The “coquecigrue” problem for Leibniz algebras is that of finding an appropriate generalization of Lie’s third theorem, that is, of finding a generalization of the notion of group such that Leibniz algebras are the corresponding tangent algebra structures. The difficulty is determining exactly what properties this generalization should have. Here we show that Lie racks, smooth left distributive...

We introduce the notion of a Hu-Liu prime ideal in the context of left commutative rngs, and establish the contravariant functor from the category of left commutative rngs into the category of topological spaces. It is well known that new points must be introduced in order to expand algebraic geometry over algebraically closed fields into Grothendieck’s scheme theory over commutative rings. We ...

Let L be a free Lie algebra over a field k, I a non-trivial proper ideal of L, n > 1 an integer. The multiplicator H2(L/I , k) of L/I is not finitely generated, and so in particular, L/I is not finitely presented, even when L/I is finite dimensional.

Algebraic scheme for constructing deformations of structure constants for associative algebras generated by a deformation driving algebras (DDAs) is proposed. An ideal of left divisors of zero plays a central role in this construction. Deformations of associative commutative three-dimensional algebras with the DDA being a three-dimensional Lie algebra and their connection with integrable system...

Algebraic scheme for constructing deformations of structure constants for associative algebras generated by a deformation driving algebras (DDAs) is discussed. An ideal of left divisors of zero plays a central role in this construction. Deformations of associative three-dimensional algebras with the DDA being a three-dimensional Lie algebra and their connection with integrable systems are studied.

In this paper we construct a deformation quantization of the algebra of polynomials of an arbitrary (regular and non regular) coadjoint orbit of a compact semisimple Lie group. The deformed algebra is given as a quotient of the enveloping algebra by a suitable ideal.

Let S = k[x1, . . . , xn] be a polynomial ring over a field k and I a monomial ideal of S. It is well known that the Poincaré series of k over S/I is rational. We describe the coefficients of the denominator of the series and study the multigraded homotopy Lie algebra of S/I.

Following I. S. Krasilshchik and A. M. Vinogradov [8], we regard PDEs as infinite-dimensional manifolds with involutive distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and Bäcklund transformations in soliton theory. We show that, similarly to usual coverings in topology, at least for some PDEs differential coverings ar...

A reductive Lie algebra g is one that can be written C(g) ⊕ [g,g], where C(g) denotes the center of g. Equivalently, for any ideal a, there is another ideal b such that g = a⊕ b. A Cartan subalgebra of g is a subalgebra h that is maximal with respect to being abelian and having ad X being semisimple for all X ∈ h. For a reductive group, h = C(g) ⊕ h′, where h′ is a Cartan subalgebra of the semi...

Simulations not seen before compare turbulent mixing rates for ideal fluids and for real immiscible fluids with experimental values for the surface tension. The simulated real fluid mixing rates lie near the center of the range of experimental values. A comparison to theoretical predictions relating the mixing rate, the bubble width, and the bubble height fluctuations based on bubble merger mod...