Decomposition of the adjoint representation of the small quantum sl2sl_2

Given a finite type root datum and a primitive root of unity $q=\sqrt[l]{1}$, G.~Lusztig has defined in [Lu] a remarkable finite dimensional Hopf algebra \fu over the cyclotomic field ${\Bbb Q}(\sqrt[l]{1})$. In this note we study the adjoint representation \ad of \fu in the simplest case of the root datum $sl_2$. The semisimple part of this representation is of big importance in the study of local systems of conformal blocks in WZW model for $\hat{sl}_2$ at level $l-2$ in arbitrary genus. The problem of distinguishing the semisimple part is closely related to the problem of integral representation of conformal blocks (see [BFS]). We find all the indecomposable direct summands of \ad with multiplicities. It appears that \ad is isomorphic to a direct sum of simple and projective modules. It can be lifted to a module over the (infinite dimensional) quantum universal enveloping algebra with divided powers $U_q(sl_2)$ which is also a direct sum of simples and projectives.Comment: 12 pages, submitted by M.Finkelberg at request of V.Ostri

Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups

We classify Lagrangian subcategories of the representation category of a twisted quantum double of a finite group. In view of results of 0704.0195v2 this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of the representation category of a twisted quantum double of a finite group G and module categories over the category of twisted G-graded vector spaces such that the dual tensor category is pointed. This can be viewed as a quantum version of V. Drinfeld's characterization of homogeneous spaces of a Poisson-Lie group in terms of Lagrangian subalgebras of the double of its Lie bialgebra. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories.Comment: 26 pages; several comments and references adde

From modular invariants to graphs: the modular splitting method

We start with a given modular invariant M of a two dimensional su(n)_k conformal field theory (CFT) and present a general method for solving the Ocneanu modular splitting equation and then determine, in a step-by-step explicit construction, 1) the generalized partition functions corresponding to the introduction of boundary conditions and defect lines; 2) the quantum symmetries of the higher ADE graph G associated to the initial modular invariant M. Notice that one does not suppose here that the graph G is already known, since it appears as a by-product of the calculations. We analyze several su(3)_k exceptional cases at levels 5 and 9.Comment: 28 pages, 7 figures. Version 2: updated references. Typos corrected. su(2) example has been removed to shorten the paper. Dual annular matrices for the rejected exceptional su(3) diagram are determine

The influence of an external magnetic field on the dynamic stress of an elastic conducting one-sided layer with a longitudinal shear crack

We study the interaction of a magnetoelastic shear wave with a curvilinear tunnel crack in an ideally conducting diamagnetic (resp. paramagnetic) one-sided (resp. two-sided) layer in the presence of an external static magnetic field. The bases of the one-sided layer are free of mechanical load, and the rim of the face is clamped or free. The corresponding linearized boundary-value problem of magnetoelasticity is reduced to a singular integrodifferential equation with subsequent implementation on a computer. We give numerical results that characterize the influence of the size of the preliminary magnetic field, the frequencies of the load, the curvature, and the orientation of the crack on the stress intensity factor. When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/2163

The interaction of a magnetoelastic shear wave with longitudinal cavities in a conducting layer

We study the influence of a strong magnetic field on the interaction of a shear wave with longitudinal cylindrical cavities in an elastic ideally conducting layer. The resulting singular integral equation of the boundary-value problem under consideration is implemented numerically for the case of a single cavity. We present the results of computation of the stresses on the edge of a circular cavity and an elliptical cavity. When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/2163

Анестезиологическое обеспечение открытых операций на плоде

Fetal surgery is a rapidly growing feld of medicine. Anesthetic provision of fetal operations is developing together with progressing surgical techniques. The fundamentals of the physiology of a pregnant woman and a fetus are considered. The features of transplacental drug transfer are described. The mechanisms of uteroplacental and fetoplacental interactions have been analyzed. The contemporary data are presented that allow judging about the method of choice and approach to anesthesia in open fetal surgery. Reviewed scientifc information, including reviews of recent years and randomized trials The perspective of an anesthesiologist is shown, involved in open fetal surgery, including perioperative and intraoperative management of two high-risk patients, i.e. a pregnant woman and fetus undergoing surgical intervention at the same time. Postoperative management of patient data. The impossibility of carrying out such operations without the well-coordinated work of a large multidisciplinary team of specialists is noted.Фетальная хирургия является быстроразвивающейся областью медицины. Анестезиологическое обеспечение фетальных операций развивается совместно с прогрессом в хирургической технике. Рассмотрены основы физиологии беременной и плода. Описаны особенности трансплацентарного переноса лекарственных средств. Проанализированы механизмы маточно-плацентарного и фетоплацентарного взаимодействия. Приведены современные данные, позволяющие судить о методе выбора и подхода к анестезии при открытой хирургии плода.Рассмотрена научная информация, включая обзоры последних лет и рандомизированные исследования. Показан взгляд анестезиолога, участвующего в проведении открытых операций на плоде, включая периоперационное и интраоперационное ведение двух пациентов высокого риска – беременной женщины и плода, которым одномоментно выполняется оперативное вмешательство, а также ведение в послеоперационном периоде. Отмечена невозможность проведения подобных операций без слаженной работы большой мультидисциплинарной команды специалистов

From conformal embeddings to quantum symmetries: an exceptional SU(4) example

We briefly discuss several algebraic tools that are used to describe the quantum symmetries of Boundary Conformal Field Theories on a torus. The starting point is a fusion category, together with an action on another category described by a quantum graph. For known examples, the corresponding modular invariant partition function, which is sometimes associated with a conformal embedding, provides enough information to recover the whole structure. We illustrate these notions with the example of the conformal embedding of SU(4) at level 4 into Spin(15) at level 1, leading to the exceptional quantum graph E4(SU(4)).Comment: 22 pages, 3 color figures. Version 2: We changed the color of figures (ps files) in such a way that they are still understood when converted to gray levels. Version 3: Several references have been adde

Character D-modules via Drinfeld center of Harish-Chandra bimodules

The category of character D-modules is realized as Drinfeld center of the abelian monoidal category of Harish-Chandra bimodules. Tensor product of Harish-Chandra bimodules is related to convolution of D-modules via the long intertwining functor (Radon transform) by a result of Beilinson and Ginzburg (Represent. Theory 3, 1–31, 1999). Exactness property of the long intertwining functor on a cell subquotient of the Harish-Chandra bimodules category shows that the truncated convolution category of Lusztig (Adv. Math. 129, 85–98, 1997) can be realized as a subquotient of the category of Harish-Chandra bimodules. Together with the description of the truncated convolution category (Bezrukavnikov et al. in Isr. J. Math. 170, 207–234, 2009) this allows us to derive (under a mild technical assumption) a classification of irreducible character sheaves over ℂ obtained by Lusztig by a different method. We also give a simple description for the top cohomology of convolution of character sheaves over ℂ in a given cell modulo smaller cells and relate the so-called Harish-Chandra functor to Verdier specialization in the De Concini–Procesi compactification.United States. Defense Advanced Research Projects Agency (grant HR0011-04-1-0031)National Science Foundation (U.S.) (grant DMS-0625234)National Science Foundation (U.S.) (grant DMS-0854764)AG Laboratory HSE (RF government grant, ag. 11.G34.31.0023)Russian Foundation for Basic Research (grant 09-01-00242)Ministry of Education and Science of the Russian Federation (grant No. 2010-1.3.1-111-017-029)Science Foundation of the NRU-HSE (award 11-09-0033)National Science Foundation (U.S.) (grant DMS-0602263