We represent a classical Maxwell-Bloch equation and related to it positive part of the AKNS hierarchy in geometrical terms. The Maxwell-Bloch evolution is given by an infinitesimal action of a nilpotent subalgebra n+ of affine Lie algebra ŝl2 on a Maxwell-Bloch phase space treated as a homogeneous space of n+. A space of local integrals of motion is described using cohomology methods. We show t...

This paper is a review of results on generalized Harish-Chandra modules in the framework of cohomological induction. The main results, obtained during the last 10 years, concern the structure of the fundamental series of (g, k)−modules, where g is a semisimple Lie algebra and k is an arbitrary algebraic reductive in g subalgebra. These results lead to a classification of simple (g, k)−modules o...

The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of a free group factor is contained in unique maximal subalgebra. This motivated by related results Popa's deformation/rigidity theory and Peterson-Thom's on L^{2}-Betti numbers. We present an approach to this terms so-called strong convergence random matrices formulating which natural generalization the Haagerup-Thorbjo...

We introduce what we call the principal subalgebra of a lattice vertex (super) algebra associated to an arbitrary Z-basis of the lattice. In the first part (to appear), the second author considered the case of positive bases and found a description of the principal subalgebra in terms of generators and relations. Here, in the most general case, we obtain a combinatorial basis of the principal s...

In this paper we study subalgebras A of the algebra D(X) of differential operators on a smooth variety X which are big in the following sense: using the order of a differential operator, the ring D(X) is equipped with a filtration. Its associated graded algebra D(X) is commutative and can be regarded as the set of regular functions on the cotangent bundle ofX . The subalgebra A inherits a filtr...

Every lown Boolean algebra, for 1 ≤ n ≤ 4, is isomorphic to a computable Boolean algebra. It is not yet known whether the same is true for n > 4. However, it is known that there exists a low5 subalgebra of the computable atomless Boolean algebra which, when viewed as a relation on the computable atomless Boolean algebra, does not have a computable copy. We adapt the proof of this recent result ...

Let S(0,1) be the *-algebra of all classes Lebesgue measurable functions on unit interval (0,1) and let a complete symmetric Δ-normed *-subalgebra S(0,1), in which simple are dense, e.g., L∞(0,1), Llog(0,1), Arens algebra Lω(0,1) equipped with their natural Δ-norms. We show that there exists no non-trivial derivation commuting dyadic translations interval. type II (or I∞) von Neumann algebra, a...

We show that, up to Morita equivalence, any standardly stratified algebra, admits an exact Borel subalgebra. In fact, we this in the more general case of finite-dimensional algebras possessing admissible homological system. This generalizes a theorem by Koenig, Külshammer, and Ovsienko, which holds for quasi-hereditary algebras. Our proof follows same scheme proposed these authors, context: ass...

If every finitely generated subalgebra of an algebra A has a property P , then A is said to have P as a local property. In this paper we study classes L(K) and M(K), where K is a class of unary algebras, L(K) consists of all algebras such that every finitely generated subalgebra is in K, and M(K) is the class of all algebras with every monogenic subalgebra in K. In particular, the cases where K...

A new twisted deformation, Uz(so(4, 2)), of the conformal algebra of the (3+1)-dimensional Minkowskian spacetime is presented. This construction is provided by a classical r-matrix spanned by ten Weyl–Poincaré generators, which generalizes non-standard quantum deformations previously obtained for so(2, 2) and so(3, 2). However, by introducing a conformal nullplane basis it is found that the twi...