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Pavel Pyatov - One of the best experts on this subject based on the ideXlab platform.

  • cayley hamilton Theorem for symplectic quantum matrix algebras
    arXiv: Quantum Algebra, 2020
    Co-Authors: Oleg Ogievetsky, Pavel Pyatov
    Abstract:

    We establish the analogue of the Cayley--Hamilton Theorem for the quantum matrix algebras of the symplectic type.

  • spectral extension of the quantum group cotangent bundle
    Communications in Mathematical Physics, 2009
    Co-Authors: A P Isaev, Pavel Pyatov
    Abstract:

    The structure of a cotangent bundle is investigated for quantum linear groups GL q (n) and SL q (n). Using a q-version of the Cayley-Hamilton Theorem we construct an extension of the algebra of differential operators on SL q (n) (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators—the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive SL q (n) type dynamical R-matrices in a surprisingly simple way. Second, we calculate an evolution operator for the model of the q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. The relation between the two operators is given by a modular functional equation for the Riemann theta function.

  • spectral extension of the quantum group cotangent bundle
    arXiv: Quantum Algebra, 2008
    Co-Authors: A P Isaev, Pavel Pyatov
    Abstract:

    The structure of a cotangent bundle is investigated for quantum linear groups GLq(n) and SLq(n). Using a q-version of the Cayley-Hamilton Theorem we construct an extension of the algebra of differential operators on SLq(n) (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators -- the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive SLq(n) type dynamical R-matrices in a surprisingly simple way. Second, we calculate an evolution operator for the model of q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. Relation between the two operators is given by a modular functional equation for Riemann theta function.

  • orthogonal and symplectic quantum matrix algebras and cayley hamilton Theorem for them
    arXiv: Quantum Algebra, 2005
    Co-Authors: Oleg Ogievetsky, Pavel Pyatov
    Abstract:

    For families of orthogonal and symplectic types quantum matrix (QM-) algebras, we derive corresponding versions of the Cayley-Hamilton Theorem. For a wider family of Birman-Murakami-Wenzl type QM-algebras, we investigate a structure of its characteristic subalgebra (the subalgebra in which the coefficients of characteristic polynomials take values). We define 3 sets of generating elements of the characteristic subalgebra and derive recursive Newton and Wronski relations between them. For the family of the orthogonal type QM-algebras, additional reciprocal relations for the generators of the characteristic subalgebra are obtained.

Wim Michiels - One of the best experts on this subject based on the ideXlab platform.

  • on the m dimensional cayley hamilton Theorem and its application to an algebraic decision problem inferred from the h2 norm analysis of delay systems
    Automatica, 2020
    Co-Authors: Marco A Gomez, Raphael M Jungers, Wim Michiels
    Abstract:

    Abstract We consider a recursion formula for multi-dimensional powers of a finite set of matrices, which can be interpreted as a natural generalization of the celebrated Cayley–Hamilton Theorem, and we show how it allows to solve an algebraic decision problem on a semigroup of matrices, which bears similarities to the observability problem of a switched linear system. This problem appears in the computation of the H 2 norm of a stable system described by a class of linear time-invariant delay differential equations (DDAEs) with multiple delays. The H 2 norm of a DDAE may not be finite even if there are seemingly no direct feedthrough terms. We show that necessary and sufficient conditions for a finite H 2 norm consist of an infinite number of linear equations to be satisfied, inducing the algebraic decision problem, and that using the generalized Cayley–Hamilton Theorem checking these conditions can be turned into a check of a finite number of equations. We conclude with some comments on the computation of the H 2 norm whenever it is finite and by stating an open problem.

Roald Hoffmann - One of the best experts on this subject based on the ideXlab platform.

  • quantum interference graphs walks and polynomials
    Chemical Reviews, 2018
    Co-Authors: Yuta Tsuji, Ernesto Estrada, Ramis Movassagh, Roald Hoffmann
    Abstract:

    In this paper, we explore quantum interference (QI) in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley–Hamilton Theorem for characteristic polynomials and the Coulson–Rushbrooke pairing Theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green’s function for electron transmission in terms of the odd powers of the vertex adjacency matrix or Huckel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are canceled by others, leading to another class of quantum interference. For nonalternant hydrocarbons, the finite Green’s function expansion may include...

  • Quantum Interference, Graphs, Walks, and Polynomials
    2018
    Co-Authors: Yuta Tsuji, Ernesto Estrada, Ramis Movassagh, Roald Hoffmann
    Abstract:

    In this paper, we explore quantum interference (QI) in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley–Hamilton Theorem for characteristic polynomials and the Coulson–Rushbrooke pairing Theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green’s function for electron transmission in terms of the odd powers of the vertex adjacency matrix or Hückel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are canceled by others, leading to another class of quantum interference. For nonalternant hydrocarbons, the finite Green’s function expansion may include both even and odd powers. Nevertheless, QI can in some circumstances come about for nonalternants from cancellation of odd- and even-length walk terms. We report some progress, but not a complete resolution, of the problem of understanding the coefficients in the expansion of the Green’s function in a power series of the adjacency matrix, these coefficients being behind the cancellations that we have mentioned. Furthermore, we introduce a perturbation theory for transmission as well as some potentially useful infinite power series expansions of the Green’s function

Marco A Gomez - One of the best experts on this subject based on the ideXlab platform.

  • on the m dimensional cayley hamilton Theorem and its application to an algebraic decision problem inferred from the h2 norm analysis of delay systems
    Automatica, 2020
    Co-Authors: Marco A Gomez, Raphael M Jungers, Wim Michiels
    Abstract:

    Abstract We consider a recursion formula for multi-dimensional powers of a finite set of matrices, which can be interpreted as a natural generalization of the celebrated Cayley–Hamilton Theorem, and we show how it allows to solve an algebraic decision problem on a semigroup of matrices, which bears similarities to the observability problem of a switched linear system. This problem appears in the computation of the H 2 norm of a stable system described by a class of linear time-invariant delay differential equations (DDAEs) with multiple delays. The H 2 norm of a DDAE may not be finite even if there are seemingly no direct feedthrough terms. We show that necessary and sufficient conditions for a finite H 2 norm consist of an infinite number of linear equations to be satisfied, inducing the algebraic decision problem, and that using the generalized Cayley–Hamilton Theorem checking these conditions can be turned into a check of a finite number of equations. We conclude with some comments on the computation of the H 2 norm whenever it is finite and by stating an open problem.

A P Isaev - One of the best experts on this subject based on the ideXlab platform.

  • spectral extension of the quantum group cotangent bundle
    Communications in Mathematical Physics, 2009
    Co-Authors: A P Isaev, Pavel Pyatov
    Abstract:

    The structure of a cotangent bundle is investigated for quantum linear groups GL q (n) and SL q (n). Using a q-version of the Cayley-Hamilton Theorem we construct an extension of the algebra of differential operators on SL q (n) (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators—the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive SL q (n) type dynamical R-matrices in a surprisingly simple way. Second, we calculate an evolution operator for the model of the q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. The relation between the two operators is given by a modular functional equation for the Riemann theta function.

  • spectral extension of the quantum group cotangent bundle
    arXiv: Quantum Algebra, 2008
    Co-Authors: A P Isaev, Pavel Pyatov
    Abstract:

    The structure of a cotangent bundle is investigated for quantum linear groups GLq(n) and SLq(n). Using a q-version of the Cayley-Hamilton Theorem we construct an extension of the algebra of differential operators on SLq(n) (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators -- the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive SLq(n) type dynamical R-matrices in a surprisingly simple way. Second, we calculate an evolution operator for the model of q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. Relation between the two operators is given by a modular functional equation for Riemann theta function.