Quantum mechanics is a mathematical language, much like calculus. This book provides an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and on. The aim of this paper is to analyze the categories of representations of quantum groups at roots of unity from this prospective. We describe the use of a single quantum bit for secure key distribution. There is an underlying, more general theory of which wave mechanics is but one mathematical manifestation or representation. An introduction to quantum computing for nonphysicists. The author also discusses the quantum plane as an object that generalizes the affine plane, namely the two variables x, y generating the plane no longer commute but instead satisfy yx q xy. Introduction to quantum groups will be of interest to mathematicians working in the representation theory of lie groups and lie algebras, knot theorists, theoretical physicists, and graduate students. This book provides an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and on drinfelds recent fundamental contributions. An introduction to quantum groups lectures at ncgoa07 christian. As i discuss in this introcjuctory section, the equations that govern the motions of electrons and of nuclei are not the familiar newton equatrons. Representations of quantum groups defined over commutative. Just as classical computers can be thought of in boolean algebra terms, quantum computers are reasoned about with quantum mechanics.
Quantum representations of braid groups and holonomy lie algebras kohno, toshitake, 2017. Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and drinfelds recent fundamental contributions. Download fulltext pdf quantum groups, squeezing, bloch, and theta functions article pdf available in modern physics letters b 0720 november 2011 with 198 reads. One of the ideas behind quantum groups is that if we consider a structure that is in a sense equivalent but larger, namely a group algebra or a universal. Since large parts of the book are independent of the theory of perverse sheaves, the work may also be used as a textbook. Kassel abstract for a group g, the notion of a ribbon gcategory was introduced in turaevhomotopy eld theory in dimension 3 and crossed groupcategories, preprint, math. Coverage also focuses on hopf algebras that produce solutions of the yangbaxter equation and here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and drinfelds recent. Quantum logic gates and nuclear magnetic resonance pulse sequences nmr quantum logic gates for homonuclear spin systems singleflux quantum logic circuits exploiting collisionbased fusion gates. Pdf introduction to quantum groups download full pdf. It presents the quantum groups attached to sl2 as well as the basic concepts of the theory of hopf algebras.
This book provides an introduction to the theory of quantum groups with emphasis. The rismenabled version of quantum espresso can be downloaded from satomichi nishiharas git repository. At the same time this quantum group can be interpreted as the quantisation of a particle moving in a curved geometry with blackholelike features. Hopf algebras, quantum groups and yangbaxter equations mdpi. The josephson junction is the basic building block of a superconducting qubit, and thus a quantum computer. Jimbo, independently, in their study of the quantum yangbaxter equation arising from 2dimensional solvable lattice models. The classical description of a particle, hilbert space formalism, group theory, lie algebra, the green function approach, the evolution operator, scattering theory, quantum mechanics in practice, dynamics and driven systems. Pdf introduction to quantum groups download full pdf book. Hopf algebras, quantum groups and topological eld theory. If youre looking for a free download links of quantum computation and quantum information pdf, epub, docx and torrent then this site is not for you. An introduction to quantum groups lectures at ncgoa07 christian kassel summary lecture 1. But the real power of quantum computation derives from the exponential state spaces of multiple quantum bits.
Braids,qbinomials, and quantum groups sciencedirect. Even single quantum bits enable interesting applications. Since the i n t ro duction of quantum groups b y drinfeld, jim b o and w orono wicz in the mideigh ties, a lot o f researc h has b een going on dev eloping the relations b et w een q special. When dxtasheet select the in this file command, acrobat will run an optical character recognition process that will convert the image to selectable text and then this will allow you to copy the text in the document. Quantum groups graduate texts in mathematics by christian kassel quantum groups graduate texts in mathematics by christian kassel here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and drinfelds recent fundamental contributions. Hopf algebras, quantum groups and topological eld theory winter term 201920. The first part presents in detail the quantum groups attached to slsubscript 2 as well as the basic concepts of the theory of hopf algebras. It begins with an introduction to the quantum theory from a computerscience perspective.
Noncommutative algebra an overview sciencedirect topics. Download quantum computation and quantum information pdf ebook. Representations of compact lie groups, theodor brocker tammo tom dieck. Coverage also focuses on hopf algebras that produce solutions of the yangbaxter equation and provides an account of drinfelds elegant treatment of the monodromy of the. At the writing of this thesis the theory of quantum groups is a young and. The theory of quantum groups is a rapidly developing area with numerous applications in mathematics and theoretical physics, e.
Between 1987 and 1995 literally thousands of papers on quantum groups have been published. The center of quantum group can be described by quantum determinant. Introduction, historical survey, hopf algebras, basic examples, algebra of regular functions on sl2. Quantum groups have given many examples of noncommutative noncocommutative bialgebras than were known before this research area had taken off. The successes of quantum mechanics have been extraordinary. Drinfeldjimbos quantum enveloping algebra u qsl2, duality with the quantum algebra of functions sl. Cyclic groups and quantum logic gates pdf free download. Drinfeld, quantum groups, in proceedings of the international congress of mathematicians, a. The phrase quantum jump or quantum leap is now in common usage, and incorrectly too.
Springer have made a bunch of books available for free. Quantum computation is the eld that investigates the computational power and other properties of computers based on quantummechanical principles. It illustrates the quantumcomputational approach with several elementary examples of quantum speedup, before moving to the major applications. By representing each qubit with a vast collection of mol ecules, one can afford to let measurements interact with a few of them. Mathematical society of japan memoirs project euclid. Download now the quantum groups discussed in this book are the quantized enveloping algebras introduced by drinfeld and jimbo in 1985, or variations thereof. Lectures on quantum groups jens carsten jantzen since its origin about ten years ago, the theory of quantum groups has become one of the most fascinating topics of modern mathematics, with numerous applications to several sometimes rather disparate areas, including lowdimensional topology and mathematical physics.
Quantum logic gates and nuclear magnetic resonance pulse sequences nmr quantum logic gates for homonuclear spin systems singlefluxquantum logic. An important objective is to nd quantum algorithms that are signi cantly faster than any classical algorithm solving the same problem. Pdf 8 lectures on quantum groups and qspecial functions. Rational homotopy theory, yves felix stephen halperin jeanclaude thomas. Lectures on quantum groups jens carsten jantzen download. The discovery of quantum groups was quite unexpected since it was known for a long time that compact groups and semisimple lie algebras are rigid objects, in other words, they cannot be deformed. Having glanced a little bit at the last two in this list, i found both of them more difficult to read.
Very useful for understanding and research in quantum groups, in particular, the chapters on the braid group action and crystal bases highly recommended to all research mathematicians working in quantum groups the writing is one of the most pleasant attributes of this book. Similarly, the quantum group u q g can be regarded as an algebra over the field qq, the field of all rational functions of an indeterminate q over q see below in the section on quantum groups at q 0. Designed as a secondsemester text book proof mathematics pdf for uptu b. Commons attribution cc by license, which allows users to download, copy and build. A fully configured ubuntu virtual machine that can be run from windowsmacos xlinuxsolaris is available on the materials cloud site. Introduction to quantum groups george lusztig springer. In fact, chemists, who have used nmr for decades to study complicated molecules, have been doing quantum computing. Graduate texts in mathematics 155, springer, berlin, 1995. By choosing representations of the braid groups, one obtains numerical or matrix realizations of these identities. You can read the quantum groups book on our website. Pdf quantum groups download full pdf book download. Hopf algebras, quantum groups and topological field theory. Just as classical physics uses calculus to explain nature, quantum physics uses quantum mechanics to explain nature.
Quantum computer science by david mermin download link. Quantum groups available for download and read online in other formats. Introduction to quantum mechanics pdf download book. Very useful for understanding and research in quantum groups, in particular, the chapters on the braid group action and crystal bases highly recommended to all research mathematicians working in quantum groups the writing is one of. We call h 2 ha a quantized function algebra qfa, resp. The quantum groups discussed in this book are the quantized enveloping algebras introduced by drinfeld and jimbo in 1985, or variations thereof. These include lectures on algebraic quantum groups, by ken brown and ken goodearl, lectures on quantum groups, by jens jantzen, introduction to quantum groups, by george lusztig, and quantum groups and their primitive ideals, by anthony joseph.
Introduction, historical survey, hopf algebras, basic. Gt0005291 with a view towards constructing 3dimensional homotopy quantum eld theories hqfts with target. Representations of quantum groups defined over commutative rings ii representations of quantum groups defined over commutative rings ii cox, ben l enright, thomas j. An introduction to the theory of quantum groups ewu digital. The theory of quantum groups has led to a new, extremely rigid structure, in which the objects of the theory are provided with canonical basis with rather remarkable properties. Quantum groups are certain families of hopf algebras that are deformations of universal enveloping algebras of kacmoody algebras. From coalgebras to drinfeld algebras steven shnider, shlomo sternberg download bok. Jan 14, 2020 the rismenabled version of quantum espresso can be downloaded from satomichi nishiharas git repository. In order to construct 3dimensional homotopy quantum eld theories hqfts, the second author introduced for a group g the notions of a ribbon gcategory and a modular ribbon gcategory. Preparation of quantum nanostructure size and dimensionality effect, fermigas.
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