Quantum Algebra Research Articles

On 2nd-stage quantization of quantum cluster algebras

Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the 2nd-stage quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its 2nd-stage quantized cluster algebras. Based on this observation, we find that a quantum cluster algebra possesses a mutually alternating quantum cluster algebra such that their 2nd-stage quantization can be essentially the same.As an example, we give the 2nd-stage quantized cluster algebra Ap,q(SL(2)) of FunC(SLq(2)) in §7.1 and show that it is a non-trivial 2nd-stage quantization, which may be realized as a parallel supplement to two parameters quantization of the general quantum group. As another example, we present a class of quantum cluster algebras with coefficients which possess a non-trivial 2nd-stage quantization. In particular we obtain a class of quantum cluster algebras from surfaces with coefficients which possess non-trivial 2nd-stage quantization.Finally, we prove that the compatible Poisson structure of a quantum cluster algebra without coefficients is always a locally standard Poisson structure. Following this, it is shown that the 2nd-stage quantization of a quantum cluster algebra without coefficients is in fact trivial.

Rigid and separable algebras in fusion 2-categories

Rigid monoidal 1-categories are ubiquitous throughout quantum algebra and low-dimensional topology. We study a generalization of this notion, namely rigid algebras in an arbitrary monoidal 2-category. Examples of rigid algebras include G-graded fusion 1-categories, and G-crossed fusion 1-categories. We explore the properties of the 2-categories of modules and of bimodules over a rigid algebra, by giving a criterion for the existence of right and left adjoints. Then, we consider separable algebras, which are particularly well-behaved rigid algebras. Specifically, given a fusion 2-category, we prove that the 2-categories of modules and of bimodules over a separable algebra are finite semisimple. Finally, we define the dimension of a connected rigid algebra in a fusion 2-category, and prove that such an algebra is separable if and only if its dimension is non-zero.

A parafermionic hypergeometric function and supersymmetric 6j-symbols

We study properties of a parafermionic generalization of the hyperbolic hypergeometric function appearing as the most important part in the fusion matrix for Liouville field theory and the Racah-Wigner symbols for the Faddeev modular double. We show that this generalized hypergeometric function is a limiting form of the rarefied elliptic hypergeometric function V(r) and derive its transformation properties and a mixed difference-recurrence equation satisfied by it. At the intermediate level we describe symmetries of a more general rarefied hyperbolic hypergeometric function. An important r=2 case corresponds to the supersymmetric hypergeometric function given by the integral appearing in the fusion matrix of N=1 super Liouville field theory and the Racah-Wigner symbols of the quantum algebra Uq(osp(1|2)). We indicate relations to the standard Regge symmetry and prove some previous conjectures for the supersymmetric Racah-Wigner symbols by establishing their different parametrizations.

Equilibrium states for the massive Sine-Gordon theory in the Lorentzian signature

In this paper we investigate the massive Sine-Gordon model in the ultraviolet finite regime in thermal states over the two-dimensional Minkowski spacetime. We combine recently developed methods of perturbative algebraic quantum field theory with techniques developed in the realm of constructive quantum field theory over Euclidean spacetimes to construct the correlation functions of the equilibrium state of the Sine-Gordon theory in the adiabatic limit. First of all, the observables of the Sine-Gordon theory are seen as functionals over the free configurations and are obtained as a suitable combination of the S−matrices of the interaction Lagrangian restricted to compact spacetime regions over the free massive theory. These S−matrices are given as power series in the coupling constant with values in the algebra of fields over the free massive theory. Adapting techniques like conditioning and inverse conditioning to spacetimes with Lorentzian signature, we prove that these power series converge when evaluated on a generic field configuration. The latter observation implies convergence in the strong operator topology in the GNS representations of the considered states. In the second part of the paper, adapting the cluster expansion technique to the Lorentzian case, we prove that the correlation functions of the interacting equilibrium state at finite temperature (KMS state) can be constructed also in the adiabatic limit, where the interaction Lagrangian is supported everywhere in space.

Approach to Equilibrium in Translation-Invariant Quantum Systems: Some Structural Results

We formulate the problem of approach to equilibrium in algebraic quantum statistical mechanics and study some of its structural aspects, focusing on the relation between the zeroth law of thermodynamics (approach to equilibrium) and the second law (increase in entropy). Our main result is that approach to equilibrium is necessarily accompanied by a strict increase in the specific (mean) energy and entropy. In the course of our analysis, we introduce the concept of quantum weak Gibbs state which is of independent interest.

New model category structures for algebraic quantum field theory

New model category structures for algebraic quantum field theory

Complex number and its discovery history

The operations of complex numbers are the main subject of the mathematical analysis area known as complex analysis. It is also known as the theory of functions of a complex variable. The primary research topic in the field of complex analysis is holomorphic functions. These functions are defined on the complex plane, have differentiable properties, and allow for negative values. The residue theorem, the Cauchy integral formula, the Laurent series expansion, etc. are a few concepts, ideas, and methods that are commonly used in research. In particular, over the years, complex analysis in mathematics, physics, and engineering has been extensively used in algebraic geometry, fluid dynamics, quantum mechanics, and other related areas. Two Italian mathematicians, Girolamo Cardano and Raphael Bombelli, discovered complex numbers in the 16th century while attempting to solve a particular algebra, and Cauchy and Riemann extended it in the 19th century. This essay begin with investigation of arithmetic propertity of comlex numbers and then fully explores history development of complex numbers.

Discovery of Complex numbers

A branch of mathematical analysis called complex analysis studies the functions of complex numbers. It is also sometimes referred to as the theory of functions of a complex variable. Complex analysis has been used extensively in mathematics, physics, and engineering over the years, particularly in the areas of algebraic geometry, fluid dynamics, quantum mechanics, and other related fields. It dates back to the 16th century, when Italian mathematicians Girolamo Cardano and Raphael Bombelli first noticed complex numbers while attempting to solve a particular algebra, and was later developed by Cauchy and Riemann in the 19th century. The development of complex numbers has a lengthy history. Mathematicians have advanced the discipline of mathematics significantly after thousands of years of development. During this time, mathematicians also found a great deal of previously unknown mathematical information and proved formulae and phenomena that had previously been impossible to verify. And it covers complex numbers as well as some of the mathematics related to them. In this paper, the complete discovery of complex numbers from cubic equation to topology of complex numbers is detailedly revealed.

Proof of Cauchy’s theorem on Disc

Mathematical analysis known as "complex analysis" examines characteristics of functions, sequences, derivatives of complex numbers, and other objects. When Italian mathematicians Girolamo Cardano and Raphael Bombelli attempted to solve a unique algebra, that is when people first learned about complex numbers. After hundreds of years of growth, complex analysis is now a crucial component of mathematics, physics, and engineering, particularly in the areas of algebraic geometry, fluid dynamics, quantum mechanics, and other related subjects. Complex analysis is a fascinating and esoteric field of study. Mathematicians have created groundbreaking work in this area during the past few centuries. Complex numbers were first defined by mathematicians, who subsequently gradually looked into their algebraic and geometric properties. After hundreds of years, they discovered and studied complex computations. Convergent power series local representations exist for holomorphic functions. This amazing discovery, made by Cauchy between 1830 and 1840, helps to explain the fascinating properties of holomorphic functions. The Cauchy integral theorem is one of the most important concepts in complex analysis. In this paper, a detailed proof of Cauchy’s theorem on a local disc is given.

An Askey-Wilson Algebra of Rank 2

An algebra is introduced which can be considered as a rank 2 extension of the Askey-Wilson algebra. Relations in this algebra are motivated by relations between coproducts of twisted primitive elements in the two-fold tensor product of the quantum algebra $\mathcal{U}_{q}(\mathfrak{sl}(2,\mathbb C))$. It is shown that bivariate $q$-Racah polynomials appear as overlap coefficients of eigenvectors of generators of the algebra. Furthermore, the corresponding $q$-difference operators are calculated using the defining relations of the algebra, showing that it encodes the bispectral properties of the bivariate $q$-Racah polynomials.

Quantization of the derivation Lie algebra over quantum torus

Recently, Lie bialgebra structures of the derivation Lie algebra over quantum torus were determined. In this paper, we use the general method of quantization by a Drinfel’d twist to quantize these algebras with their Lie bialgebra structures and present the explicit formulae.

A Curriculum of Table-Top Quantum Optics Experiments to Teach Quantum Physics

The rise of quantum information as a viable technology requires appropriate instructional curricula for preparing a future workforce. Key concepts that are the basis of quantum information involve fundamentals of quantum mechanics, such as superposition, entanglement and measurement. To complement modern initiatives to teach quantum physics to the emerging workforce, lab experiences are needed. We have developed a curriculum of quantum optics experiments to teach quantum mechanics fundamentals and quantum algebra. These laboratories provide hands-on experimentation of optical components on a table-top. We have also created curricular materials, manuals, tutorials, parts and price lists for instructors. Automation of the apparatus offers the flexibility of using the apparatus remotely and for giving access to a greater number of students with a single setup.

Strictification theorems for the homotopy time-slice axiom

It is proven that the homotopy time-slice axiom for many types of algebraic quantum field theories (AQFTs) taking values in chain complexes can be strictified. This includes the cases of Haag–Kastler-type AQFTs on a fixed globally hyperbolic Lorentzian manifold (with or without time-like boundary), locally covariant conformal AQFTs in two spacetime dimensions, locally covariant AQFTs in one spacetime dimension, and the relative Cauchy evolution. The strictification theorems established in this paper prove that, under suitable hypotheses that hold true for the examples listed above, there exists a Quillen equivalence between the model category of AQFTs satisfying the homotopy time-slice axiom and the model category of AQFTs satisfying the usual strict time-slice axiom.

Multistrand Eigenvalue Conjecture and Racah Symmetries

Racah matrices of quantum algebras are of great interest at present time. These matrices have a relation with $\mathcal{R}$-matrices, which are much simpler than the Racah matrices themselves. This relation is known as the eigenvalue conjecture. In this paper we study symmetries of Racah matrices which follow from the eigenvalue conjecture for multistrand braids.

Algebraic Symplectic Reduction and Quantization of Singular Spaces

The algebraic method of singular reduction is applied for non regular group action on manifolds which provides singular symplectic spaces. The problem of deformation quantization of the singular surfaces is the focus. For some examples of singular Poisson spaces the deformation quantization is explicitly constructed. In is shown that for the flat phase space with the classical moment map and the orthogonal group action the deformation quantization converges for the entire arguments of exponential type.

Wedge domains in non-compactly causal symmetric spaces

This article is part of an ongoing project aiming at the connections between causal structures on homogeneous spaces, Algebraic Quantum Field Theory, modular theory of operator algebras and unitary representations of Lie groups. In this article we concentrate on non-compactly causal symmetric spaces G/H. This class contains de Sitter space but also other spaces with invariant partial ordering. The central ingredient is an Euler element h in the Lie algebra of $${{\mathfrak {g}}}$$ . We define three different kinds of wedge domains depending on h and the causal structure on G/H. Our main result is that the connected component containing the base point eH of these seemingly different domains all agree. Furthermore we discuss the connectedness of those wedge domains. We show that each of these spaces has a natural extension to a non-compactly causal symmetric space of the form $$G_{{\mathbb {C}}}/G^c$$ where $$G^c$$ is a certain real form of the complexification $$G_{{\mathbb {C}}}$$ of G. As $$G_{{\mathbb {C}}}/G^c$$ is non-compactly causal, it also contains three types of wedge domains. Our results says that the intersection of these domains with G/H agree with the wedge domains in G/H.

Derivations of quantum and involution generalized Weyl algebras

In this paper, we classify the derivations of degree-one generalized Weyl algebras over a univariate Laurent polynomial ring. In particular, our results cover the Weyl–Hayashi algebra, a quantization of the first Weyl algebra arising as a primitive factor algebra of [Formula: see text], and a family of algebras which localize to the group algebra of the infinite group with generators [Formula: see text] and [Formula: see text], subject to the relation [Formula: see text].

Homotopy theory of net representations

The homotopy theory of representations of nets of algebras over a (small) category with values in a closed symmetric monoidal model category is developed. We illustrate how each morphism of nets of algebras determines a change-of-net Quillen adjunction between the model categories of net representations, which is furthermore, a Quillen equivalence when the morphism is a weak equivalence. These techniques are applied in the context of homotopy algebraic quantum field theory with values in cochain complexes. In particular, an explicit construction is presented that produces constant net representations for Maxwell [Formula: see text]-forms on a fixed oriented and time-oriented globally hyperbolic Lorentzian manifold.

Entanglement entropy and non-local duality: Quantum channels and quantum algebras

We investigate the transformation of entanglement entropy under dualities, using the Kramers–Wannier duality present in the transverse field Ising model as our example. Entanglement entropy between local spin degrees of freedom is not generically preserved by the duality; instead, entangled states may be mapped to states with no local entanglement. To understand the fate of this entanglement, we consider two quantitative descriptions of degrees of freedom and their transformation under duality. The first involves Kraus operators implementing the partial trace as a quantum channel, while the second utilizes the algebraic approach to quantum mechanics, where degrees of freedom are encoded in subalgebras. Using both approaches, we show that entanglement of local degrees of freedom is not lost; instead it is transferred to non-local degrees of freedom by the duality transformation.

Rp,q-multivariate discrete probability distributions

We construct the multivariate probability distributions (Pólya, inverse Pólya, hypergeometric and negative hypergeometric) from the generalized quantum deformed algebras. Moreover, we derive the corresponding bivariate probability distributions and determine their properties (Rp,q-factorial moments and covariance). Besides, we deduce particular cases of probability distributions from the quantum algebras known in the literature.