The Experts below are selected from a list of 123 Experts worldwide ranked by ideXlab platform

Nicolás Andruskiewitsch - One of the best experts on this subject based on the ideXlab platform.

  • On the quiver-theoretical quantum Yang–Baxter equation
    Selecta Mathematica, 2005
    Co-Authors: Nicolás Andruskiewitsch
    Abstract:

    Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum Yang–Baxter equation, or more properly the braid equation, is investigated in this setting. A Solution of the braid equation in this category is called a “Solution” for short. Results of Etingof–Schedler–Soloviev, Lu–Yan–Zhu and Takeuchi on the set-theoretical quantum Yang–Baxter equation are generalized to the context of quivers, with groupoids playing the role of groups. The notion of “braided groupoid” is introduced. Braided groupoids are Solutions and are characterized in terms of bijective 1-cocycles. The structure groupoid of a non-Degenerate Solution is defined; it is shown that it is a braided groupoid. The reduced structure groupoid of a non-Degenerate Solution is also defined. Non-Degenerate Solutions are classified in terms of representations of matched pairs of groupoids. By linearization we construct star-triangular face models and realize them as modules over quasitriangular quantum groupoids introduced in papers by M. Aguiar, S. Natale and the author.

  • On the quiver-theoretical quantum Yang-Baxter equation
    Selecta Mathematica-new Series, 2005
    Co-Authors: Nicolás Andruskiewitsch
    Abstract:

    Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum Yang–Baxter equation, or more properly the braid equation, is investigated in this setting. A Solution of the braid equation in this category is called a “Solution” for short. Results of Etingof–Schedler–Soloviev, Lu–Yan–Zhu and Takeuchi on the set-theoretical quantum Yang–Baxter equation are generalized to the context of quivers, with groupoids playing the role of groups. The notion of “braided groupoid” is introduced. Braided groupoids are Solutions and are characterized in terms of bijective 1-cocycles. The structure groupoid of a non-Degenerate Solution is defined; it is shown that it is a braided groupoid. The reduced structure groupoid of a non-Degenerate Solution is also defined. Non-Degenerate Solutions are classified in terms of representations of matched pairs of groupoids. By linearization we construct star-triangular face models and realize them as modules over quasitriangular quantum groupoids introduced in papers by M. Aguiar, S. Natale and the author.

  • On the quiver-theoretical quantum Yang-Baxter equation
    arXiv: Quantum Algebra, 2004
    Co-Authors: Nicolás Andruskiewitsch
    Abstract:

    Quivers over a fixed base set form a monoidal category with tensor product given by pullback. The quantum Yang-Baxter equation, or more properly the braid equation, is investigated in this setting. A Solution of the braid equation in this category is called a "Solution" for short. Results of Etingof-Schedler-Soloviev, Lu-Yan-Zhu and Takeuchi on the set-theoretical quantum Yang-Baxter equation are generalized to the context of quivers, with groupoids playing the role of groups. The notion of "braided groupoid" is introduced. Braided groupoids are Solutions and are characterized in terms of bijective 1-cocycles. The structure groupoid of a non-Degenerate Solution is defined; it is shown that it is braided groupoid. The reduced structure groupoid of a non-Degenerate Solution is also defined. Non-Degenerate Solutions are classified in terms of representations of matched pairs of groupoids. By linearization we construct star-triangular face models and realize them as modules over quasitriangular quantum groupoids introduced in recent papers by M. Aguiar, S. Natale and the author.

Benedetta Morini - One of the best experts on this subject based on the ideXlab platform.

Xianda Zhang - One of the best experts on this subject based on the ideXlab platform.

  • nonorthogonal joint diagonalization free of Degenerate Solution
    IEEE Transactions on Signal Processing, 2007
    Co-Authors: Xilin Li, Xianda Zhang
    Abstract:

    The problem of approximate joint diagonalization of a set of matrices is instrumental in numerous statistical signal processing applications. For nonorthogonal joint diagonalization based on the weighted least-squares (WLS) criterion, the trivial (zero) Solution can simply be avoided by adopting some constraint on the diagonalizing matrix or penalty terms. However, the resultant algorithms may converge to some undesired Degenerate Solutions (nonzero but singular or ill-conditioned Solutions). This paper discusses and analyzes the problem of Degenerate Solutions in detail. To solve this problem, a novel nonleast-squares criterion for approximate nonorthogonal joint diagonalization is proposed and an efficient algorithm, called fast approximate joint diagonalization (FAJD), is developed. As compared with the existing nonorthogonal diagonalization algorithms, the new algorithm can not only avoid the trivial Solution but also any Degenerate Solutions. Theoretical analysis shows that the FAJD algorithm has some advantages over the existing nonorthogonal diagonalization algorithms. Simulation results are presented to demonstrate the efficiency of this paper's algorithm

Stefania Bellavia - One of the best experts on this subject based on the ideXlab platform.

Xilin Li - One of the best experts on this subject based on the ideXlab platform.

  • nonorthogonal joint diagonalization free of Degenerate Solution
    IEEE Transactions on Signal Processing, 2007
    Co-Authors: Xilin Li, Xianda Zhang
    Abstract:

    The problem of approximate joint diagonalization of a set of matrices is instrumental in numerous statistical signal processing applications. For nonorthogonal joint diagonalization based on the weighted least-squares (WLS) criterion, the trivial (zero) Solution can simply be avoided by adopting some constraint on the diagonalizing matrix or penalty terms. However, the resultant algorithms may converge to some undesired Degenerate Solutions (nonzero but singular or ill-conditioned Solutions). This paper discusses and analyzes the problem of Degenerate Solutions in detail. To solve this problem, a novel nonleast-squares criterion for approximate nonorthogonal joint diagonalization is proposed and an efficient algorithm, called fast approximate joint diagonalization (FAJD), is developed. As compared with the existing nonorthogonal diagonalization algorithms, the new algorithm can not only avoid the trivial Solution but also any Degenerate Solutions. Theoretical analysis shows that the FAJD algorithm has some advantages over the existing nonorthogonal diagonalization algorithms. Simulation results are presented to demonstrate the efficiency of this paper's algorithm