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Register a new user2-dimensional algebras. Application to Jordan, G-associative and Hom-associative algebras
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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.
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We introduce the notion of weakly associative algebra and its relations with the notion of nonassociative Poisson algebras.
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.
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.
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.
It is known that there are Lie algebras with non-semigroup gradings, i.e. such that the binary operation on the grading set is not associative. We provide a similar example in the class of associative algebras.
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