368 research outputs found
Complete families of commuting functions for coisotropic Hamiltonian actions
Let G be an algebraic group over a field F of characteristic zero, with Lie
algebra g=Lie(G). The dual space g^* equipped with the Kirillov bracket is a
Poisson variety and each irreducible G-invariant subvariety X\subset g^*
carries the induced Poisson structure. We prove that there is a family of
algebraically independent polynomial functions {f_1,...f_l} on X, which
pairwise commute with respect to the Poisson bracket and such that l=(dim
X+tr.deg F(X)^G)/2. We also discuss several applications of this result to
complete integrability of Hamiltonian systems on symplectic Hamiltonian
G-varieties of corank zero and 2.Comment: Changed presentatio
Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras
This is a continuation of our "Lecture on Kac--Moody Lie algebras of the
arithmetic type" \cite{25}.
We consider hyperbolic (i.e. signature ) integral symmetric bilinear
form (i.e. hyperbolic lattice), reflection group
, fundamental polyhedron \Cal M of and an acceptable
(corresponding to twisting coefficients) set P({\Cal M})\subset M of vectors
orthogonal to faces of \Cal M (simple roots). One can construct the
corresponding Lorentzian Kac--Moody Lie algebra {\goth g}={\goth
g}^{\prime\prime}(A(S,W,P({\Cal M}))) which is graded by .
We show that \goth g has good behavior of imaginary roots, its denominator
formula is defined in a natural domain and has good automorphic properties if
and only if \goth g has so called {\it restricted arithmetic type}. We show
that every finitely generated (i.e. P({\Cal M}) is finite) algebra {\goth
g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1))) may be embedded to {\goth
g}^{\prime\prime}(A(S,W,P({\Cal M}))) of the restricted arithmetic type. Thus,
Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a
natural class to study.
Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the
best automorphic properties for the denominator function if they have {\it a
lattice Weyl vector }. Lorentzian Kac--Moody Lie algebras of the
restricted arithmetic type with generalized lattice Weyl vector are
called {\it elliptic}Comment: Some corrections in Sects. 2.1, 2.2 were done. They don't reflect on
results and ideas. 31 pages, no figures. AMSTe
Noiseless Quantum Circuits for the Peres Separability Criterion
In this Letter we give a method for constructing sets of simple circuits that
can determine the spectrum of a partially transposed density matrix, without
requiring either a tomographically complete POVM or the addition of noise to
make the spectrum non-negative. These circuits depend only on the dimension of
the Hilbert space and are otherwise independent of the state.Comment: 4 pages RevTeX, 7 figures encapsulated postscript. v5: title changed
slightly, more-or-less equivalent to the published versio
Triangular de Rham Cohomology of Compact Kahler Manifolds
We study the de Rham 1-cohomology H^1_{DR}(M,G) of a smooth manifold M with
values in a Lie group G. By definition, this is the quotient of the set of flat
connections in the trivial principle bundle by the so-called gauge
equivalence. We consider the case when M is a compact K\"ahler manifold and G
is a solvable complex linear algebraic group of a special class which contains
the Borel subgroups of all complex classical groups and, in particular, the
group of all triangular matrices. In this case, we get a
description of the set H^1_{DR}(M,G) in terms of the 1-cohomology of M with
values in the (abelian) sheaves of flat sections of certain flat Lie algebra
bundles with fibre (the Lie algebra of G) or, equivalently, in terms
of the harmonic forms on M representing this cohomology
Affine spherical homogeneous spaces with good quotient by a maximal unipotent subgroup
For an affine spherical homogeneous space G/H of a connected semisimple
algebraic group G, we consider the factorization morphism by the action on G/H
of a maximal unipotent subgroup of G. We prove that this morphism is
equidimensional if and only if the weight semigroup of G/H satisfies some
simple condition.Comment: v2: title and abstract changed; v3: 16 pages, minor correction
Cyclic elements in semisimple lie algebras
We develop a theory of cyclic elements in semisimple Lie algebras. This notion was introduced by Kostant, who associated a cyclic element with the principal nilpotent and proved that it is regular semisimple. In particular, we classfiy all nilpotents giving rise to semisimple and regular semisimple cyclic elements. As an application, we obtain an explicit construction of all regular elements in Weyl groups
Hypermatrix factors for string and membrane junctions
The adjoint representations of the Lie algebras of the classical groups
SU(n), SO(n), and Sp(n) are, respectively, tensor, antisymmetric, and symmetric
products of two vector spaces, and hence are matrix representations. We
consider the analogous products of three vector spaces and study when they
appear as summands in Lie algebra decompositions. The Z3-grading of the
exceptional Lie algebras provide such summands and provides representations of
classical groups on hypermatrices. The main natural application is a formal
study of three-junctions of strings and membranes. Generalizations are also
considered.Comment: 25 pages, 4 figures, presentation improved, minor correction
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