples and noncommutative Poisson bialgebras. Using quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the introduction of the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra. Rota-Baxter operators, more generally O-operators on noncommutative Poisson algebras, and noncommutative pre-Poisson algebras are introduced, by which we construct skew-symmetric solutions of the Poisson Yang-Baxter equation in some special noncommutative Poisson algebras obtained from these structures.

In this paper we propose a procedure for a noncommutative derived Poisson reduction, in the spirit of the Kontsevich-Rosenberg principle: "a noncommutative structure of some kind on A should give an analogous commutative structure on all schemes Rep n (A) ". We use double Poisson structures as noncommutative Poisson structures and noncommutative Hamiltonian spaces -- as first introduced by M. Van den Bergh -- to define (derived) zero loci of Hamiltonian actions and a noncommutative Chevalley-Eilenberg and BRST constructions, showing how we recover the corresponding commutative constructions using the representation functor. In a dedicated final short section we highlight how the categorical properties of the representation functor lead to the natural introduction of new interesting notions, such as noncommutative group schemes, group actions, or Poisson-group schemes, which could help to understand the previous results in a different light, and in future research generalise them into a broader, clearer correspondence between noncommutative and commutative equivariant geometry.

In our previous publications we have introduced analogs of partial derivatives on the algebras U(gl(N)). In the present paper we compare two methods of introducing these analogs: via the so-called quantum doubles and by means of a coalgebraic structure. In the case N=2 we extend the quantum partial derivatives from U(u(2)) (the compact form of the algebra U(gl(2))) on a bigger algebra, constructed in two steps. First, we define the derivatives on a central extension of this algebra, then we prolongate them on some elements of the corresponding skew-field by using the Cayley-Hamilton identities for certain matrices with noncommutative entries. Eventual applications of this differential calculus are discussed.

In this paper we shall show the equivalence of algebraic and analytic localisation for algebras of smooth deformation quantization for several situations. The proofs are based on old work by Whitney, Malgrange and Tougeron on the commutative algebra of smooth function rings from the 60's and 70's.

In analogy with classical submanifold theory, we introduce morphisms of real metric calculi together with noncommutative embeddings. We show that basic concepts, such as the second fundamental form and the Weingarten map, translate into the noncommutative setting and, in particular, we prove a noncommutative analogue of Gauss equations for the curvature of a submanifold. Moreover, the mean curvature of an embedding is readily introduced, giving a natural definition of a noncommutative minimal embedding, and we illustrate the novel concepts by considering the noncommutative torus as a minimal surface in the noncommutative 3-sphere.

We prove several results in the theory of fusion categories using the product (norm) and sum (trace) of Galois conjugates of formal codegrees. First, we prove that finitely-many fusion categories exist up to equivalence whose global dimension has a fixed norm. Furthermore, with two exceptions, all formal codegrees of spherical fusion categories with square-free norm are rational integers. This implies, with three exceptions, that every spherical braided fusion category whose global dimension has prime norm is pointed. The reason exceptions occur is related to the classical Schur-Siegel-Smyth problem of describing totally positive algebraic integers of small absolute trace.

We construct a supercategory that can be seen as a skew version of (thickened) KLR algebras for the type A quiver. We use our supercategory to construct homological invariants of tangles and show that for every link our invariant gives a link homology theory supercategorifying the Jones polynomial. Our homology is distinct from even Khovanov homology and we present evidence supporting the conjecture that it is isomorphic to odd Khovanov homology. We also show that cyclotomic quotients of our supercategory give supercategorifications of irreducible finite-dimensional representations of gl n of level 2.

We extend the covering of even and odd Khovanov link homology to tangles, using arc algebras. For this, we develop the theory of quasi-associative algebras and bimodules graded over a category with a 3-cocycle. Furthermore, we show that a covering version of a level 2 cyclotomic half 2-Kac--Moody algebra acts on the bicategory of quasi-associative bimodules over the covering arc algebras, relating our work to a construction of Vaz.