The quantum orthogonal ck groups are realized as the hopf algebras of the noncommutative functions over an. Introduction, historical survey, hopf algebras, basic examples, algebra of. Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and drinfelds recent fundamental contributions. Kassel, quantum groups, graduate texts in mathematics. Braided rotabaxter algebras, quantum quasishuffle algebras. Hopf algebras and topological field theories have applications in representation theory, topology, string theory, quantum gravity and quantum information theory. Graduate texts in mathematics 155, springer, berlin, 1995. In this wellwritten presentation, motivated by numerous examples and problems, the authors introduce the basic theory of braid groups, highlighting several definitions that show their equivalence. Quantum groups and hopf algebras, for fall term 2015.
Discussion of knots, links and tangles and graphical calculus in. The analogous statement for modular categories would be for a. Lectures on quantum groups jens carsten jantzen download. Chapter 2 life of a particle in real time we introduce quantum theory for a single, spinless particle of mass m0. An important objective is to nd quantum algorithms that are signi cantly faster than any classical algorithm solving the same problem. Quantum groups by christian kassel springer gtm 155 a quantum groups primer by shahn majid london math. Lecture note series 292 this book has real insight and is nicely divided into individual lectures of reasonable size. Download quantum computation and quantum information pdf. Vladimir georgievich turaev, born in 1954 is a russian mathematician, specializing in topology turaev received in 1979 from the steklov institute of mathematics his candidate of sciences degree phd under oleg viro. An introduction to quantum groups lectures at ncgoa07 christian. 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.
It presents the quantum groups attached to sl2 as well as the basic concepts of the theory of hopf algebras. On one hand also semiconductor quantum dot structures are applied, however, which are very often integrated in micro or nano cavities enhancing drastically the lightmatter interaction. A path to current algebra is written for the reader at ease with at least one such structure and keen to learn the latest algebraic concepts and techniques. This is a contribution to the project of quiver approaches to quasi quantum groups initiated in arxiv. 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. Hopf algebras, quantum groups and topological field theory. A quantum group approach 3 kind of quantum multiplication as the outcome is a superposition. Quantum groups have been proposed as qanalog extensions of the spacetime symmetry groups such as the poincare group 15, 16. 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. Braided structures provide effective ways of quantization such as for quantum groups. Introduction to hopf algebras and representations, spring 2011. 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. Quantum groups have given many examples of noncommutative noncocommutative bialgebras than were known before this research area had taken off. Several minicourses on a variety of topics from noncommutative geometry, operator algebras and related areas will be given by.
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. Hopf algebras, quantum groups and topological eld theory. 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 enveloping algebra. We assume that the particle moves in euclidean space with coordinates xand of dimension s d. Vanderbilt mathematics, noncommutative geometry and. Having glanced a little bit at the last two in this list, i found both of them more difficult to read.
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. The center of quantum group can be described by quantum determinant. With those techniques also the interaction and control of single particles is possible, such as photons and charge carriers. Other applications of quantum groups to physics are those involving quantum gravity 17, 18, chernsimons gauge theory 19, 20, hidden quantum group symmetries 21. In fact, chemists, who have used nmr for decades to study complicated molecules, have been doing quantum computing. Quantum computation is the eld that investigates the computational power and other properties of computers based on quantummechanical principles. Discussion of knots, links and tangles and graphical calculus in finite dimensional modules over braided hopf algebras. Quantum representations of braid groups and holonomy lie algebras kohno, toshitake, 2017. The josephson junction is the basic building block of a superconducting qubit, and thus a quantum computer. Pdf introduction to quantum groups download full pdf. An introduction to quantum groups lectures at ncgoa07 christian kassel summary lecture 1. 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.
Math 846 syllabus syllabus lecture 1, w 92 introduction lecture 2, f 94 first examples lecture 3, w 99. Introduction to hopf algebras and representations, spring. University, which followed the book quantum groups by kassel. The first part presents in detail the quantum groups attached to sl2 as well as the basic concepts of the theory of hopf. Oct 24, 2016 cohomological invariant for groups coming from quantum group theory, oberseminar topologie, universitat bonn, germany 23 november 2010 abstract. The prounipotent completion of a braided category 85 8. A key to understanding these new developments is categorical duality. Quantum logic gates and nuclear magnetic resonance pulse sequences nmr quantum logic gates for homonuclear spin systems singlefluxquantum logic circuits exploiting collisionbased fusion gates. The study of hopf algebras sometimes also known as quantum groups is a very active field, relating algebra, representation theory and mathematical physics.
Christian kassel, marc rosso, vladimir turaev, mathematiker. As i discuss in this introcjuctory section, the equations that govern the motions of electrons and of nuclei are not the familiar newton equatrons. Better still, umn students have online access to the book through the library website. 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 field theory 24 may, 2005 at 7. This is a contribution to the project of quiver approaches to quasiquantum groups initiated in arxiv. We classify majid bimodules over groups with 3cocycles by virtue of projective. Actually the outcomes xk are endowed with an intrinsic probability distribution related to the quantum dimensions of the simple objects. The successes of quantum mechanics have been extraordinary. 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.
If this is the first time you use this feature, you will be asked to authorise cambridge core to connect with your account. There is an underlying, more general theory of which wave mechanics is but one mathematical manifestation or representation. Quantum groups graduate texts in mathematics 155 by. The phrase quantum jump or quantum leap is now in common usage, and incorrectly too. Quantum groups and their primitive ideals modern surveys in mathematics 29 by anthony joseph. By representing each qubit with a vast collection of mol ecules, one can afford to let measurements interact with a few of them. As one can relate algebraic groups with commutative hopf algebras via group schemes, it is also agreed that the category of quantum groups should correspond to the opposite category of the category of hopf algebras. Lie algebra g and universal enveloping algebra of g. This book provides an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and on. Cyclic groups and quantum logic gates pdf free download. Springer made a bunch of books available for free, these were the direct links springerfreemathsbooks. Part of the graduate texts in mathematics book series gtm, volume 155. Report on joint work with pierre guillot, cohomology of invariant drinfeld twists on group algebras, imrn 2010 2010, 18941939 arxiv. Introduction, historical survey, hopf algebras, basic examples, algebra of regular functions on sl2.
Continuing recent study relating the two structures, this paper considers rotabaxter algebras and dendriform. Pdf from projective representations to quasiquantum groups. With those techniques also the interaction and control of single particles is. A promising technology is the quantum computers, and this paper gives a general overview about this subject. Publication date 1995 topics hopf algebras, mathematical physics, quantum groups, topology publisher. Lecture course hopf algebras, quantum groups and topological. The fifth annual spring institute on noncommutative geometry and operator algebras is a collaborative learning and research conference in which participants will exchange ideas, consult on topics, and lead discussions. Turaev was a professor at the university of strasbourg and then became a professor at indiana university. Springer made a bunch of books available for free, these were. The extension of frt quantization theory for the nonsemisimple ck groups is suggested. 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.
Drinfeldjimbos quantum enveloping algebra u qsl2, duality with the quantum algebra of functions sl. Hopf algebras, quantum groups and topological eld theory winter term 201920. Part of the graduate texts in mathematics book series gtm, volume 155 log in to check access. An introduction tony hey department of electronics and computer science, university of southampton, southampton, united kingdom so17 1bj. Building qubits phase across junction energy maximum 0 energy energy minimum energy diagram of a junction electrons weak link superconductor what are the basic principles. Quantum groups graduate texts in mathematics 155 by christian kassel. At the same time this quantum group can be interpreted as the quantisation of a particle moving in a curved geometry with blackholelike features.