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Journal of Geometric Mechanics
American Institute of Mathematical Sciences
Journal of Geometric Mechanics

American Institute of Mathematical Sciences

1941-4889

Journal of Geometric Mechanics/Journal Journal of Geometric MechanicsSCIISTP
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    GENERAL THEORY OF LIE GROUPOIDS AND LIE ALGEBROIDS

    Voronov, Theodore
    7页
      原文链接:
    • NSTL
    • Amer Inst Mathematical Sciences-Aims

    SYMPLECTIC Z(2)(n)-MANIFOLDS

    Bruce, Andrew JamesGrabowski, Janusz
    27页
    查看更多>>摘要：Roughly speaking, Z(2)(n)-manifolds are 'manifolds' equipped with Z(2)(n)-graded commutative coordinates with the sign rule being determined by the scalar product of their Z(2)(n)-degrees. We examine the notion of a symplectic Z(2)(n)-manifold, i.e., a Z(2)(n)-manifold equipped with a symplectic two-form that may carry non-zero Z(2)(n)-degree. We show that the basic notions and results of symplectic geometry generalise to the 'higher graded' setting, including a generalisation of Darboux's theorem.
      原文链接:
    • NSTL
    • Amer Inst Mathematical Sciences-Aims

    ON TWISTOR ALMOST COMPLEX STRUCTURES

    Cahen, MichelGutt, SimoneRawnsley, John
    19页
    查看更多>>摘要：In this paper we look at the question of integrability, or not, of the two natural almost complex structures J(del)(+/-) defined on the twistor space J(M, g) of an even-dimensional manifold M with additional structures g and del a g-connection. We measure their non-integrability by the dimension of the span of the values of N-J del +/-. We also look at the question of the compatibility of J(del)(+/-) with a natural closed 2-form omega(J(M,g,del)) defined on J(M, g). For (M, g) we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civita connection or a symplectic manifold with a given symplectic connection del. In all cases J(M, g) is a bundle of complex structures on the tangent spaces of M compatible with g. In the case M is oriented we require the orientation of the complex structures to be the given one. In the symplectic case the complex structures are positive.
      原文链接:
    • NSTL
    • Amer Inst Mathematical Sciences-Aims

    SCHWINGER'S PICTURE OF QUANTUM MECHANICS: 2-GROUPOIDS AND SYMMETRIES

    Ciaglia, Florio M.DI Cosmo, FabioIbort, AlbertoMarmo, Giuseppe...
    22页
    查看更多>>摘要：Starting from the groupoid approach to Schwinger's picture of Quantum Mechanics, a proposal for the description of symmetries in this framework is advanced. It is shown that, given a groupoid G paired right arrows Omega associated with a (quantum) system, there are two possible descriptions of its symmetries, one "microscopic", the other one "global". The microscopic point of view leads to the introduction of an additional layer over the grupoid G, giving rise to a suitable algebraic structure of 2-groupoid. On the other hand, taking advantage of the notion of group of bisections of a given groupoid, the global perspective allows to construct a group of symmetries out of a 2-groupoid. The latter notion allows to introduce an analog of the Wigner's theorem for quantum symmetries in the groupoid approach to Quantum Mechanics.
      原文链接:
    • NSTL
    • Amer Inst Mathematical Sciences-Aims

    LOCAL AND GLOBAL INTEGRABILITY OF LIE BRACKETS

    Fernandes, R. U. I. L.Zhang, Yuxuan
    30页
    查看更多>>摘要：We survey recent results on the local and global integrability of a Lie algebroid, as well as the integrability of infinitesimal multiplicative geometric structures on it.
      原文链接:
    • NSTL
    • Amer Inst Mathematical Sciences-Aims

    ON THE HISTORY OF LIE BRACKETS, CROSSED MODULES, AND LIE-RINEHART ALGEBRAS

    Huebschmann, Johannes
    18页
    查看更多>>摘要：This is an overview of ideas related to brackets in early homotopy theory, crossed modules, the obstruction 3-cocycle for the nonabelian extension problem, the Teichmu center dot ller cocycle, Lie-Rinehart algebras, Lie algebroids, and differential algebra.
      原文链接:
    • NSTL
    • Amer Inst Mathematical Sciences-Aims

    TRANSITIVE DOUBLE LIE ALGEBROIDS VIA CORE DIAGRAMS

    Lean, Madeleine JotzMackenzie, Kirill C. H.
    55页
    查看更多>>摘要：The core diagram of a double Lie algebroid consists of the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core-anchors are surjective, then the double Lie algebroid and its core diagram are called transitive. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram. The comma double Lie algebroid associated to a morphism of Lie algebroids is defined. If the latter morphism is one of the core-anchors of a transitive core diagram, then the comma double algebroid can be quotiented out by the second core-anchor, yielding a transitive double Lie algebroid, which is the one that is equivalent to the transitive core diagram. Brown's and Mackenzie's equivalence of transitive core diagrams (of Lie groupoids) with transitive double Lie groupoids is then used in order to show that a transitive double Lie algebroid with integrable sides and core is automatically integrable to a transitive double Lie groupoid.
      原文链接:
    • NSTL
    • Amer Inst Mathematical Sciences-Aims

    FROM SCHOUTEN TO MACKENZIE: NOTES ON BRACKETS

    Kosmann-Schwarzbach, Yvette
    18页
    查看更多>>摘要：In this paper, dedicated to the memory of Kirill Mackenzie, I relate the origins and early development of the theory of graded Lie brackets, first in the publications on differential geometry of Schouten, Nijenhuis, and Frolicher- Nijenhuis, then in the work of Gerstenhaber and Nijenhuis-Richardson in co-homology theory.
      原文链接:
    • NSTL
    • Amer Inst Mathematical Sciences-Aims

    LOCAL CONVEXITY FOR SECOND ORDER DIFFERENTIAL EQUATIONS ON A LIE ALGEBROID

    Marrero, Juan CarlosDe Diego, David MartinMartinez, Eduardo
    23页
    查看更多>>摘要：A theory of local convexity for a second order differential equation (SODE) on a Lie algebroid is developed. The particular case when the sode is homogeneous quadratic is extensively discussed.
      原文链接:
    • NSTL
    • Amer Inst Mathematical Sciences-Aims

    BRACKETS BY ANY OTHER NAME

    Stasheff, J. I. M.
    16页
    查看更多>>摘要：Brackets by another name Whitehead or Samelson products have a history parallel to that in Kosmann-Schwarzbach's "From Schouten to Mackenzie: notes on brackets". Here I sketch the development of these and some of the other brackets and products and braces within homotopy theory and homological algebra and with applications to mathematical physics.
      原文链接:
    • NSTL
    • Amer Inst Mathematical Sciences-Aims