인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

/pdf/encyclopedia-of-complexity-and-systems-science-4on813lf4k.pdf

Encyclopedia of Complexity and Systems Science

In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does not satisfy the Jacobi identity except on certain subspaces. In this paper we systematize the properties of this bracket in the definition of a Courant algebroid. This structure on a vector bundle $E\rightarrow M$, consists of an antisymmetric bracket on the sections of $E$ whose ``Jacobi anomaly'' has an explicit expression in terms of a bundle map $E\rightarrow TM$ and a field of symmetric bilinear forms on $E$. When $M$ is a point, the definition reduces to that of a Lie algebra carrying an invariant nondegenerate symmetric bilinear form. For any Lie bialgebroid $(A,A^{*})$ over $M$ (a notion defined by Mackenzie and Xu), there is a natural Courant algebroid structure on $A\oplus A^{*}$ which is the Drinfel'd double of a Lie bialgebra when $M$ is a point. Conversely, if $A$ and $A^*$ are complementary isotropic subbundles of a Courant algebroid $E$, closed under the bracket (such a bundle, with dimension half that of $E$, is called a Dirac structure), there is a natural Lie bialgebroid structure on $(A,A^{*})$ whose double is isomorphic to $E$. The theory of Manin triples is thereby extended from Lie algebras to Lie algebroids. Our work gives a new approach to bihamiltonian structures and a new way of combining two Poisson structures to obtain a third one. We also take some tentative steps toward generalizing Drinfel'd's theory of Poisson homogeneous spaces from groups to groupoids.

/pdf/manin-triples-for-lie-bialgebroids-2do61urqws.pdf

Manin Triples for Lie Bialgebroids

UNIVERSAL ENVELOPING ALGEBRAS Algebraic constructions The Poincare-Birkhoff-Witt theorem POISSON GEOMETRY Poisson structures Normal forms Local Poisson geometry POISSON CATEGORY Poisson maps Hamiltonian actions DUAL PAIRS Operator algebras Dual pairs in Poisson geometry Examples of symplectic realizations GENERALIZED FUNCTIONS Group algebras Densities GROUPOIDS Groupoids Groupoid algebras Extended groupoid algebras ALGEBROIDS Lie algebroids Examples of Lie algebroids Differential geometry for Lie algebroids DEFORMATIONS OF ALGEBRAS OF FUNCTIONS Algebraic deformation theory Weyl algebras Deformation quantization.

/pdf/geometric-models-for-noncommutative-algebras-3yqtarz6qx.pdf

Geometric Models for Noncommutative Algebras

We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian, reductions, the Guillemin-Sternberg symplectic cro

/pdf/lie-group-valued-moment-maps-cn6yjmun8e.pdf

Lie group valued moment maps

Given a Lie group G, a G-manifold M, and a point b of M with compact stabilizer, we construct slices for the lifted tangent and cotangent actions at a pre-image of b in terms of a slice for the G-action on M at the point b. We interpret the slice for the lifted cotangent action in terms of a symplectic slice and in terms of a Witt-Artin-decomposition.

Slices for lifted tangent and cotangent actions

Let K be a compact Lie group, endowed with a bi-invariant Riemannian metric, which we denote by κ. The complexification K of K inherits a Kähler structure having twice the kinetic energy of the metric as its potential; let ε denote the symplectic volume form. Left and right translation turn the Hilbert space HL2(KC, e−κ/tηε) of square-integrable holomorphic functions on K relative to a suitable measure written as e−κ/tηε into a unitary (K × K)-representation; here η is an additional term coming from the metaplectic correction, and t > 0 is a real parameter. In the physical interpretation, this parameter amounts to Planck ’s constant ~. We establish the statement of the Peter-Weyl theorem for the Hilbert space HL2(KC, e−κ/tηε) to the effect that (i) HL2(KC, e−κ/tηε) contains the vector space of representative functions on K as a dense subspace and that (ii) the assignment to a holomorphic function of its Fourier coefficients yields an isomorphism of Hilbert algebras from the convolution algebra HL2(KC, e−κ/tηε) onto an algebra of the kind b ⊕End(V ). Here V ranges over the irreducible rational representations of K and b ⊕ refers to a suitable completion of the direct sum algebra ⊕End(V ). Consequences are: (i) the existence of a uniquely determined unitary isomorphism between L2(K, dx) (where dx refers to Haar measure on K) and the Hilbert space HL2(KC, e−κ/tηε), and (ii) a proof that this isomorphism coincides with the Blattner-Kostant-Sternberg pairing map from L2(K, dx) to HL2(KC, e−κ/tηε), multiplied by (4πt)− dim(K)/4. Among our crucial tools is Kirillov’s character formula. Our methods are geometric, rely on the orbit method, and are independent of heat kernel harmonic analysis, which is used by B. C. Hall to obtain many of these results [J. of Funct. Anal. 122 (1994), 103–151], [Comm. in Math. Physics 226 (2002), 233–268]. 2000 Mathematics Subject Classification. Primary: 17B63 17B81 22E30 22E46 22E70 32W30 53D50 81S10; Secondary: 14L35 17B65 17B66 32Q15 53D17 53D20.

2 O ct 2 00 8 KIRILLOV ’ S CHARACTER FORMULA

We construct a solution of the master equation by means of standard tools from homological perturbation theory under just the hypothesis that the ground field be of characteristic zero, thereby avoiding the formality assumption of the relevant Lie algebra. To this end we endow the homology H(g) of any differential graded Lie algebra g with an sh-Lie structure such that g and H(g) are sh-equivalent. We discuss our solution of the master equation in the context of deformation theory. Given the extra structure appropriate to the extended moduli space of complex structures on a Calabi-Yau manifold, the known solutions result as a special case.

Formal solution of the master equation via HPT and deformation theory

Exploiting a notion of Kaehler structure on a stratified space introduced elsewhere we show that, in the Kaehler case, reduction after quantization coincides with quantization after reduction: Key tools developed for that purpose are stratified polarizations and stratified prequantum modules, the latter generalizing prequantum bundles. These notions encapsulate, in particular, the behaviour of a polarization and that of a prequantum bundle across the strata. Our main result says that, for a positive Kaehler manifold with a hamiltonian action of a compact Lie group, when suitable additional conditions are imposed, reduction after quantization coincides with quantization after reduction in the sense that not only the reduced and unreduced quantum phase spaces correspond but the (invariant) unreduced and reduced quantum observables as well. Over a stratified space, the appropriate quantum phase space is a costratified Hilbert space in such a way that the costratified structure reflects the stratification. Examples of stratified Kaehler spaces arise from the closures of holomorphic nilpotent orbits including angular momentum zero reduced spaces, and from representations of compact Lie groups. For illustration, we carry out Kaehler quantization on various spaces of that kind including singular Fock spaces.

pdf/kaehler-quantization-and-reduction-2g19uey5ay.pdf

Kaehler quantization and reduction

A Chern-Weil construction for extensions of Lie-Rinehart algebras is introduced. This generalizes the classical Chern-Weil construction in differential geometry and yields characteristic classes for arbitrary extensions of Lie-Rinehart algebras. Some examples arising from spaces with singularities and from foliations are given that cannot be treated by means of the classical Chern-Weil construction.

Extensions of Lie-Rinehart algebras and the Chern-Weil construction

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