相关论文: Introduction to Quantum Group Theory

Affine quantum groups are certain pseudo-quasitriangular Hopf algebras that arise in mathematical physics in the context of integrable quantum field theory, integrable quantum spin chains, and solvable lattice models. They provide the…

We introduce and study the concept of a bornological quantum group. This generalizes the theory of algebraic quantum groups in the sense of van Daele from the algebraic setting to the framework of bornological vector spaces. Working with…

These notes summarise a talk surveying the combinatorial or Hamiltonian quantisation of three dimensional gravity in the Chern-Simons formulation, with an emphasis on the role of quantum groups and on the way the various physical constants…

We give a basic overview of computational complexity, query complexity, and communication complexity, with quantum information incorporated into each of these scenarios. The aim is to provide simple but clear definitions, and to highlight…

This is an introduction to quantum algebra, from a geometric perspective. The classical spaces $X$, such as the Lie groups, homogeneous spaces, or more general manifolds, are described by various algebras $A$, defined over various fields…

In the present paper we propose a new approach to quantum fields in terms of category algebras and states on categories. We define quantum fields and their states as category algebras and states on causal categories with partial involution…

Starting with a four-dimensional gauge theory approach to rational, elliptic, and trigonometric solutions of the Yang-Baxter equation, we determine the corresponding quantum group deformations to all orders in $\hbar$ by deducing their RTT…

This paper is a short account of the construction of a new class of the infinite-dimensional representations of the quantum groups. The examples include finite-dimensional quantum groups $U_q(\mathfrak{g})$, Yangian $Y(\mathfrak{g})$ and…

The notion of Fourier transform is among the more important tools in analysis, which has been generalized in abstract harmonic analysis to the level of abelian locally compact groups. The aim of this paper is to further generalize the…

This is an introduction to the finite groups, with focus on the groups of permutations and reflections, and more generally, on the finite groups of unitary matrices. We first discuss the basics of group theory, featuring the cyclic,…

Lie groups and quantum algebras are connected through their common universal enveloping algebra. The adjoint action of Lie group on its algebra is naturally extended to related q-algebra and q-coalgebra. In such a way, quantum structure can…

These are lecture notes of a mini-course given by the first author in Moscow in July 2019, taken by the second author and then edited and expanded by the first author. They were also a basis of the lectures given by the first author at the…

In this paper, we give an alternative approach to the theory of locally compact quantum groups, as developed by Kustermans and Vaes. We develop the theory completely within the von Neumann algebra framework. At various points, we also do…

We obtain two related characterizations of discrete quantum groups and discrete quantum groups of Kac type as allegorical group objects in the symmetric monoidal dagger category of quantum sets and relations, of interest to quantum…

In a recent article, we gave a definition of partition C*-algebras. These are universal C*-algebras based on algebraic relations which are induced from partitions of sets. In this follow up article, we show that often we can associate a…

The article gives the second part of the treatise on Regular Algebraic $K$-theory (Sections V & VI) of the author. Regular algebraic $K$-theory for groups is a homology theory for discrete groups closely connected to (but different from)…

These notes give an elementary introduction to Lie groups, Lie algebras, and their representations. Designed to be accessible to graduate students in mathematics or physics, they have a minimum of prerequisites. Topics include definitions…

Quantum Theory is one of the pillars of modern science developed over the last hundred years. In this review paper we introduce, step by step, the quantum theory understood as a mathematical model describing quantum experiments. We start…

The program of a physical concept of information is outlined in the framework of quantum theory. A proposal is made for how to avoid the introduction of axiomatic observables. The conventional (collapse) and the Everett interpretations of…

In 1987, Woronowicz gave a definition of compact matrix quantum groups generalizing compact Lie groups in the setting of noncommutative geometry. About twenty years later, Banica and Speicher isolated a class of compact matrix quantum…