Koszul duality in deformation quantization and Tamarkin's approach to Kontsevich formality

Let $\alpha$ be a quadratic Poisson bivector on a vector space $V$. Then one can also consider $\alpha$ as a quadratic Poisson bivector on the vector space $V^*[1]$. Fixed a universal deformation quantization (prediction some weights to all Kontsevich graphs [K97]), we have deformation quantization of the both algebras $S(V^*)$ and $\Lambda(V)$. These are graded quadratic algebras, and therefore Koszul algebras. We prove that for some universal deformation quantization, independent on $\alpha$, these two algebras are Koszul dual. We characterize some deformation quantizations for which this theorem is true in the framework of the Tamarkin's theory [T1].Comment: 49 pages, 2 figure

On L-infinity morphisms of cyclic chains

Recently the first two authors constructed an L-infinity morphism using the S^1-equivariant version of the Poisson Sigma Model (PSM). Its role in deformation quantization was not entirely clear. We give here a "good" interpretation and show that the resulting formality statement is equivalent to formality on cyclic chains as conjectured by Tsygan and proved recently by several authors.Comment: 11 page

The Shapovalov determinant for the Poisson superalgebras

Among simple Z-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C_{2}: these are the Lie superalgebra k^L(1|6) of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra sh(0|2k) of special Hamiltonian fields in 2k odd indeterminates, and the Kac--Moody version of sh(0|2k). Using C_{2} we compute N. Shapovalov determinant for k^L(1|6) and sh(0|2k), and for the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov described irreducible finite dimensional representations of po(0|n) and sh(0|n); we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over po(0|2k) and sh(0|2k)

The Hopf Algebra of Renormalization, Normal Coordinates and Kontsevich Deformation Quantization

Using normal coordinates in a Poincar\'e-Birkhoff-Witt basis for the Hopf algebra of renormalization in perturbative quantum field theory, we investigate the relation between the twisted antipode axiom in that formalism, the Birkhoff algebraic decomposition and the universal formula of Kontsevich for quantum deformation.Comment: 21 pages, 15 figure

Клинико-функциональные нарушения сердечно-сосудистой системы у пациентов с хронической обструктивной болезнью легких при наличии хронической сердечной недостаточности

The study was aimed to investigate clinical and functional particularities of cardiovascular system in patients with COPD and chronic heart failure (CHF). We examined 1737 patients, of them, 498 with COPD (the 1st group), 721 with COPD and concomitant class I to III stable angina (the 2nd group), and 519 with class I to III stable angina (the 3rd group). All the patients also had CHF. The study included clinical assessment using modified R. Cody's scale, ECG, Holter monitoring, lung function testing, Doppler echocardiography, 6-minute walk test (6MWT). Obstructive disorders were found in 424 of the 2nd group patients (58.8 %) and 268 of the 3rd group patients (51.7 %). Number of patients with class II to IV chronic cor pulmonale was similar in all groups. The class III was determined more often in the 2nd group (36.5 %; р < 0.05), the class I prevailed in the 1st group (17.8 %; р < 0.05). The stage I CHF predominated in the 1st group (31.9 %; р < 0.05), the stages IIА and IIБ were seen mainly in the 2nd group (51.6 % and 26.4 % respectively; р < 0.05). The majority of the 3rd group patients (57.3 %) had the stage IIA CHF. According to most clinical signs, CHF was more severe in the 2nd group. 6MWT was 308.08 ± 6.03 m in the 1st group, 156.4 ± 4.38 m in the 2nd group, and 212.8 ± 5.65 m in the 3rd group (р < 0.05). Low left ventricular (LV) ejection fraction was seen significantly more often in COPD patients with concomitant angina. This group as well as COPD patients had significantly higher mean pulmonary artery pressure (mPAP) (24.7 ± 0.41 mm Hg and 29.9 ± 0.27 mm Hg respectively) compared to 21.7 ± 0.07 mm Hg in the 3rd group patients. Right ventricular (RV) diastolic dysfunction was revealed in 154 patients of the 1st group (83.7 %), 194 patients of the 2nd group (91.9 %) and 72 patients of the 3rd group (43.1 %). In conclusion, all the patients had RV and LV remodeling signs. COPD patients had more prominent RV hypertrophy and dilation, angina patients had enlargement of left heart and moderate decrease in LV contractility. The patients with COPD and angina had the greatest increase in size and wall thickness of both LV and RV and low pump function of LV.Целью исследования явилось изучение особенностей клинико-функциональных нарушений сердечно-сосудистой системы у больных ХОБЛ при наличии хронической сердечной недостаточности (ХСН). Обследовано 1 737 пациентов: 498 – с ХОБЛ (1-я группа), 721 с сочетанием ХОБЛ и стенокардии напряжения I–III функциональных классов (ФК) (2-я группа), 519 – со стенокардией напряжения I–III ФК (3-я группа). У всех больных ХСН проводили оценку клинического состояния по шкале R.Cody в модификации В.Ю.Мареева (ШОКС), ЭКГ, суточное мониторирование ЭКГ по Холтеру, функции внешнего дыхания, эходопплерокардиографию, тест с 6-минутной ходьбой (6-МШТ). Обструктивные нарушения легочной вентиляции разной степени выраженности имели 424 (58,8 %) больных 2-й группы и 268 (51,7 %) больных 3-й группы. Количество больных с хроническим легочным сердцем II и IV ФК было примерно одинаковым в группах. Больных с III ФК было больше во 2-й группе (36,5 %; р < 0,05), I ФК – в 1-й группе (17,8 %; р < 0,05). ХСН I стадии чаще выявлялась в 1-й группе (31,9 %; р < 0,05), IIА и IIБ стадий – во 2-й группе (51,6 и 26,4 % соответственно; р < 0,05). Более чем у1 / 2 пациентов 3-й группы (57,3 %) была диагностирована IIА стадия ХСН. По большинству клинических проявлений у больных 2-й группы ХСН была более тяжелой. Дистанция в 6-МШТ составила 308,08 ± 6,03 м в 1-й группе, 156,4 ± 4,38 м во 2-й группе и 212,8 ± 5,65 м в 3й группе (р < 0,05). У больных с сочетанием ХОБЛ и ишемической болезни сердца (ИБС) низкая фракция выброса левого желудочка (ЛЖ) выявлялась достоверно чаще. В группах больных ХОБЛ и с сочетанием ХОБЛ и ИБС среднее давление в легочной артерии (СрДЛА) было значимо выше (24,7 ± 0,41 и 29,9 ± 0,27 мм рт. ст. соответственно) по сравнению с 21,7 ± 0,07 мм рт. ст. в 3-й группе. Диастолическая дисфункция правого желудочка (ПЖ) выявлена у 154 (83,7 %) больных 1-й группы, 194 (91,9 %) больных 2-й группы и 72 (43,1 %) больных 3-й группы. Таким образом, у больных всех групп наблюдались процессы ремоделирования ЛЖ и ПЖ. У пациентов с ХОБЛ были более выражены признаки гипертрофии и дилатации ПЖ, у больных ИБС – увеличение левых отделов сердца, а также некоторое снижение показателей сократимости ЛЖ, при сочетании ХОБЛ и ИБС наиболее выраженным было увеличение размеров и толщины стенок обоих желудочков и снижение насосной функции ЛЖ

M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra

We show that the zeroth cohomology of M. Kontsevich's graph complex is isomorphic to the Grothendieck-Teichmueller Lie algebra grt_1. The map is explicitly described. This result has applications to deformation quantization and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber operad. They are parameterized by grt_1, up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) E_n operads may be expressed through the cohomology of a suitable graph complex. Our methods also give a second proof of a result of H. Furusho, stating that the pentagon equation for grt_1-elements implies the hexagon equation

Symmetry, Gravity and Noncommutativity

We review some aspects of the implementation of spacetime symmetries in noncommutative field theories, emphasizing their origin in string theory and how they may be used to construct theories of gravitation. The geometry of canonical noncommutative gauge transformations is analysed in detail and it is shown how noncommutative Yang-Mills theory can be related to a gravity theory. The construction of twisted spacetime symmetries and their role in constructing a noncommutative extension of general relativity is described. We also analyse certain generic features of noncommutative gauge theories on D-branes in curved spaces, treating several explicit examples of superstring backgrounds.Comment: 52 pages; Invited review article to be published in Classical and Quantum Gravity; v2: references adde

Effective Batalin--Vilkovisky theories, equivariant configuration spaces and cyclic chains

Kontsevich's formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifold M is L-infinity-quasi-isomorphic to its cohomology. The construction of the L-infinity map is given in terms of integrals of differential forms on configuration spaces of points in the upper half-plane. Here we consider configuration spaces of points in the disk and work equivariantly with respect to the rotation group. This leads to considering the differential graded Lie algebra of multivector fields endowed with a divergence operator. In the case of R^d with standard volume form, we obtain an L-infinity morphism of modules over this differential graded Lie algebra from cyclic chains of the algebra of functions to multivector fields. As a first application we give a construction of traces on algebras of functions with star-products associated with unimodular Poisson structures. The construction is based on the Batalin--Vilkovisky quantization of the Poisson sigma model on the disk and in particular on the treatment of its zero modes.Comment: 27 page