arXiv subjects
Publications and source records attributed to Boris Kadets.
We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II$_{1}$ factor as Fraïssé limits of suitable classes of structures. Moreover by means of Fraïssé theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.
In this paper we continue to study Belavin-Drinfeld cohomology introduced in arXiv:1303.4046 [math.QA] and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra. Here we compute Belavin-Drinfeld cohomology for all non-skewsymmetric $r$-matrices from the Belavin-Drinfeld list for simple Lie algebras of type $B$, $C$, and $D$.
The aim of this paper is to provide an overview of the results about classification of quantum groups that were obtained in arXiv:1303.4046 [math.QA] and arXiv:1502.00403 [math.QA].
In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra $\mathfrak{g}$. This problem reduces to the classification of all Lie bialgebra structures on $\mathfrak{g}(\mathbb{K})$, where $\mathbb{K}=\mathbb{C}((\hbar))$. The associated classical double is of the form $\mathfrak{g}(\mathbb{K})\otimes_{\mathbb{K}} A$, where $A$ is one of the following: $\mathbb{K}[ε]$, where $ε^{2}=0$, $\mathbb{K}\oplus \mathbb{K}$ or $\mathbb{K}[j]$ where $j^{2}=\hbar$. The first case relates to quasi-Frobenius Lie algebras. In the second and third cases we introduce a theory of Belavin-Drinfeld cohomology associated to any non-skewsymmetric $r$-matrix from the Belavin-Drinfeld list. We prove a one-to-one correspondence between gauge equivalence classes of Lie bialgebra structures on $\mathfrak{g}(\mathbb{K})$ and cohomology classes (in case II) and twisted cohomology classes (in case III) associated to any non-skewsymmetric $r$-matrix.