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991.
Consider the Hopf algebra (A, ) of regular functions on a compact quantum group. Let (A
o
,) denote its maximal dual Hopf algebra. We show that the tensor product Hopf algebra (H
2,2) of (A
o
,) and its opposite Hopf algebra is endowed with a modular pair (,) in involution; a notion introduced by A. Connes and J. Moscovici, who associate canonically a cocyclic object to such Hopf algebras. Denote the Hopf cyclic cohomology thus obtained by HC
*
(,)(H
2). Next we define an action of H
2),2 on A and show that the Haar state of (A, ) is a -invariant -trace on A with respect to this action. This gives us a canonical map from HC
*
(,)(H
2) to the ordinary cyclic cohomology of A. 相似文献
992.
Summary. {Equilibrium solutions of systems of parameterized ordinary differential equations \dot x = f(x, α) , x ∈ R
n
, α∈ R
m
can be characterized by their parametric distance to manifolds of critical solutions at which the behavior of the system
changes qualitatively. Critical points of interest are bifurcation points and points at which state variable constraints or
output constraints are violated. We use normal vectors on manifolds of critical points to measure the distance between these
manifolds and equilibrium solutions as suggested in I. Dobson [J. Nonlinear Sci., 3:307-327, 1993], where systems of equations
to calculate normal vectors on codimension-1 bifurcations were presented. We present a scheme to derive systems of equations
to calculate normal vectors on manifolds of critical points which (i) generalizes to bifurcations of arbitrary codimension,
(ii) can be applied to state variable constraints and output constraints, (iii) implies that the normal vector defining system
of equations is of size c
1
n+ c
2
m+ c
3
, c
i
∈ R , i.e., no bilinear terms nm or higher-order terms occur, (iv) reduces the number of equations for normal vectors on Hopf bifurcation manifolds compared
to previous work, and (v) simplifies the proof of regularity of the normal vector system. As an application of this scheme,
we present systems of equations for normal vectors to manifolds of output/state variable constraints, to manifolds of saddle-node,
Hopf, cusp, and isola bifurcations, and we give illustrative examples of their use in engineering applications.}
Received September 27, 2000; accepted December 10, 2001 Online publication March 11, 2002 Communicated by Y. G. Kevrekidis
Communicated by Y. G. Kevrekidis
rid=" 相似文献
993.
For a polynomial algebra
in several variables over a commutative ring R with a Hopf algebra structure
the existence of the dual Hopf algebra
is proved. 相似文献
994.
The parameter dependence of the number and type of the stationary points of an ODE is considered. The number of the stationary points is determined by the saddle-node (SN) bifurcation set and their type (e.g., stability) is given by another bifurcation diagram (e.g., Hopf bifurcation set). The relation between these bifurcation curves on the parmeter plane is investigated. It is shown that the cross-shaped diagram, when the Hopf bifurcation curve makes a loop around a cusp point of the SN curve, is typical in some sense. It is proved that the two bifurcation curves meet tangentially at their common points (Takens–Bogdanov point), and these common points persist as a third parameter is varied. An example is shown that exhibits two different types of 3-codimensional degenerate Takens–Bogdanov bifurcation. 相似文献
995.
本文对专著“Hopf Algebra”(E.Abe)中若干基本结论或其证明的不妥或错误之进行了修正和改进。 相似文献
996.
Justin M. Mauger 《Transactions of the American Mathematical Society》2004,356(8):3301-3323
We study the cohomology of a locally finite, connected, cocommutative Hopf algebra over . Specifically, we are interested in those algebras for which is generated as an algebra by and . We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras with monogenic and semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for to be semi-Koszul. Special attention is given to the case in which is the restricted universal enveloping algebra of the Lie algebra obtained from the mod- lower central series of a -group. We show that the algebras arising in this way from extensions by of an abelian -group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 -groups, and it is shown that these are all semi-Koszul for .
997.
We define an equivariant K
0-theory for Yetter–Drinfeld algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a pairing, generalizing Connes pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory. 相似文献
998.
Shuanhong Wang Dingguo Wang Zhongping Yao 《Southeast Asian Bulletin of Mathematics》2000,24(1):105-113
The smash coproduct coalgebra has been generalized to crossed coproduct coalgebra in [3]. It is natural to replace the smash coproduct by the crossed coproduct and consider the conditions under which the smash product algebra structure and the crossed coproduct coalgebra structure will inherit a bialgebra structure or a Hopf algebra structure. We derive necessary and sufficient conditions for this problem. This generalizes the corresponding results in [7]. Finally, we characterize this new structure by introducing a concept of (H, )-comodule and prove that Heisenberg double [4] and smash coproduct do not make a bialgebra.AMS Subject Classification (1991): 16S40 16W30The first and second authors were partially supported by NNSF of China. The first author was also supported by the NSF of Henan Province and the second author by the YSF of Shandong Province (No. Q98A05113). 相似文献
999.
Peng-ming Zhang Xi-guo Lee Shao-feng Wu Yi-shi Duan 《International Journal of Theoretical Physics》2007,46(7):1753-1762
We study the wave dislocations with an induced gauge potential. The topological current characterized the wave dislocations
is constructed with the dual of Abelian gauge field. And the topological charges and locations of the wave dislocations are
determined by the φmapping topological current theory. Furthermore, it is shown that the knotted wave dislocations can be
described with a Hopf invariant in the wave field. At last we discussed the evolution of the knotted wave dislocations.
PACS 02.10.Kn, 02.40.-k, 11.15.-q 相似文献
1000.
The Multiple-Scale Method is applied directly to a one-dimensional continuous model to derive the equations governing the asymptotic dynamic of the system around a bifurcation point. The theory is illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped viscoelastic device and loaded by a follower force. Nonlinear, integro-differential equations of motion are derived and expanded up to cubic terms in the transversal displacements and velocities of the beam. They are put in an operator form incorporating the mechanical boundary conditions, which account for the lumped viscoelastic device; the problem is thus governed by mixed algebraic-integro-differential operators. The linear stability of the trivial equilibrium is first studied. It reveals the existence of divergence, Hopf and double-zero bifurcations. The spectral properties of the linear operator and its adjoint are studied at the bifurcation points by obtaining closed-form expressions. Notably, the system is defective at the double-zero point, thus entailing the need to find a generalized eigenvector. A multiple-scale analysis is then performed for the three bifurcations and the relevant bifurcation equations are derived directly in their normal forms. Preliminary numerical results are illustrated for the double-zero bifurcation. 相似文献