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

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

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.

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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.

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Geometric Models for Noncommutative Algebras

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.

Let R be a commutative ring, and let A be a Poisson algebra over R. We construct an (R,A)-Lie algebra structure, in the sense of Rinehart, on the A-module of Kahler differentials of A depending naturally on A and the Poisson bracket. This gives rise to suitable algebraic notions of Poisson homology and cohomology for an arbitrary Poisson algebra. A geometric version thereof includes the canonical homology and Poisson cohomology of a Poisson manifold introduced by Brylinski, Koszul, and Lichnerowicz, and absorbes the latter in standard homological algebra by expressing them as Tor and Ext groups, respectively, over a suitable algebra of differential operators. Furthermore, the Poisson structure determines a closed 2-form in the complex computing Poisson cohomology. This 2-form generalizes the 2-form defining a symplectic structure on a smooth manifold; moreover, the class of that 2-form in Poisson cohomology generalizes the class in de Rham cohomology of a symplectic structure on a smooth manifold and appears as a crucial ingredient for the construction of suitable linear representations of A, viewed as a Lie algebra; representations of this kind occur in quantum theory. To describe this class and to construct the representations, we relate formal concepts of connection and curvature generalizing the classical ones with extensions of Lie algebras. We illustrate our results with a number of examples of Poisson algebras and with a quantization procedure for a relativistic particle with zero rest mass and spin zero.

Poisson cohomology and quantization

Small models for chain algebras

Identities among relations

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Johannes Huebschmann

Papers

Poisson cohomology and quantization

Small models for chain algebras

Identities among relations

Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras

Crossedn-fold extensions of groups and cohomology