Frontiers of Mathematics in China, ISSN 1673-3452, 8/2018, Volume 13, Issue 4, pp. 999 - 1011

For a Poisson algebra, the category of Poisson modules is equivalent to the module category of its Poisson enveloping algebra, where the Poisson enveloping...

16E65 | 17B63 | deformation quantization | Mathematics, general | Mathematics | 16S80 | Poisson enveloping algebra | quantized universal enveloping algebra | MATHEMATICS | MANIFOLDS | COHOMOLOGY | Algebra | Deformation | Poisson distribution | Mathematical analysis

16E65 | 17B63 | deformation quantization | Mathematics, general | Mathematics | 16S80 | Poisson enveloping algebra | quantized universal enveloping algebra | MATHEMATICS | MANIFOLDS | COHOMOLOGY | Algebra | Deformation | Poisson distribution | Mathematical analysis

Journal Article

Journal of Algebra, ISSN 0021-8693, 03/2015, Volume 426, pp. 92 - 136

For a Poisson algebra A, by exploring its relation with Lie–Rinehart algebras, we prove a Poincaré–Birkhoff–Witt theorem for its universal enveloping algebra...

Universal enveloping algebras | Poisson Hopf algebras | Hopf algebras | Lie–Rinehart algebras | Lie-Rinehart algebras | MATHEMATICS | DIMENSION | Algebra

Universal enveloping algebras | Poisson Hopf algebras | Hopf algebras | Lie–Rinehart algebras | Lie-Rinehart algebras | MATHEMATICS | DIMENSION | Algebra

Journal Article

Communications in Algebra, ISSN 0092-7872, 11/2018, Volume 46, Issue 11, pp. 4891 - 4904

It is proved that the Poisson enveloping algebra of a double Poisson-Ore extension is an iterated double Ore extension. As an application, properties that are...

Poisson enveloping algebra | Double Ore extension | double Poisson-Ore extension | MATHEMATICS | BIRKHOFF-WITT THEOREM | Algebra

Poisson enveloping algebra | Double Ore extension | double Poisson-Ore extension | MATHEMATICS | BIRKHOFF-WITT THEOREM | Algebra

Journal Article

Acta Mathematica Sinica, English Series, ISSN 1439-8516, 1/2013, Volume 29, Issue 1, pp. 105 - 118

We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative...

Poisson module | 17B63 | 16S40 | Mathematics, general | Mathematics | Non-commutative Poisson algebra | Poisson enveloping algebra | MATHEMATICS | MATHEMATICS, APPLIED | MODULES | LEIBNIZ PAIRS | POISSON ALGEBRAS | Algebra | Studies | Poisson distribution | Mathematical analysis | Equivalence | Categories | Modules

Poisson module | 17B63 | 16S40 | Mathematics, general | Mathematics | Non-commutative Poisson algebra | Poisson enveloping algebra | MATHEMATICS | MATHEMATICS, APPLIED | MODULES | LEIBNIZ PAIRS | POISSON ALGEBRAS | Algebra | Studies | Poisson distribution | Mathematical analysis | Equivalence | Categories | Modules

Journal Article

Algebras and Representation Theory, ISSN 1386-923X, 2019

Journal Article

Journal of Algebra, ISSN 0021-8693, 09/2017, Volume 485, pp. 166 - 198

Poisson algebras are, just like Lie algebras, particular cases of Lie–Rinehart algebras. The latter were introduced by Rinehart in his seminal 1963 paper,...

Poisson structures | Singularities | Enveloping algebras | MATHEMATICS | Algebra | Mathematics | Rings and Algebras

Poisson structures | Singularities | Enveloping algebras | MATHEMATICS | Algebra | Mathematics | Rings and Algebras

Journal Article

Proceedings of the American Mathematical Society, ISSN 0002-9939, 11/2015, Volume 143, Issue 11, pp. 4633 - 4645

We prove that the universal enveloping algebra of a Poisson-Ore extension is a length two iterated Ore extension of the original universal enveloping algebra....

Ore extension | Universal enveloping algebra | Poisson algebra | Mathematics - Rings and Algebras

Ore extension | Universal enveloping algebra | Poisson algebra | Mathematics - Rings and Algebras

Journal Article

中国科学：数学英文版, ISSN 1674-7283, 2016, Volume 59, Issue 5, pp. 849 - 860

We introduce the notions of differential graded（DG） Poisson algebra and DG Poisson module. Let A be any DG Poisson algebra. We construct the universal...

UE | 微分几何 | 代数结构 | 包络代数 | 范畴同构 | 同调代数 | 代数和 | 模块 | differential graded Poisson algebras | differential graded algebras | differential graded Hopf algebras | differential graded Lie algebras | universal enveloping algebras | monoidal category | Algebra | Construction | Tensors | Categories | Mathematical analysis | Modules | Paper | Homology | Mathematics - Rings and Algebras

UE | 微分几何 | 代数结构 | 包络代数 | 范畴同构 | 同调代数 | 代数和 | 模块 | differential graded Poisson algebras | differential graded algebras | differential graded Hopf algebras | differential graded Lie algebras | universal enveloping algebras | monoidal category | Algebra | Construction | Tensors | Categories | Mathematical analysis | Modules | Paper | Homology | Mathematics - Rings and Algebras

Journal Article

Communications in Algebra, ISSN 0092-7872, 06/2018, Volume 46, Issue 6, pp. 2714 - 2729

In this paper, the so-called differential graded (DG for short) Poisson Hopf algebra is introduced, which can be considered as a natural extension of Poisson...

differential graded Poisson algebras | Differential graded Hopf algebras | differential graded Poisson Hopf algebras | universal enveloping algebras | MATHEMATICS | Algebra

differential graded Poisson algebras | Differential graded Hopf algebras | differential graded Poisson Hopf algebras | universal enveloping algebras | MATHEMATICS | Algebra

Journal Article

Bulletin of the Malaysian Mathematical Sciences Society, ISSN 0126-6705, 11/2019, Volume 42, Issue 6, pp. 3343 - 3377

For any differential graded (DG for short) Poisson algebra A given by generators and relations, we give a “formula” for computing the universal enveloping...

Universal enveloping algebras | 16S10 | 17B35 | 17B63 | Simple differential graded Poisson module | Mathematics, general | Differential graded Poisson algebras | Mathematics | Applications of Mathematics | PBW-basis | 16E45 | MATHEMATICS | Polynomials | Algebra

Universal enveloping algebras | 16S10 | 17B35 | 17B63 | Simple differential graded Poisson module | Mathematics, general | Differential graded Poisson algebras | Mathematics | Applications of Mathematics | PBW-basis | 16E45 | MATHEMATICS | Polynomials | Algebra

Journal Article

Journal of Algebra, ISSN 0021-8693, 03/2012, Volume 354, Issue 1, pp. 77 - 94

Let k be an arbitrary field of characteristic 0. It is shown that for any n⩾1 the universal enveloping algebras of the Poisson symplectic algebra Pn(k) and the...

Poisson algebras | Universal enveloping algebras | Derivations | Left dependence | Automorphisms | Secondary | Primary | LIE-ALGEBRAS | POLYNOMIALS | MATHEMATICS | RINGS | JACOBIAN CONJECTURE | WEYL ALGEBRA | ENDOMORPHISMS

Poisson algebras | Universal enveloping algebras | Derivations | Left dependence | Automorphisms | Secondary | Primary | LIE-ALGEBRAS | POLYNOMIALS | MATHEMATICS | RINGS | JACOBIAN CONJECTURE | WEYL ALGEBRA | ENDOMORPHISMS

Journal Article

Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8121, 07/2010, Volume 43, Issue 29, p. 293001

This paper reviews the properties and applications of certain n-ary generalizations of Lie algebras in a self-contained and unified way. These generalizations...

GAUGE-THEORIES | PHYSICS, MULTIDISCIPLINARY | UNITARY GROUP | ENVELOPING-ALGEBRAS | LEIBNIZ ALGEBRAS | DEFORMATION QUANTIZATION | SCHOUTEN-NIJENHUIS BRACKET | GRADED LIE-ALGEBRAS | NAMBU-MECHANICS | PHYSICS, MATHEMATICAL | STRING THEORY | GENERALIZED POISSON

GAUGE-THEORIES | PHYSICS, MULTIDISCIPLINARY | UNITARY GROUP | ENVELOPING-ALGEBRAS | LEIBNIZ ALGEBRAS | DEFORMATION QUANTIZATION | SCHOUTEN-NIJENHUIS BRACKET | GRADED LIE-ALGEBRAS | NAMBU-MECHANICS | PHYSICS, MATHEMATICAL | STRING THEORY | GENERALIZED POISSON

Journal Article

Siberian Mathematical Journal, ISSN 0037-4466, 7/2013, Volume 54, Issue 4, pp. 759 - 768

Under study are the centralizers of 3-dimensional simple Lie subalgebras in the universal enveloping algebra of a 7-dimensional simple Malcev algebra. We find...

alternative algebra | algebra of octonions | Mathematics, general | Poisson bracket | Mathematics | universal enveloping algebra | Lie algebra | Malcev algebra | MATHEMATICS | Public contracts | Algebra

alternative algebra | algebra of octonions | Mathematics, general | Poisson bracket | Mathematics | universal enveloping algebra | Lie algebra | Malcev algebra | MATHEMATICS | Public contracts | Algebra

Journal Article

Journal of Algebra, ISSN 0021-8693, 10/2017, Volume 488, pp. 244 - 281

The paper naturally continues series of works on identical relations of group rings, enveloping algebras, and other related algebraic structures. Let L be a...

Poisson algebras | Restricted Lie algebras | Symmetric algebras | Identical relations | Nilpotent Lie algebras | Solvable Lie algebras | Truncated symmetric algebras | RESTRICTED ENVELOPING-ALGEBRAS | GROUP-RINGS | POLYNOMIAL-IDENTITIES | IDEALS | MATHEMATICS | NILPOTENT ASSOCIATIVE ALGEBRAS | SMASH PRODUCTS | SUPERALGEBRAS | INDEXES | Algebra | Mathematics - Rings and Algebras

Poisson algebras | Restricted Lie algebras | Symmetric algebras | Identical relations | Nilpotent Lie algebras | Solvable Lie algebras | Truncated symmetric algebras | RESTRICTED ENVELOPING-ALGEBRAS | GROUP-RINGS | POLYNOMIAL-IDENTITIES | IDEALS | MATHEMATICS | NILPOTENT ASSOCIATIVE ALGEBRAS | SMASH PRODUCTS | SUPERALGEBRAS | INDEXES | Algebra | Mathematics - Rings and Algebras

Journal Article

Journal of Algebra, ISSN 0021-8693, 05/2017, Volume 477, pp. 95 - 146

Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We provide necessary and also some sufficient conditions...

Semi-invariants | Polynomiality | Poisson center | Dixmier's fourth problem | Enveloping algebra | QUOTIENT DIVISION RING | INVARIANTS | THEOREM | BIPARABOLIC SEAWEED ALGEBRAS | MATHEMATICS | DIMENSION | GELFAND-KIRILLOV CONJECTURE | COMMUTATIVE SUBALGEBRAS | INDEX | Polynomiality Enveloping algebra | ENVELOPING ALGEBRA | Algebra | Mathematics - Representation Theory

Semi-invariants | Polynomiality | Poisson center | Dixmier's fourth problem | Enveloping algebra | QUOTIENT DIVISION RING | INVARIANTS | THEOREM | BIPARABOLIC SEAWEED ALGEBRAS | MATHEMATICS | DIMENSION | GELFAND-KIRILLOV CONJECTURE | COMMUTATIVE SUBALGEBRAS | INDEX | Polynomiality Enveloping algebra | ENVELOPING ALGEBRA | Algebra | Mathematics - Representation Theory

Journal Article

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, ISSN 0002-9939, 11/2015, Volume 143, Issue 11, pp. 4633 - 4645

We prove that the universal enveloping algebra of a Poisson-Ore extension is a length two iterated Ore extension of the original universal enveloping algebra....

MATHEMATICS | Ore extension | MATHEMATICS, APPLIED | Poisson algebra | universal enveloping algebra

MATHEMATICS | Ore extension | MATHEMATICS, APPLIED | Poisson algebra | universal enveloping algebra

Journal Article

Letters in Mathematical Physics, ISSN 0377-9017, 9/2017, Volume 107, Issue 9, pp. 1715 - 1740

In this paper, we show that the twisted Poincaré duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives...

Calabi–Yau algebra | Theoretical, Mathematical and Computational Physics | Complex Systems | Poisson (co)homology | Physics | Hochschild (co)homology | Geometry | Dualizing complex | 16E40 | 17B35 | 17B63 | Group Theory and Generalizations | Poisson algebra | HOPF-ALGEBRAS | RINGS | Poisson (co) homology | POINCARE-DUALITY | PHYSICS, MATHEMATICAL | DUALIZING COMPLEXES | Hochschild (co) homology | UNIVERSAL ENVELOPING-ALGEBRAS | COHOMOLOGY | Calabi-Yau algebra | MANIFOLD | Algebra | Mathematics - Rings and Algebras

Calabi–Yau algebra | Theoretical, Mathematical and Computational Physics | Complex Systems | Poisson (co)homology | Physics | Hochschild (co)homology | Geometry | Dualizing complex | 16E40 | 17B35 | 17B63 | Group Theory and Generalizations | Poisson algebra | HOPF-ALGEBRAS | RINGS | Poisson (co) homology | POINCARE-DUALITY | PHYSICS, MATHEMATICAL | DUALIZING COMPLEXES | Hochschild (co) homology | UNIVERSAL ENVELOPING-ALGEBRAS | COHOMOLOGY | Calabi-Yau algebra | MANIFOLD | Algebra | Mathematics - Rings and Algebras

Journal Article

International Journal of Theoretical Physics, ISSN 0020-7748, 2/2008, Volume 47, Issue 2, pp. 311 - 332

We describe enveloping algebras of finite-dimensional Lie algebras which are formal in the sense that their Hochschild complex as a differential graded Lie...

Quantum Physics | Physics, general | Mathematical and Computational Physics | Physics | Elementary Particles, Quantum Field Theory | LIE-ALGEBRAS | POISSON MANIFOLDS | COHOMOLOGY | PHYSICS, MULTIDISCIPLINARY | QUANTIZATION | STAR-PRODUCT

Quantum Physics | Physics, general | Mathematical and Computational Physics | Physics | Elementary Particles, Quantum Field Theory | LIE-ALGEBRAS | POISSON MANIFOLDS | COHOMOLOGY | PHYSICS, MULTIDISCIPLINARY | QUANTIZATION | STAR-PRODUCT

Journal Article

Journal of Algebra and its Applications, ISSN 0219-4988, 12/2016, Volume 15, Issue 10

Let k be an arbitrary field of characteristic 0. We prove that the group of automorphisms of a free Poisson field P(x, y) in two variables x, y over k is...

Poisson algebras | left dependence | universal enveloping algebras | automorphisms | MATHEMATICS | MATHEMATICS, APPLIED | ENVELOPING-ALGEBRAS | RINGS | FREE ASSOCIATIVE ALGEBRA

Poisson algebras | left dependence | universal enveloping algebras | automorphisms | MATHEMATICS | MATHEMATICS, APPLIED | ENVELOPING-ALGEBRAS | RINGS | FREE ASSOCIATIVE ALGEBRA

Journal Article

Journal of Algebra and its Applications, ISSN 0219-4988, 03/2018, Volume 17, Issue 3

We prove that when Kontsevich's deformation quantization is applied on weight homo-geneous Poisson structures, the operators in the *-product formula are...

Deformation quantization | semisimple Lie algebras | transverse Poisson structures | W -algebras | Slodowy slice | MATHEMATICS, APPLIED | COISOTROPIC SUBMANIFOLDS | REPRESENTATIONS | SLODOWY SLICES | MATHEMATICS | W-algebras | SYMMETRICAL SPACES | ENVELOPING-ALGEBRAS | INVARIANT DIFFERENTIAL-OPERATORS | QUANTIZATION

Deformation quantization | semisimple Lie algebras | transverse Poisson structures | W -algebras | Slodowy slice | MATHEMATICS, APPLIED | COISOTROPIC SUBMANIFOLDS | REPRESENTATIONS | SLODOWY SLICES | MATHEMATICS | W-algebras | SYMMETRICAL SPACES | ENVELOPING-ALGEBRAS | INVARIANT DIFFERENTIAL-OPERATORS | QUANTIZATION

Journal Article

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