I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type

We introduce a notion of I-factorial quantum torsor, which consists of an integrable ergodic action of a locally compact quantum group on a type I-factor such that also the crossed product is a type I-factor. We show that any such I-factorial quantum torsor is at the same time a I-factorial quantum torsor for the dual locally compact quantum group, in such a way that the construction is involutive. As a motivating example, we show that quantized compact semisimple Lie groups, when amplified via a crossed product construction with the function algebra on the associated weight lattice, admit I-factorial quantum torsors, and give an explicit realization of the dual quantum torsor in terms of a deformed Heisenberg algebra for the Borel part of a quantized universal enveloping algebra.     

The Canonical Basis for the Quantum Group of Type B2

We use the Ringel-Hall algebra approach to study the canonical basis elements for the quantum group of type B₂ which are characterized in Xi [12]. However, our approach simplifies several computations there.     

Absolute Valued Strongly Power-Associative Triple Systems

Absolute valued triple system are the natural ternary extension of absolute valued algebras. In this article we classify strongly power-associative absolute valued triple systems.     

Enveloping Algebras of Solvable Malcev Algebras of Dimension Five

We study the universal enveloping algebras of the one-parameter family of solvable 5-dimensional non-Lie Malcev algebras. We explicitly determine the universal nonassociative enveloping algebras (in the sense of Pérez-Izquierdo and Shestakov) and the centers of the universal enveloping algebras. We also determine the universal alternative enveloping algebras.     

Relative matched pairs of finite groups from depth two inclusions of von Neumann algebras to quantum groupoids

In this work we give a generalization of matched pairs of (finite) groups to describe a general class of depth two inclusions of factor von Neumann algebras and the C^∗-quantum groupoids associated with, using double groupoids.