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Weakly associative algebras, Poisson algebras and quantization

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 Added by Elisabeth Remm
 Publication date 2020
  fields
and research's language is English




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We introduce the notion of weakly associative algebra and its relations with the notion of nonassociative Poisson algebras.



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130 - Elisabeth Remm 2020
We study a special class of weakly associative algebras: the symmetric Leibniz algebras. We describe the structure of the commutative and skew symmetric algebras associated with the polarization-depolarization principle. We also give a structure theorem for the symmetric Leibniz algebras and we study formal deformations in the context of deformation quantization.
70 - Elisabeth Remm 2020
We study deformation quantization of nonassociative algebras whose associator satisfies some symmetric relations. This study is expanded to a larger class of nonassociative algebras includind Leibniz algebras. We apply also to this class the rule of polarization-depolarization.
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 group S_3. The classification is then given by the study of the orbits of this group on a 3-dimensional plane, viewed as a Fano plane. As applications, we establish classifications of Jordan algebras, algebras of Lie type or Hom-Associative algebras.
We show that the invariants of a free associative algebra of finite rank under a linear action of a finite-dimensional Hopf algebra generated by group-like and skew-primitive elements form a finitely generated algebra exactly when the action is scalar. This generalizes an analogous result for group actions by automorphisms obtained by Dicks and Formanek, and Kharchenko.
95 - Edward S. Letzter 2019
In 1992, following earlier conjectures of Lichtman and Makar-Limanov, Klein conjectured that a noncommutative domain must contain a free, multiplicative, noncyclic subsemigroup. He verified the conjecture when the center is uncountable. In this note we consider the existence (or not) of free subsemigroups in associative $k$-algebras $R$, where $k$ is a field not algebraic over a finite subfield. We show that $R$ contains a free noncyclic subsemigroup in the following cases: (1) $R$ satisfies a polynomial identity and is noncommutative modulo its prime radical. (2) $R$ has at least one nonartinian primitive subquotient. (3) $k$ is uncountable and $R$ is noncommutative modulo its Jacobson radical. In particular, (1) and (2) verify Kleins conjecture for numerous well known classes of domains, over countable fields, not covered in the prior literature.
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