1,188 research outputs found
Affine Hecke algebras of type D and generalisations of quiver Hecke algebras
We define and study cyclotomic quotients of affine Hecke algebras of type D.
We establish an isomorphism between (direct sums of blocks of) these cyclotomic
quotients and a generalisation of cyclotomic quiver Hecke algebras which are a
family of Z-graded algebras closely related to algebras introduced by Shan,
Varagnolo and Vasserot. To achieve this, we first complete the study of
cyclotomic quotients of affine Hecke algebras of type B by considering the
situation when a deformation parameter p squares to 1. We then relate the two
generalisations of quiver Hecke algebras showing that the one for type D can be
seen as fixed point subalgebras of their analogues for type B, and we carefully
study how far this relation remains valid for cyclotomic quotients. This allows
us to obtain the desired isomorphism. This isomorphism completes the family of
isomorphisms relating affine Hecke algebras of classical types to
(generalisations of) quiver Hecke algebras, originating in the famous result of
Brundan and Kleshchev for the type A.Comment: 26 page
Airy, Beltrami, Maxwell, Morera, Einstein and Lanczos potentials revisited
The main purpose of this paper is to revisit the well known potentials,
called stress functions, needed in order to study the parametrizations of the
stress equations, respectively provided by G.B. Airy (1863) for 2-dimensional
elasticity, then by E. Beltrami (1892), J.C. Maxwell (1870) and G. Morera
(1892) for 3-dimensional elasticity, finally by A. Einstein (1915) for
4-dimensional elasticity, both with a variational procedure introduced by C.
Lanczos (1949,1962) in order to relate potentials to Lagrange multipliers.
Using the methods of Algebraic Analysis, namely mixing differential geometry
with homological algebra and combining the double duality test involved with
the Spencer cohomology, we shall be able to extend these results to an
arbitrary situation with an arbitrary dimension n. We shall also explain why
double duality is perfectly adapted to variational calculus with differential
constraints as a way to eliminate the corresponding Lagrange multipliers. For
example, the canonical parametrization of the stress equations is just
described by the formal adjoint of the n2(n2 -- 1)/12 components of the
linearized Riemann tensor considered as a linear second order differential
operator but the minimum number of potentials needed in elasticity theory is
equal to n(n -- 1)/2 for any minimal parametrization. Meanwhile, we can provide
all the above results without even using indices for writing down explicit
formulas in the way it is done in any textbook today. The example of
relativistic continuum mechanics with n = 4 is provided in order to prove that
it could be strictly impossible to obtain such results without using the above
methods. We also revisit the possibility (Maxwell equations of electromag-
netism) or the impossibility (Einstein equations of gravitation) to obtain
canonical or minimal parametrizations for various other equations of physics.
It is nevertheless important to notice that, when n and the algorithms
presented are known, most of the calculations can be achieved by using
computers for the corresponding symbolic computations. Finally, though the
paper is mathematically oriented as it aims providing new insights towards the
mathematical foundations of elasticity theory and mathematical physics, it is
written in a rather self-contained way
Homalg: A meta-package for homological algebra
The central notion of this work is that of a functor between categories of
finitely presented modules over so-called computable rings, i.e. rings R where
one can algorithmically solve inhomogeneous linear equations with coefficients
in R. The paper describes a way allowing one to realize such functors, e.g.
Hom, tensor product, Ext, Tor, as a mathematical object in a computer algebra
system. Once this is achieved, one can compose and derive functors and even
iterate this process without the need of any specific knowledge of these
functors. These ideas are realized in the ring independent package homalg. It
is designed to extend any computer algebra software implementing the
arithmetics of a computable ring R, as soon as the latter contains algorithms
to solve inhomogeneous linear equations with coefficients in R. Beside
explaining how this suffices, the paper describes the nature of the extensions
provided by homalg.Comment: clarified some points, added references and more interesting example
Gabor analysis over finite Abelian groups
The topic of this paper are (multi-window) Gabor frames for signals over
finite Abelian groups, generated by an arbitrary lattice within the finite
time-frequency plane. Our generic approach covers simultaneously
multi-dimensional signals as well as non-separable lattices. The main results
reduce to well-known fundamental facts about Gabor expansions of finite signals
for the case of product lattices, as they have been given by Qiu, Wexler-Raz or
Tolimieri-Orr, Bastiaans and Van-Leest, among others. In our presentation a
central role is given to spreading function of linear operators between
finite-dimensional Hilbert spaces. Another relevant tool is a symplectic
version of Poisson's summation formula over the finite time-frequency plane. It
provides the Fundamental Identity of Gabor Analysis.In addition we highlight
projective representations of the time-frequency plane and its subgroups and
explain the natural connection to twisted group algebras. In the
finite-dimensional setting these twisted group algebras are just matrix
algebras and their structure provides the algebraic framework for the study of
the deeper properties of finite-dimensional Gabor frames.Comment: Revised version: two new sections added, many typos fixe
Banach algebras of pseudodifferential operators and their almost diagonalization
We define new symbol classes for pseudodifferntial operators and investigate
their pseudodifferential calculus. The symbol classes are parametrized by
commutative convolution algebras. To every solid convolution algebra over a
lattice we associate a symbol class. Then every operator with such a symbol is
almost diagonal with respect to special wave packets (coherent states or Gabor
frames), and the rate of almost diagonalization is described precisely by the
underlying convolution algebra. Furthermore, the corresponding class of
pseudodifferential operators is a Banach algebra of bounded operators on . If a version of Wiener's lemma holds for the underlying convolution algebra,
then the algebra of pseudodifferential operators is closed under inversion. The
theory contains as a special case the fundamental results about Sj\"ostrand's
class and yields a new proof of a theorem of Beals about the H\"ormander class
of order 0.Comment: 28 page
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