User xuexing lu - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:15:38Z http://mathoverflow.net/feeds/user/4155 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/16024/what-is-the-relation-between-quantum-symmetry-and-quantum-groups What is the relation between quantum symmetry and quantum groups? Xuexing Lu 2010-02-22T03:07:14Z 2010-06-23T03:05:56Z <p>What kind of role do quantum groups play in modern physics ? Do quantum groups naturally arise in quantum mechanics or quantum field theories? What should quantum symmetry refer to ? Can we say that the "symmetry" of a noncommutative space (quantum phase space) should be a quantum group? Do quantum groups describe "extended symmetry" ?</p> http://mathoverflow.net/questions/23456/about-classification-of-split-group-extensions About classification of split group extensions Xuexing Lu 2010-05-04T17:07:21Z 2010-05-04T17:24:20Z <p>Are the equivalent class of split extension of G by K really in one to one correspondence with homomorphisms from G to Aut(K)? When I am trying to proof it, I find it may be not the case. I only get that $1\to K\to K\rtimes_{\rho_1}G\to G\to 1$ and $1\to K\to K\rtimes_{\rho_2}G\to G\to 1$ are equivalent if and only if there is a nonabelian 1-cocycle $\beta:G\to K$ such that $\rho_1=Ad_{\beta}\circ \rho_2$. When $\rho_1=\rho_2$, $\beta$ is an abelian 1-cocycle.Thus, the automrophism group of $1\to K\to K\rtimes_{\rho}G\to G\to 1$is isomorphic to $Z^{1}_{\rho}(G,C_K)$.</p> http://mathoverflow.net/questions/16399/what-is-the-relation-between-characters-of-a-group-and-its-lie-algebra What is the relation between characters of a group and its lie algebra? Xuexing Lu 2010-02-25T14:51:46Z 2010-04-01T12:30:03Z <p>What is the relation between characters of a group and its lie algebra?</p> <p>Roughly,I know that there is a one to one correspondence between representations of a lie algebra and its simple connected lie group by the exp map,and two irreducible representations of a lie group are unequivalent if and only if their characters are different. Now,then,I think the above statement will still hold for representations of lie algebras, but I am not sure about it . If someone can give some advises?</p> <p>Thanks Kevin Buzzard. For represatatons of general lie algebras the character (trace) don't work .<br> But for compact lie group,characters still work,so I restrict my question on compact lie groups and their lie algebras.<br> In representation theory of semisimple lie algebras ,we have formal characters/algebra characters which are sums of some formal elements over the weights. I wonder whether we can realize these algebra characters to be real characters ?<br> Or can we make them to be functions on cartan subalgebras or maximal torus? </p> http://mathoverflow.net/questions/16668/does-any-tensor-category-correspond-to-a-bialgebra Does any tensor category correspond to a bialgebra ? Xuexing Lu 2010-02-28T10:18:05Z 2010-02-28T18:22:00Z <p>I wonder how strong the power of Tanaka philosophy is,and if we accept that a tensor category is a generalized bialgebra,what difficulties we will come up against ?</p> <p>edit: whether most tensor categoryes are representable ,or whether for every "good enough" tensor category there exist a bialgebra with its module category isomorphic to the given category?</p> http://mathoverflow.net/questions/16584/about-state-field-correspondence about state-field correspondence Xuexing Lu 2010-02-27T08:51:39Z 2010-02-27T16:16:32Z <p>In the defination of vertex algebra,we call the vertex operator state-field correspondence,does that mean that it is an injective map?? Are there some physical intepretations about state-field correspondence ?.Or why we need state-field correspondence in physical viewpoint?? Does it have some relations to highest weight representations?</p> http://mathoverflow.net/questions/16392/how-to-interpret-the-sugawara-construction-from-a-physical-or-mathematical-viewp How to interpret the Sugawara construction from a physical or mathematical viewpoint? Xuexing Lu 2010-02-25T12:52:43Z 2010-02-25T19:08:20Z <p>In theoretical physics, the Sugawara theory is a set of formulae and theorems that allow one to construct a stress-energy tensor of a specific type of conformal field theory from a bilinear expression involving currents.</p> <p>How to interpret the Sugawara construction from a physical or mathematical viewpoint? </p> <p>Sugawara construction is a kind of mathod to embed varasoro algebra into completions of universal enveloping algebras of affine algras ? what special properties do this kind of embeding have? In soliton theorey, I know there is a boson-fermion correspondence which realize free boson algebra in the completion of free fermion algebra.<br> I wonder if there are some common principles under them? </p> http://mathoverflow.net/questions/16423/about-vertex-algebra-mode-expansion About vertex algebra ,mode expansion Xuexing Lu 2010-02-25T17:46:10Z 2010-02-25T18:48:52Z <p>A vertex operator is a linear map associating every state to a operator-valued distributions(quantum field) on a algebra curve,which is also called operator-state correspondence.<br> Chose a local complex coordinate, we can locally expand quantum fields as operator valued formal laurent series,this process is called mode expansion(?), the coefficients are called fourier coefficients. I confuse the terminology fourier coefficients,why people give them this name,does mode expansion relate to fourier transformation?</p> http://mathoverflow.net/questions/16425/introduction-to-modular-property-of-affine-alegebra-and-conformal-vertex-algebra Introduction to modular property of affine alegebra and conformal vertex algebra Xuexing Lu 2010-02-25T17:59:25Z 2010-02-25T18:43:56Z <p>I wonder how modular property naturally arises in conformal theory. Is it obvious from physical viewpoint?</p> http://mathoverflow.net/questions/16668/does-any-tensor-category-correspond-to-a-bialgebra/16674#16674 Comment by Xuexing Lu Xuexing Lu 2010-03-01T15:56:32Z 2010-03-01T15:56:32Z @Akhil Mathew: tensor equivalence,because we are considering tensor categories. http://mathoverflow.net/questions/16668/does-any-tensor-category-correspond-to-a-bialgebra/16709#16709 Comment by Xuexing Lu Xuexing Lu 2010-03-01T15:49:38Z 2010-03-01T15:49:38Z Twist equivalent sounds interesting! The endomorphisms of identity functor or the natural transformations from identity functor to itself form a algebra.Is it isomorphic to the algebra of endomorphisms of the bialgebra ? Is the isomorphism canonical in some sense ? http://mathoverflow.net/questions/16668/does-any-tensor-category-correspond-to-a-bialgebra/16709#16709 Comment by Xuexing Lu Xuexing Lu 2010-03-01T04:46:00Z 2010-03-01T04:46:00Z So a tensor category with a fiber functor can give us a bialgebra defined to be the endomorphisms of the fiber functor, and various fiber functors give different bialgebra, and they are morita equivalent . but if we consider the endomorphisms of identical functor which still a algebra /bialgbra,there may be some realation between the two algebra. http://mathoverflow.net/questions/16668/does-any-tensor-category-correspond-to-a-bialgebra/16674#16674 Comment by Xuexing Lu Xuexing Lu 2010-02-28T14:34:56Z 2010-02-28T14:34:56Z sorry, the generalized bialgebra is not a formal terminology ,I am trying to express such an idea that whether most tensor categoryes are representable ,or whether for every &quot;good enough&quot; tensor category there exist a bialgebra with its module category isomorphic to the given category. http://mathoverflow.net/questions/16024/what-is-the-relation-between-quantum-symmetry-and-quantum-groups Comment by Xuexing Lu Xuexing Lu 2010-02-23T13:25:28Z 2010-02-23T13:25:28Z Thanks Pavel Etingof! But what is a partical ? how to difine it ? Are symmetries needed to difine particals? Should they be the irreducible representions? http://mathoverflow.net/questions/16024/what-is-the-relation-between-quantum-symmetry-and-quantum-groups Comment by Xuexing Lu Xuexing Lu 2010-02-23T13:18:29Z 2010-02-23T13:18:29Z Theo Johnson-Freyd said : the short answer is that &quot;quantum groups&quot; were invented in the study of quantum integrable systems. quantum integrable systems,as far as my best understand from mathmatical viewpoint,are just algbras of observables with sufficent symmetries and the hilbert space of states is the module of algbra of observables. My questions are: 1 Are algbras of observables quantum groups? 2 What are the symmetry of these systems? the symmetry of algebra of observables or the symmetry of the module ? 3 What is the relation between symmetry and quantum group? http://mathoverflow.net/questions/16024/what-is-the-relation-between-quantum-symmetry-and-quantum-groups Comment by Xuexing Lu Xuexing Lu 2010-02-23T12:47:58Z 2010-02-23T12:47:58Z I know that quantum groups are just hopf algebtras which are bialgebras with antipode and some compatible conditions and coproducts are essential to define tensor product of modules so that the moudle category become a tensor category. But I don't know how to interpret tensor product in physical language.