• Corpus ID: 117709606

Associative and Lie deformations of Poisson algebras

@article{Remm2011AssociativeAL,
  title={Associative and Lie deformations of Poisson algebras},
  author={Elisabeth Remm},
  journal={Communications in Mathematics},
  year={2011},
  volume={20},
  pages={117-136}
}
  • Elisabeth Remm
  • Published 13 May 2011
  • Mathematics
  • Communications in Mathematics
Considering a Poisson algebra as a nonassociative algebra satisfying the Markl-Remm identity, we study deformations of Poisson algebras as deformations of this nonassociative algebra. We give a natural interpretation of deformations which preserve the underlying associative structure and of deformations which preserve the underlying Lie algebra and we compare the associated cohomologies with the Poisson cohomology parametrizing the general deformations of Poisson algebras. 

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References

SHOWING 1-10 OF 27 REFERENCES

Poisson structures on associated with rigid Lie algebras

We present the classical Poisson-Lichnerowicz cohomology for the Poisson algebra of polynomials $\mathbb{C}[X_{1},..., X_{n}]$ using exterior calculus. After presenting some non homogeneous Poisson

Valued Deformations of Algebras

We develop the notion of deformations using a valuation ring as ring of coefficients. This permits to consider in particular the classical Gerstenhaber deformations of associative or Lie algebras as

Poisson structures on C[X1, . . . , Xn] associated with rigid Lie algebras

We present the classical Poisson-Lichnerowicz cohomology for the Poisson algebra of polynomials C[X1; : : : ;Xn] using exterior calculus. After presenting some non-homogenous Poisson brackets on this

Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the

A Class of Nonassociative Algebras Including Flexible and Alternative Algebras, Operads and Deformations

There exists two types of nonassociative algebras whose associator satisfies a symmetric relation associated with a 1-dimensional invariant vector space with respect to the natural action of the

2-dimensional algebras. Application to Jordan, G-associative and Hom-associative algebras

We classify, up to isomorphism, the 2-dimensional algebras over a field K. We focuse also on the case of characteristic 2, identifying the matrices of GL(2,F_2) with the elements of the symmetric

Deformation Theory (lecture notes)

First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the

On algebras obtained by tensor product