Sofic equivalence relations

We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes' Embedding Conjecture as well as the Measurable Determinant Conjecture of Lück, Sauer and Wegner hold for treeable equivalence relations.     

广义仿射李代数上的非交换Poisson代数结构

非交换的Poisson代数同时具有(未必交换的)结合代数和李代数两种代数结构,且结合代数和李代数之间满足所谓的Leibniz法则.本文确定了一般广义仿射李代数上所有的Poisson代数结构.     

Rieffel deformation of homogeneous spaces

Let G₁⊂G be a closed subgroup of a locally compact group G and let X=G/G₁ be the quotient space of left cosets. Let X=(C₀(X),ΔX) be the corresponding G-C^∗-algebra where G=(C₀(G),Δ). Suppose that Γ is a closed abelian subgroup of G₁ and let Ψ be a 2-cocycle on the dual group . Let GΨ be the Rieffel deformation of G. Using the results of the previous paper of the author we may construct GΨ-C^∗-algebra XΨ - the Rieffel deformation of X. On the other hand we may perform the Rieffel deformation of the subgroup G₁ obtaining the closed quantum subgroup , which in turn, by the results of S. Vaes, leads to the GΨ-C^∗-algebra . In this paper we show that . We also consider the case where Γ⊂G is not a subgroup of G₁, for which we cannot construct the subgroup . Then generically XΨ cannot be identified with a quantum quotient. What may be shown is that it is a GΨ-simple object in the category of GΨ-C^∗-algebras.     

On sofic actions and equivalence relations

The notion of sofic equivalence relation was introduced by Gabor Elek and Gabor Lippner. Their technics employ some graph theory. Here we define this notion in a more operator algebraic context, starting from Connes? Embedding Problem, and prove the equivalence of these two definitions. We introduce a notion of sofic action for an arbitrary group and prove that an amalgamated product of sofic actions over amenable groups is again sofic. We also prove that an amalgamated product of sofic groups over an amenable subgroup is again sofic.     

Structures of the twisted N = 1 Schrödinger–Neveu–Schwarz algebra

In this paper, we determine the universal central extension, derivation algebra and automorphism group of the twisted N = 1 Schrödinger–Neveu–Schwarz algebra. Furthermore, we generalize these results to the generalized twisted N = 1 Schrödinger–Neveu–Schwarz algebra in the final section.     

Composition algebras and outer automorphisms of algebraic groups via triality

In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form n over a field k of characteristic not two, and a category arising from an action of the projective similarity group of n on certain pairs of automorphisms of the group scheme PGO⁺(n) defined over k. This extends results recently obtained in the same direction for symmetric composition algebras. We also derive known results on composition algebras from our equivalence.     

Derived equivalences between triangular matrix algebras

In this paper, we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras and describe methods to construct tilting modules and tilting complexes inducing derived equivalences between them.     

A note on Bridgeland Hall algebras

In this note, let 𝒜 be a finitary hereditary abelian category with enough projectives. By using the associativity formula of Hall algebras, we give a new proof of the main theorem in [17 Yanagida, S. (2016). A note on Bridgeland’s Hall algebra of two-periodic complexes. Math. Z. 282(3):973–991.[Crossref], [Web of Science ®] , [Google Scholar]], which states that the Bridgeland Hall algebra of 2-cyclic complexes of projective objects in 𝒜 is isomorphic to the Drinfeld double Hall algebra of 𝒜.