246 research outputs found
A categorical approach to Weyl modules
Global and local Weyl Modules were introduced via generators and relations in
the context of affine Lie algebras in a work by the first author and Pressley
and were motivated by representations of quantum affine algebras. A more
general case was considered by Feigin and Loktev by replacing the polynomial
ring with the coordinate ring of an algebraic variety. We show that there is a
natural definition of the local and global modules via homological properties.
This characterization allows us to define the Weyl functor from the category of
left modules of a commutative algebra to the category of modules for a simple
Lie algebra. As an application we are able to understand the relationships of
these functors to tensor products, generalizing previous results. Finally an
analysis of the fundamental Weyl modules proves that the functors are not left
exact in general, even for coordinate rings of affine varieties.Comment: 29 page
Mass function of haloes: scale invariant models
Press-Schechter theory gives a simple, approximate functional form of the
mass function of dark matter haloes. Sheth and Tormen (ST) refined this mass
function to give an improved analytical fit to results of N-body simulations.
These forms of the halo mass function are universal (independent of cosmology
and power spectrum) when scaled in suitable variables. Using large suites of
LCDM N-body simulations, studies in the last few years have shown that this
universality is only approximate. We explore whether some of the deviations
from universality can be attributed to the power spectrum by computing the mass
function in N-body simulations of various scale-free models in an Einstein-de
Sitter cosmology. This choice of cosmology does not introduce any scale into
the problem. These models have the advantage of being self-similar, hence
stringent checks can be imposed while running these simulations. This set of
numerical experiments is designed to isolate any power spectrum dependent
departures from universality of mass functions. We show explicitly that the
best fit ST parameters have a clear dependence on power spectrum. Our results
also indicate that an improved analytical theory with more parameters is
required in order to provide better fits to the mass function.Comment: 8 pages, four figure
High redshift supermassive blackholes: accretion through cold flows
We use zoom-in techniques to re-simulate three high-redshift (z > 5.5) halos
which host 10^9 solar mass blackholes from the ~ Gpc volume, MassiveBlack
cosmological hydrodynamic simulation. We examine a number of factors
potentially affecting supermassive blackhole growth at high redshift in
cosmological simulations. These include numerical resolution, feedback
prescriptions and formulation of smoothed particle hydrodynamics. We find that
varying the size of the region over which feedback energy is deposited
directly, either for fixed number of neighbours or fixed volume makes very
little difference to the accretion history of blackholes. Changing mass
resolution by factors of up to 64 also does not change the blackhole growth
history significantly. We find that switching from the density-entropy
formulation to the pressure-entropy formulation of smoothed particle
hydrodynamics slightly increases the accretion rate onto blackholes. In general
numerical details appear to have small effects on the main fueling mechanism
for blackholes at these high redshifts. We examine the fashion by which this
occurs, finding that the insensitivity to simulation technique seems to be a
hallmark of the cold flow feeding picture of these high-z supermassive
blackholes. We show that the gas that participates in critical accretion
phases, in these massive objects at z > 6~7 is in all cases colder, denser, and
forms more coherent streams than the average gas in the halo. This is also
mostly the case when the blackhole accretion is feedback regulated (z < 6),
however the distinction is less prominent. For our resimulated halos, cold
flows appear to be a viable mechanism for forming the most massive blackholes
in the early universe, occurring naturally in LambdaCDM models of structure
formation. Not requiring fine tuning of numerical parameters, they seem to be
physically inevitable in these objects.Comment: 15 pages, 12 figure
Demazure Filtrations of Tensor Product Modules and Character Formula
We study the structure of the finite-dimensional representations of
, the current Lie algebra type of , which are obtained
by taking tensor products of special Demazure modules. We show that these
representations admit a Demazure flag and obtain a closed formula for the
graded multiplicities of the level 2 Demazure modules in the filtration of the
tensor product of two local Weyl modules for . Furthermore,
we derive an explicit expression for graded character of the tensor product of
a local Weyl module with an irreducible module. In
conjunction with the results of \cite{MR3210603}, our findings provide evidence
for the conjecture in \cite{9} that the tensor product of Demazure modules of
levels m and n respectively has a filtration by Demazure modules of level m +
n
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