195 research outputs found
Box splines and the equivariant index theorem
In this article, we start to recall the inversion formula for the convolution
with the Box spline. The equivariant cohomology and the equivariant K-theory
with respect to a compact torus G of various spaces associated to a linear
action of G in a vector space M can be both described using some vector spaces
of distributions, on the dual of the group G or on the dual of its Lie algebra.
The morphism from K-theory to cohomology is analyzed and the multiplication by
the Todd class is shown to correspond to the operator (deconvolution) inverting
the semidiscrete convolution with a box spline. Finally, the multiplicities of
the index of a G-transversally elliptic operator on M are determined using the
infinitesimal index of the symbol.Comment: 44 page
Polynomial Relations in the Centre of U_q(sl(N))
When the parameter of deformation q is a m-th root of unity, the centre of
U_q(sl(N))$ contains, besides the usual q-deformed Casimirs, a set of new
generators, which are basically the m-th powers of all the Cartan generators of
U_q(sl(N)). All these central elements are however not independent. In this
letter, generalising the well-known case of U_q(sl(2)), we explicitly write
polynomial relations satisfied by the generators of the centre. Application to
the parametrization of irreducible representations and to fusion rules are
sketched.Comment: 8 pages, minor TeXnical revision to allow automatic TeXin
Braid Group Action and Quantum Affine Algebras
We lift the lattice of translations in the extended affine Weyl group to a
braid group action on the quantum affine algebra. This action fixes the
Heisenberg subalgebra pointwise. Loop like generators are found for the algebra
which satisfy the relations of Drinfeld's new realization. Coproduct
formulas are given and a PBW type basis is constructed.Comment: 15 page
Centre and Representations of U_q(sl(2|1)) at Roots of Unity
Quantum groups at roots of unity have the property that their centre is
enlarged. Polynomial equations relate the standard deformed Casimir operators
and the new central elements. These relations are important from a physical
point of view since they correspond to relations among quantum expectation
values of observables that have to be satisfied on all physical states. In this
paper, we establish these relations in the case of the quantum Lie superalgebra
U_q(sl(2|1)). In the course of the argument, we find and use a set of
representations such that any relation satisfied on all the representations of
the set is true in U_q(sl(2|1)). This set is a subset of the set of all the
finite dimensional irreducible representations of U_q(sl(2|1)), that we
classify and describe explicitly.Comment: Minor corrections, References added. LaTeX2e, 18 pages, also
available at http://lapphp0.in2p3.fr/preplapp/psth/ENSLAPP583.ps.gz . To
appear in J. Phys. A: Math. Ge
Colored Vertex Models, Colored IRF Models and Invariants of Trivalent Colored Graphs
We present formulas for the Clebsch-Gordan coefficients and the Racah
coefficients for the root of unity representations (-dimensional
representations with ) of . We discuss colored vertex
models and colored IRF (Interaction Round a Face) models from the color
representations of . We construct invariants of trivalent colored
oriented framed graphs from color representations of .Comment: 39 pages, January 199
Hidden Symmetry of the Differential Calculus on the Quantum Matrix Space
A standard bicovariant differential calculus on a quantum matrix space is considered. The principal result of this work is in observing
that the is in fact a
-module differential algebra.Comment: 5 page
Permutonestohedra
There are several real spherical models associated with a root arrangement, depending on the choice of a building set. The connected components of these models are manifolds with corners which can be glued together to obtain the corresponding real De Concini–Procesi models. In this paper, starting from any root system with finite Coxeter group W and any W -invariant building set, we describe an explicit realization of the real spherical model as a union of polytopes (nestohedra) that lie inside the chambers of the arrangement. The main point of this realization is that the convex hull of these nestohedra is a larger polytope, a permutonestohedron, equipped with an action of W or also, depending on the building set, of Aut ( ). The permutonestohedra are natural generalizations of Kapranov’s permutoassociahedra
Polynomial Realization of and Fusion Rules at Exceptional Values of
Representations of the algebra are constructed in the space of
polynomials of real (complex) variable for . The spin addition rule
based on eigenvalues of Casimir operator is illustrated on few simplest cases
and conjecture for general case is formulated
Representations of U_q(sl(N)) at Roots of Unity
The Gelfand--Zetlin basis for representations of is improved to
fit better the case when is a root of unity. The usual -deformed
representations, as well as the nilpotent, periodic (cyclic), semi-periodic
(semi-cyclic) and also some atypical representations are now described with the
same formalism.Comment: 18 pages, Plain TeX, Macros harvmac.tex and epsf needed 3 figures in
a uuencoded tar separate file. Some references are added. Also available at
http://lapphp0.in2p3.fr/preplapp/psth/uqsln.ps.g
Composition of Kinetic Momenta: The U_q(sl(2)) case
The tensor products of (restricted and unrestricted) finite dimensional
irreducible representations of \uq are considered for a root of unity.
They are decomposed into direct sums of irreducible and/or indecomposable
representations.Comment: 27 pages, harvmac and tables macros needed, minor TeXnical revision
to allow automatic TeXin
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