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...

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...

Let Uq(sl2) be the quantized enveloping algebra associated to the simple Lie algebra sl2. In this paper, we study the quantum double Dq of the Borel subalgebra Uq((sl2) ) of Uq(sl2). We construct an analogue of Kostant–Lusztig Z[v, v]-form for Dq and show that it is a Hopf subalgebra. We prove that, over an algebraically closed field, every simple Dq-module is the pullback of a simple Uq(sl2)-m...

If g is a real reductive Lie algebra and h ⊂ g is a subalgebra, then (g, h) is called real spherical provided that g = h + p for some choice of a minimal parabolic subalgebra p ⊂ g. This paper concludes the classification of real spherical pairs (g, h), where h is a reductive real algebraic subalgebra. A preceding paper treated the case where g is simple. The present work builds on that case an...

We study real Smirnov functions and investigate a certain ∗-closed subalgebra of the Smirnov class N+ containing them. Motivated by a result of Aleksandrov, we provide an explicit representation for the space Hp ∩ Hp. This leads to a natural analog of the Riesz projection on a certain quotient space of Lp for p ∈ (0, 1). We also study a Herglotz-like integral transform for singular measures on ...

Let (W, I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [BK] constructed a monoid structure on the set of all subsets of I using unipotent χ-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxe...

We consider an inductive scheme for quantized enveloping algebras, arising from certain inclusions of the associated root data. These inclusions determine an algebra-subalgebra pair with the subalgebra also a quantized enveloping algebra, and we want to understand the structure of the “difference” between the algebra and the subalgebra. Our point of view treats the background field and quantiza...

The resulting algebra is known as the Hall algebra HA of A. Hall algebras first appeared in the work of Steinitz [S] and Hall [H] in the case where A is the category of Abelian p-groups. They reemerged in the work of Ringel [R1]-[R3], who showed in [R1] that when A is the category of quiver representations of an A-D-E quiver ~ Q over a finite field Fq, the Hall algebra of A provides a realizati...