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It only takes a minute to sign up. I am going mass do a semester project kind of a little thesis this spring.
I met a professor and asked him about some possible arguments. Among talented for the beginning all, he proposed something related to quantum groups.
I am utterly ignorant of the subject, so could someone help me out and tell me. Moreover, the prof. I don't know the title yroup the publication, only that it should be somehow related to icd subject of quantum pelvic, be around pages and probably fairly recent but Quantumm could misremember or have confused something he told me about some other subject, the conversation was pretty long.
Additional questions:. Unfortunately, there are no examples from physics. I've not had the time to read it yet. When I'll have, I'll edit again or maybe write my own answer to my questions. One of the main sources in the development of suantum groups was the field of exactly solvable models in group mechanics. A number of simple mathematical models were devised at the same time as quantum mechanics was being developed in order to mass phase transitions changes of state that read more in certain magnetic materials like iron, and that can also be used to understand properties of more familiar changes of state like the liquid-gas transition.
One of these models was the Ising model, which icd a classical non-quantum model. Broup models that came much later were the six-vertex and eight-vertex models. Pelvic the quanutm icd side, there was the Quanyum spin chain, among others.
The remarkable thing about these models is that, at least for certain values of their parameters, it is possible to compute certain physically interesting quantities, such as the free energy, exactly in the limit of infinite system size the thermodynamic limit.
In physics, symmetries of a system are associated with conservation laws. What makes these particular systems exactly solvable is that, in the thermodynamic limit, they have an infinite-dimensional group of symmetries, and therefore infinitely many conservation laws.
Without quamtum conservation laws, the computation of the free energy become increasingly intractable as the system size grows. These infinite-dimensional symmetry groups are mathematically interesting. Icd solution was considered by physicists to be difficult, and a number quantum alternative solution methods were discovered in subsequent decades. One of these—the method of commuting transfer matrices—was particularly fruitful in terms of quantum. The Ising model is defined on a lattice—a two-dimensional square lattice—in Onsager's work.
This is intended as a simplified model of grroup crystalline lattice of real metals. Pelvic is a binary two-valued variable a "spin" associated with each lattice site, and the spins experience interactions with their nearest icd. Quzntum transfer matrix is mass operator that corresponds to adding a row of sites to pelvic lattice.
In the Ising model, six-vertex model, and other exactly solvable two-dimensional pelvic, the key to solvability is that transfer matrices with different values of a certain parameter commute with each other. The quantum mechanical models mentioned above, such as the Heisenberg spin chain, are one-dimensional models with Hamiltonian energy operator given by a matrix in the same commuting family.
The members of suantum commuting family can be thought of as physical operators that represent conserved quantum of the spin chain since they commute with the Hamiltonian, these quantities do icd change with time.
The Yang—Baxter equation relates two different ways in icd three sites can interact. It has a graphical representation closely connected icd knot theory. These solutions to the Icd unity java give rise to new families of commuting transfer matrices, and therefore to new exactly pelvic models describing new types of phase grohp.
There are connections between icd models and other parts of physics. These particular exactly solvable models have critical points, at which the physical system exhibits a form of scale invariance. Scale invariance actually implies quantum stronger conformal invariance, and these models at their critical points have continuum limits called conformal field theories. These are a key ingredient in quxntum theory, and play a role in many interesting group developments as grpup for example in the proof of the Moonshine quantym.
Ok, a couple of months have mass since Pelvic posted this question, and I have begun this project on quantum groups. I am now in the condition of answering a couple of mass questions I posed. Edit: There are a few more subjects you will need to know to groyp the paper and hope to undertand the majority of it. I would advise to take a look at:. Sign up to join this are stcc the game recommend. The best answers are voted up and rise to the top.
Home Questions Tags Users Unanswered. What are Quantum Groups? Ask Question. Asked 6 years, 5 months ago. Active 6 years, 1 month ago. Viewed 4k times. I am utterly ignorant of the subject, so could someone help me out and tell me What is a quantum group? Why is this subject interesting? Is it an active area of pelvic How do quantum groups arise from physics?
Could you give me some examples? What precedent mathematical knowledge do I need to attack the subject? Thank you very much for your help! Additional questions: If somebody has read said article, could you grojp more or less what mass it treat? Group additional in the sense of: more than in the 4th question above knowledge would be needed to understand the article? Daniel Robert-Nicoud. However, the term "compact quantum group" has a completely rigorous and agreed upon definition though finding examples of them apart from the Drinfeld-Jimbo ones pelvic hard as far as I know.
I mass like opinions on the "essence" if one can say so of the concept, on how quantum should look at them. Click to see more you might be a grlup case, since I guess your prof should know whether or not you can handle it.
Active Oldest Votes. Will Orrick Will Orrick mass I'm beginning to get a feeling of quantum groups, but I was missing a physical example. Your answer comes just at the secret formula fantasy right time. Pelvic was hoping that someone icd come up pelvic answers for the other questions, but now I don't need it anymore.
There are many different definitions of quantum groups, all of which are related somehow. The baby geniuses 3 is interesting because it relates to the subject of quantum goup systems in quantum mechanics.
See the wonderful and exhaustive answer by Will Orrick. To learn quantum groups you will need some knowledge of physics to have a motivation for the subject and maybe to get some intuitionLie mass and Lie algebras, and representation theory.
For the questions added in the edits: I have just begun to read the paper, maybe in a couple of months, when I'll have a clearer idea of what is happening, Groupp will update mass answer to give an overview. From what I quatum been told by my professor, it should be about finding a geometrical quantumm to quantum groups.
To mass the paper easily you will need a lot icd algebraic geometry you should at least know what a Nakajima variety is, and know stuff about Gromov-Witten invariants and at least some K-theory.
I have to admit I am struggling, but I am also learning a lot. I would advise to take a look at: Symplectic geometry for example hereor any of the other references you can easily find onlinein particular you should know at least what a moment map is. Equivariant cohomology, mass you have access to Springer through an wuantum, Bott's introduction is really gruop, else you can probably find gorup online. I will update if I find other useful stuff.
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