The Power Matrix, coadjoint action and quadratic differentials

The coefficients of a quadratic differential which is changing under the Loewner flow satisfy a well-known differential system studied by Schiffer, Schaeffer and Spencer, and others. By work of Roth, this differential system can be interpreted as Hamilton's equations. We apply the power matrix to interpret this differential system in terms of the coadjoint action of the matrix group on the dual of its Lie algebra. As an application, we derive a set of integral invariants of Hamilton's equations which is in a certain sense complete. In function theoretic terms, these are expressions in the coefficients of the quadratic differential and Loewner map which are independent of the parameter in the Loewner flow.     

Transition matrices for quantum waveguides with impurities

We consider the electron propagation in a cylindrical quantum waveguide where D is a bounded domain in described by the Dirichlet problem for the Schrödinger operator where x=(x₁, x₂), , is the transversal confinement potential, and is the impurity potential.  We construct the left and right transition matrices and give an numerical algorithm for their calculations based on the spectral parameter power series method.     

Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra

The paper develops a Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra and analyzes the resulting monodromy representation of the Weyl group.Oblatum 9-IX-1993 & 15-IV-1995The author is supported by a grant from NSERC Canada.     

Anderson localization for quasi-periodic CMV matrices and quantum walks

We consider CMV matrices, both standard and extended, with analytic quasi-periodic Verblunsky coefficients and prove Anderson localization in the regime of positive Lyapunov exponents. This establishes the CMV analog of a result Bourgain and Goldstein proved for discrete one-dimensional Schrödinger operators. We also prove a similar result for quantum walks on the integer lattice with suitable analytic quasi-periodic coins.     

Scattering matrices for the quantum body problem

Let be a generalized body Schrödinger operator with very short range potentials. Using Melrose's scattering calculus, it is shown that the free channel `geometric' scattering matrix, defined via asymptotic expansions of generalized eigenfunctions of , coincides (up to normalization) with the free channel `analytic' scattering matrix defined via wave operators. Along the way, it is shown that the free channel generalized eigenfunctions of Herbst-Skibsted and Jensen-Kitada coincide with the plane waves constructed by Hassell and Vasy and if the potentials are very short range.

     

Coinvariants and the regular representation of a cyclic P-group

We consider an indecomposable representation of a cyclic p-group ${Z_{p^r}}$ over a field of characteristic p. We show that the top degree of the corresponding ring of coinvariants is less than ${\frac{(r^2+3r)p^r}{2}}$ . This bound also applies to the degrees of the generators for the invariant ring of the regular representation.     

Picard Groups of Rings of Coinvariants

Let k be a field, H a Hopf k-algebra with bijective antipode, A a right H-comodule algebra and C a Hopf algebra with bijective antipode which is also a right H-module coalgebra. Under some appropriate assumptions, and assuming that the set of grouplike elements G(A ⊗ C) of the coring A ⊗ C is a group, we show how to calculate, via an exact sequence, the Picard group of the subring of coinvariants in terms of the Picard group of A and various subgroups of G(A ⊗ C). Presented by: Claus Ringel.     

Kirwan polyhedron of holomorphic coadjoint orbits

Let G be a simple, noncompact, connected, real Lie group with finite center, and K a maximal compact subgroup of G. We assume that G/K is Hermitian. Using GIT methods derived from the generalized eigenvalue problem, we compute a set of inequalities describing the moment polyhedron of the projection G ? ?? $ \subset \;{{\mathfrak{g}}^*} \to {{\mathfrak{t}}^*} $ for holomorphic coadjoint orbits of G.     

On the Dimension of Coinvariants of Permutation Representations

Let be a univariate, separable polynomial of degree n with roots x ₁,…,x _n in some algebraic closure of the ground field . It is a classical problem of Galois theory to find all the relations between the roots. It is known that the ideal of all such relations is generated by polynomials arising from G-invariant polynomials, where G is the Galois group of f(Z). Namely: The action of G on the ordered set of roots induces an action on by permutation of the coordinates and each defines a relation P − P(x ₁,…,x _n ) called a G-invariant relation. These generate the ideal of all relations. In this note we show that the ideal of relations admits an H-basis of G-invariant relations if and only if the algebra of coinvariants has dimension ‖G‖ over . To complete the picture we then show that the coinvariant algebra of a transitive permutation representation of a finite group G has dimension ‖G‖ if and only if G = Σ _n acting via the tautological permutation representation.     

Spectral synthesis for coadjoint orbits of nilpotent Lie groups

We determine the space of primary ideals in the group algebra \(L^{1}(G) \) of a connected nilpotent Lie group by identifying for every \(\pi \in \widehat{G} \) the family \(\mathcal I^\pi \) of primary ideals with hull \(\{\pi \} \) with the family of invariant subspaces of a certain finite dimensional sub-space \(\mathcal P_Q^\pi \) of the space of polynomials \(\mathcal P(G) \) on G.     

Accelerated method for optimization over density matrices in quantum state estimation

In this paper, a class of minimization problems over density matrices arising in the quantum state estimation is investigated. By making use of the Nesterov’s accelerated strategies, we introduce a modified augmented Lagrangian method to solve it, where the subproblem is tackled by the projected Barzilai–Borwein method with nonmonotone line search. Compared with the existing projected gradient method, several numerical examples are tested to show that the proposed method is efficient and promising.     

Investigation of the geodesic flow on an infinite-dimensional Lie group by means of the coadjoint action operator

We consider a representation of the Euler equations as the geodesic flow on an infinite-dimensional Lie group. In these terms, we establish properties of solutions, which are provided by local existence and uniqueness theorems, at a limit point.     

Entropy,stochastic matrices,and quantum operations

The goal of the present paper is to derive some conditions on saturation of (strong) subadditivity inequality for the stochastic matrices. The notion of relative entropy of stochastic matrices is introduced by mimicking quantum relative entropy. Some properties of this concept are listed, and the connection between the entropy of the stochastic quantum operations and that of stochastic matrices are discussed.