We introduce the notion of λ-double Lie algebra, which coincides with usual double algebra when λ=0. show that every for λ≠0 provides structure modified Poisson on free associative algebra. In particular, it confirms conjecture S. Arthamonov (2017). prove there are no simple finite-dimensional algebras.

Theory of conformal algebras (see [K1], [K2], [K3]) is a relatively new branch of algebra closely related to mathematical physics. The general categorial approach in this theory leads to the notion of pseudotensor category [BD] (also known as multicategory [La]). Algebras in these categories (so called pseudoalgebras, see [BDK]) allow to get a common presentation of various features of usual an...

We prove an analogue of the Poincaré-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a B∞-algebra. We construct a universal enveloping functor U2 from B∞-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form...

The product of two Heisenberg-Weil algebras contains the Jordan-Schwinger representation of su(2). This Algebra is quotiented by the square-root of the Casimir to produce a non-associative algebra denoted by Ψ. This algebra may be viewed as the right-module over one of its associative subalgebras which corresponds to the algebra of scalar fields on the noncommutative sphere. It is now possible ...

In commutative algebra, a Weitzenböck derivation is a nonzero triangular linear derivation of the polynomial algebra K[x1, . . . , xm] in several variables over a field K of characteristic 0. The classical theorem of Weitzenböck states that the algebra of constants is finitely generated. (This algebra coincides with the algebra of invariants of a single unipotent transformation.) In this paper ...

Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corresponding relativel...

When n is odd, a cohomology of type Hochschild for n-ary partially associative algebras has been defined in Gnedbaye’s thesis. Unfortunately, the cohomology definition is not valid when n is even. This fact is found again in the computations of the n-ary partially associative free algebra. In this work, we define in a first time two approachs of an Hochschild cohomology for n-ary partially asso...

A Cartan subalgebra of a finite-dimensional Lie algebra L is a nilpotent subalgebra H of L that coincides with its normalizer NL H . Such subalgebras occupy an important place in the structure theory of finite-dimensional Lie algebras and their properties have been explored in many articles (see, e.g., Barnes, 1967; Benkart, 1986; Wilson, 1977; Winter, 1969). In general (more precisely, when th...