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

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

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

Cyclotomic integers, fusion categories, and subfactors

Dimensions of objects in fusion categories are cyclotomic integers, hence number theoretic results have implications in the study of fusion categories and finite depth subfactors. We give two such applications. The first application is determining a complete list of numbers in the interval (2, 76/33) which can occur as the Frobenius-Perron dimension of an object in a fusion category. The smallest number on this list is realized in a new fusion category which is constructed in the appendix written by V. Ostrik, while the others are all realized by known examples. The second application proves that in any family of graphs obtained by adding a 2-valent tree to a fixed graph, either only finitely many graphs are principal graphs of subfactors or the family consists of the A_n or D_n Dynkin diagrams. This result is effective, and we apply it to several families arising in the classification of subfactors of index less then 5.Comment: 47 pages, with an appendix by Victor Ostri

From boundary to bulk in logarithmic CFT

The analogue of the charge-conjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy `identity brane'). We apply the general method to the c_1,p triplet models and reproduce the previously known bulk theory for p=2 at c=-2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a by-product we obtain a logarithmic version of the Verlinde formula for the c_1,p triplet models.Comment: 35 pages, 2 figures; v2: minor corrections, version to appear in J.Phys.