Central extensions of current groups in two dimensions

In this paper we generalize some of these results for loop algebras and groups as well as for the Virasoro algebra to the two-dimensional case. We define and study a class of infinite dimensional complex Lie groups which are central extensions of the group of smooth maps from a two dimensional orientable surface without boundary to a simple complex Lie group G. These extensions naturally correspond to complex curves. The kernel of such an extension is the Jacobian of the curve. The study of the coadjoint action shows that its orbits are labelled by moduli of holomorphic principal G-bundles over the curve and can be described in the language of partial differential equations. In genus one it is also possible to describe the orbits as conjugacy classes of the twisted loop group, which leads to consideration of difference equations for holomorphic functions. This gives rise to a hope that the described groups should possess a counterpart of the rich representation theory that has been developed for loop groups. We also define a two-dimensional analogue of the Virasoro algebra associated with a complex curve. In genus one, a study of a complex analogue of Hill's operator yields a description of invariants of the coadjoint action of this Lie algebra. The answer turns out to be the same as in dimension one: the invariants coincide with those for the extended algebra of currents in sl(2).Comment: 17 page

Manin matrices and Talalaev's formula

We study special class of matrices with noncommutative entries and demonstrate their various applications in integrable systems theory. They appeared in Yu. Manin's works in 87-92 as linear homomorphisms between polynomial rings; more explicitly they read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: $[M_{ij}, M_{kl}]=[M_{kj}, M_{il}]$ (e.g. $[M_{11}, M_{22}]=[M_{21}, M_{12}]$). We claim that such matrices behave almost as well as matrices with commutative elements. Namely theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, the Newton identities and so on and so forth) holds true for them. On the other hand, we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability. For instance, Manin matrices (and their q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and the so--called Cartier-Foata matrices. Also, they enter Talalaev's hep-th/0404153 remarkable formulas: $det(\partial_z-L_{Gaudin}(z))$, det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show that theorems of linear algebra, after being established for such matrices, have various applications to quantum integrable systems and Lie algebras, e.g in the construction of new generators in $Z(U(\hat{gl_n}))$ (and, in general, in the construction of quantum conservation laws), in the Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also discuss applications to the separation of variables problem, new Capelli identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints e.g. in Newton id-s fixed, normal ordering convention turned to standard one, refs. adde

Current Trends in Moscow Settlement Pattern Development: A Multiscale Approach

The article studies current trends in Moscow population in context of socio- economic polarization strengthening between the capital city and other regions of the country. The study applies multiscale approach covering Moscow influence on Central Russia and other regions, interaction with the Moscow oblast and the level of internal population distribution within Moscow and particular settlements and villages in New Moscow territories. The gap in development is significantly noticeable for expanding Moscow and Moscow oblast against the background of depopulation in Central Russia regions and cities. Within the boundaries of Moscow the continuing model of extensive spatial growth of population has led to the most rapid growth of its periphery zone. Areas similar to bedroom communities in Old Moscow are forming in the municipalities of New Moscow located along the Moscow ring road (MKAD) and main radial highways, while large part of the new territories remain a typical countryside with villages and summer residents. Analysis of New Moscow suburban areas reveals the actual land use mosaics obscured by the official delimitation of Moscow and Moscow oblast and the formal division of population into urban and rural

TRANSFORMATION OF ENVIRONMENTAL PROBLEMS IN MOSCOW: SOCIOLOGICAL DIMENSION

The paper assesses transformation of environmental situation in Moscow and citizens’ attitude toward those changes. It analyzes a mass poll of 800 Moscovites conducted in June–July 2015. The research was aimed at identifying the correlation between subjective perception of residents and objective spatial and environmental differentiation in Moscow as well as assessing the potential of Moscovites’ involvement in solution of environmental problems. Air pollution caused by production enterprises and cars, solid household waste and waste incineration plants were given special consideration. The article analyzes how Moscovites perceive problems of the whole city and of their own districts