Bounding the dimensions of rational cohomology groups

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple simply-connected algebraic group over $k$ that is defined and split over the prime field $\mathbb{F}_p$. In this paper we investigate situations where the dimension of a rational cohomology group for $G$ can be bounded by a constant times the dimension of the coefficient module. We then demonstrate how our results can be applied to obtain effective bounds on the first cohomology of the symmetric group. We also show how, for finite Chevalley groups, our methods permit significant improvements over previous estimates for the dimensions of second cohomology groups.Comment: 13 page

On dual canonical bases

The dual basis of the canonical basis of the modified quantized enveloping algebra is studied, in particular for type $A$. The construction of a basis for the coordinate algebra of the $n\times n$ quantum matrices is appropriate for the study the multiplicative property. It is shown that this basis is invariant under multiplication by certain quantum minors including the quantum determinant. Then a basis of quantum SL(n) is obtained by setting the quantum determinant to one. This basis turns out to be equivalent to the dual canonical basis

Quantum spin coverings and statistics

SL_q(2) at odd roots of unity q^l =1 is studied as a quantum cover of the complex rotation group SO(3,C), in terms of the associated Hopf algebras of (quantum) polynomial functions. We work out the irreducible corepresentations, the decomposition of their tensor products and a coquasitriangular structure, with the associated braiding (or statistics). As an example, the case l=3 is discussed in detail.Comment: 15 page

Coherent States for Quantum Compact Groups

Coherent states are introduced and their properties are discussed for all simple quantum compact groups. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit and interpret the coherent state as a holomorphic function on this orbit with values in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra. Using Gauss decomposition, the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compact R--matrix formulation (generalizing this way the $q$--deformed Grassmann and flag manifolds). The antiholomorphic realization of the irreducible representations of a compact quantum group (the analogue of the Borel--Weil construction) are described using the concept of coherent state. The relation between representation theory and non--commutative differential geometry is suggested.}Comment: 25 page

Urban energy consumption and CO2 emissions in Beijing: current and future

This paper calculates the energy consumption and CO2 emissions of Beijing over 2005–2011 in light of the Beijing’s energy balance table and the carbon emission coefficients of IPCC. Furthermore, based on a series of energy conservation planning program issued in Beijing, the Long-range Energy Alternatives Planning System (LEAP)-BJ model is developed to study the energy consumption and CO2 emissions of Beijing’s six end-use sectors and the energy conversion sector over 2012–2030 under the BAU scenario and POL scenario. Some results are found in this research: (1) During 2005–2011, the energy consumption kept increasing, while the total CO2 emissions fluctuated obviously in 2008 and 2011. The energy structure and the industrial structure have been optimized to a certain extent. (2) If the policies are completely implemented, the POL scenario is projected to save 21.36 and 35.37 % of the total energy consumption and CO2 emissions than the BAU scenario during 2012 and 2030. (3) The POL scenario presents a more optimized energy structure compared with the BAU scenario, with the decrease of coal consumption and the increase of natural gas consumption. (4) The commerce and service sector and the energy conversion sector will become the largest contributor to energy consumption and CO2 emissions, respectively. The transport sector and the industrial sector are the two most potential sectors in energy savings and carbon reduction. In terms of subscenarios, the energy conservation in transport (TEC) is the most effective one. (5) The macroparameters, such as the GDP growth rate and the industrial structure, have great influence on the urban energy consumption and carbon emissions

The need to research refractory breathlessness

High-quality research is needed to improve quality of life for people with chronic refractory breathlessness in COP